ESAT Physics Mock Module 2 Worked Solutions

Correct Answer: E. The particles oscillate parallel to the direction of energy transfer

Fastest Approach (🚀):
Longitudinal = along. Transverse = across.

Matches Option E.

Step-by-Step Breakdown:

1. The defining comparison

Transverse: particles oscillate perpendicular to the direction the energy travels. Light, water ripples and waves on a rope.
Longitudinal: particles oscillate parallel to it, producing regions of compression and rarefaction. Sound is the standard example.

2. What no wave does

In neither type do the particles travel with the wave. They oscillate about fixed positions while the energy moves on — which is why a floating cork bobs up and down as a ripple passes rather than being carried to the shore.

3. What can cross a vacuum

Only electromagnetic waves, which are transverse. Longitudinal waves are mechanical and need a medium, which is why there is no sound in space. Option D reverses this completely.

4. Both have every wave property

Wavelength, frequency, amplitude, speed and $v = f\lambda$ apply to both types. For a longitudinal wave the wavelength is the distance from one compression to the next.

Matches Option E.

Why the Other Options Are Wrong (❌):

• A. The particles oscillate at right angles to the direction of energy transfer — Definitions Swapped
  This defines a transverse wave.
• B. It has no wavelength — Property Denied
  Longitudinal waves have wavelength, measured compression to compression.
• C. The particles travel along with the wave — Matter Transported
  No wave transports matter; only energy travels.
• D. It can travel through a vacuum but not through a solid — Reversed
  Backwards — longitudinal waves need a medium and cannot cross a vacuum.

Common Mistake (⚠️):
Swapping the two definitions. The word itself carries the meaning: longitudinal means 'along', so the oscillation is along the direction of travel.

Takeaway (📌):
Transverse oscillates across the travel direction, longitudinal along it. In neither case does matter travel with the wave.

Correct Answer: C. $143$

Fastest Approach (🚀):
$235-92 = 143$.

Matches Option C.

Step-by-Step Breakdown:

1. What each number means

The upper number is the mass number $A$: the total count of protons and neutrons, together called nucleons.
The lower number is the atomic number $Z$: the number of protons, which alone determines the element.

2. Subtract

$\text{neutrons} = A-Z = 235-92 = 143$

3. The electrons

A neutral uranium atom also has $92$ electrons, matching the protons. Electrons are not counted in either number, because their mass is negligible and they sit outside the nucleus entirely.

4. Isotopes

$^{238}_{\ 92}\text{U}$ is the same element with $146$ neutrons. Changing the neutron count changes the isotope and its nuclear stability, but not its chemistry — the chemistry follows the electrons, and those follow the protons.

Matches Option C.

Why the Other Options Are Wrong (❌):

• A. $92$ — Wrong Quantity
  The number of protons, not neutrons.
• B. $235$ — Wrong Quantity
  The total number of nucleons.
• D. $327$ — Operation Error
  Adding the two numbers, which double-counts the protons.
• E. $51$ — Extra Step
  Computing $143-92$.

Common Mistake (⚠️):
Giving $92$, the proton count, or adding the two numbers. The mass number already includes the protons, so it must be a subtraction.

Takeaway (📌):
$A$ on top counts nucleons, $Z$ below counts protons, and neutrons are $A-Z$. Changing $Z$ changes the element; changing only $A$ changes the isotope.

Correct Answer: B. $1\ \text{m/s}$

Fastest Approach (🚀):
Shell momentum $= 4\times 200 = 800\ \text{kg m/s}$.
$v = \dfrac{800}{800} = 1\ \text{m/s}$.

Matches Option B.

Step-by-Step Breakdown:

1. Total momentum before

Nothing is moving, so the total momentum of cannon and shell together is zero.

2. Total momentum after must also be zero

$m_{\text{shell}}v_{\text{shell}}+m_{\text{cannon}}v_{\text{cannon}} = 0$
$4\times 200+800v = 0$
$800+800v = 0$
$v = -1\ \text{m/s}$

The minus sign says the cannon moves the opposite way to the shell. Its speed is $1\ \text{m/s}$.

3. Read the ratio

The cannon is $200$ times heavier than the shell, so it recoils $200$ times more slowly: $\dfrac{200}{200} = 1\ \text{m/s}$. That ratio shortcut answers most recoil questions in one line.

4. Energy is not shared equally

shell: $\tfrac12(4)(200^{2}) = 80\,000\ \text{J}$
cannon: $\tfrac12(800)(1^{2}) = 400\ \text{J}$

Equal and opposite momenta, but the light object carries almost all the kinetic energy. That asymmetry is why a rifle can be fired from the shoulder while the bullet is lethal, and it is the point recoil questions are really built to make.

Matches Option B.

Why the Other Options Are Wrong (❌):

• A. $2\ \text{m/s}$ — Ratio Error
  Using a mass ratio of $400$, or doubling the answer.
• C. $0.5\ \text{m/s}$ — Ratio Error
  Halving the correct answer.
• D. $200\ \text{m/s}$ — Wrong Object
  Giving the shell's speed rather than the cannon's.
• E. $4\ \text{m/s}$ — Wrong Quantity
  Dividing $800$ by $200$, or quoting the shell's mass.

Common Mistake (⚠️):
Assuming the cannon and shell move off at the same speed, or dividing the masses the wrong way round so the heavy object moves faster.

Takeaway (📌):
From rest, total momentum stays zero: the momenta are equal and opposite, so speeds are in inverse ratio to the masses.

Correct Answer: A. $6\ \text{m}^{3}$

Fastest Approach (🚀):
Pressure halves, so volume doubles.
$3\times 2 = 6\ \text{m}^{3}$.

Matches Option A.

Step-by-Step Breakdown:

1. State the law

At constant temperature, for a fixed mass of gas,
$p_1V_1 = p_2V_2$

2. Predict the direction before computing

The pressure is falling and the gas is expanding, so the volume must increase. Any answer below $3\ \text{m}^{3}$ is wrong before the arithmetic starts, which eliminates two options at once.

3. Substitute

$200\times 3 = 100\times V_2$
$600 = 100V_2$
$V_2 = 6\ \text{m}^{3}$

4. Units cancel, so kPa is fine

Both pressures are in the same unit, so it appears on both sides and cancels. There is no need to convert to pascals — a rare case where mixed-unit anxiety costs time for nothing.

Matches Option A.

Why the Other Options Are Wrong (❌):

• B. $1.5\ \text{m}^{3}$ — Ratio Inverted
  Halving the volume as the pressure halves — the relationship is inverse.
• C. $3\ \text{m}^{3}$ — No Change
  Assuming the volume is unaffected.
• D. $600\ \text{m}^{3}$ — Wrong Quantity
  Giving $p_1V_1 = 600$ as a volume.
• E. $0.67\ \text{m}^{3}$ — Ratio Error
  Dividing $3$ by the pressure ratio incorrectly.

Common Mistake (⚠️):
Inverting the ratio and halving the volume. Pressure and volume are inversely proportional, so as one falls the other rises.

Takeaway (📌):
$p_1V_1 = p_2V_2$ at constant temperature. State whether the answer should be bigger or smaller before substituting.

Correct Answer: C. The magnetic field around a long straight current-carrying wire consists of concentric circles centred on the wire

Fastest Approach (🚀):
Straight wire $\Rightarrow$ circular field lines.

Matches Option C.

Step-by-Step Breakdown:

1. The field around a straight wire

Concentric circles centred on the wire, lying in planes perpendicular to it. The field gets weaker further out, so the circles are drawn further apart. Grip the wire with your right hand, thumb along the current, and your fingers curl the way the field points.

2. Field line direction on a magnet

Outside a magnet, field lines run from north to south. Inside they return south to north, so every line forms a closed loop. Option B reverses this.

3. What a solenoid does

A solenoid produces a field like a bar magnet's, with a strong uniform field inside and a north and south pole at its ends — so it certainly has an external field, and Option E is wrong.

Reverse the current and the field reverses with it, swapping the poles end for end. That is exactly what makes electromagnets useful and permanent magnets not, so Option D is wrong.

4. Why the circular field matters

It is the reason a wire in another magnetic field feels a force: the two fields reinforce on one side and cancel on the other, pushing the wire towards the weaker side. That is the motor effect, and it starts from this picture.

Matches Option C.

Why the Other Options Are Wrong (❌):

• A. The magnetic field around a long straight current-carrying wire consists of straight lines parallel to the wire — Field Direction
  The field circles the wire; it is never parallel to it.
• B. Magnetic field lines run from the south pole to the north pole outside a bar magnet — Direction Reversed
  Reversed — outside a magnet, lines run north to south.
• D. Reversing the current in a solenoid leaves its poles unchanged — Effect Denied
  Reversing the current reverses the field and swaps the poles.
• E. A solenoid produces no external magnetic field — Effect Denied
  A solenoid behaves like a bar magnet, with poles and an external field.

Common Mistake (⚠️):
Drawing the field along the wire rather than around it. The field is always perpendicular to the current, never parallel to it.

Takeaway (📌):
Straight wire: concentric circles, right-hand grip rule. Solenoid: bar-magnet field, poles reverse with the current. Outside any magnet, lines run north to south.

Correct Answer: D. $2\ \text{m}$

Fastest Approach (🚀):
$\lambda = \dfrac{340}{170} = 2\ \text{m}$.

Matches Option D.

Step-by-Step Breakdown:

1. Rearrange the wave equation

$v = f\lambda \implies \lambda = \dfrac{v}{f}$

2. Substitute

$\lambda = \dfrac{340}{170} = 2\ \text{m}$

3. Check by reasoning rather than arithmetic

$170\ \text{Hz}$ means $170$ complete waves pass a point every second, and together they occupy $340\ \text{m}$ of air. So each one is $\dfrac{340}{170} = 2\ \text{m}$ long. That reading of the equation is worth having, because it makes the rearrangement unnecessary.

4. What changes at a boundary

Enter a different medium and the speed changes. The frequency is fixed by the source and cannot change, so the wavelength must adjust to keep $v = f\lambda$ true. Sound speeds up in water, so its wavelength there is longer for the same note.

Matches Option D.

Why the Other Options Are Wrong (❌):

• A. $0.5\ \text{m}$ — Inverted
  Computing $\dfrac{170}{340}$.
• B. $170\ \text{m}$ — Wrong Quantity
  Quoting the frequency as a length.
• C. $57\,800\ \text{m}$ — Operation Error
  Multiplying speed by frequency.
• E. $1.7\ \text{m}$ — Decimal Slip
  A factor of ten slip in the division.

Common Mistake (⚠️):
Multiplying instead of dividing, giving $57\,800\ \text{m}$. A wavelength of tens of kilometres for an audible note should be rejected instantly.

Takeaway (📌):
$v = f\lambda$. At a boundary the frequency stays fixed and the wavelength changes with the speed.

Correct Answer: A. mass $234$, atomic number $90$

Fastest Approach (🚀):
Alpha removes $4$ nucleons and $2$ protons.
$234$ and $90$.

Matches Option A.

Step-by-Step Breakdown:

1. What an alpha particle is

Two protons and two neutrons — a helium nucleus, $^{4}_{2}\text{He}$.

2. Subtract from each number

mass number: $238-4 = 234$
atomic number: $92-2 = 90$

$^{238}_{\ 92}\text{U} \rightarrow\ ^{234}_{\ 90}\text{Th}+^{4}_{2}\text{He}$

3. Check the balance

Top: $234+4 = 238$ ✓
Bottom: $90+2 = 92$ ✓

Both sides must balance separately, and that check catches almost every error in these questions.

4. Contrast with beta-minus decay

A beta-minus particle is an electron, written $^{\ 0}_{-1}\text{e}$, produced when a neutron turns into a proton. The mass number is therefore unchanged and the atomic number goes up by one. Alpha reduces both; beta-minus moves only the bottom number, and upwards.

Matches Option A.

Why the Other Options Are Wrong (❌):

• B. mass $234$, atomic number $92$ — Half Applied
  Mass number reduced but the proton count left unchanged.
• C. mass $238$, atomic number $90$ — Half Applied
  Proton count reduced but the mass number left unchanged.
• D. mass $236$, atomic number $91$ — Wrong Particle
  Subtracting $2$ and $1$ instead of $4$ and $2$.
• E. mass $234$, atomic number $93$ — Wrong Decay Type
  Adding to the atomic number, as in beta decay.

Common Mistake (⚠️):
Changing the mass number but leaving the atomic number alone, which is beta decay applied to the wrong row. Alpha decay changes both.

Takeaway (📌):
Alpha: $A-4$, $Z-2$. Beta-minus: $A$ unchanged, $Z+1$. Balance both rows across the arrow as a check.

Correct Answer: A. $20\ \text{m/s}$

Fastest Approach (🚀):
$v = 8+12 = 20\ \text{m/s}$.

Matches Option A.

Step-by-Step Breakdown:

1. List what you have

$u = 8\ \text{m/s}$, $a = 3\ \text{m/s}^{2}$, $t = 4\ \text{s}$, $v = ?$, and $s$ is not involved.

2. Choose the equation without s

$v = u+at$

3. Substitute

$v = 8+(3\times 4) = 8+12 = 20\ \text{m/s}$

4. Order of operations

Multiplication before addition. Computing $(8+3)\times 4 = 44$ treats the equation as $(u+a)t$, which is a different relationship entirely and appears as Option D.

5. Read the result physically

Gaining $3\ \text{m/s}$ every second for four seconds adds $12\ \text{m/s}$ to the starting speed. That reasoning gives the answer without the formula, and it confirms the formula was used correctly.

Matches Option A.

Why the Other Options Are Wrong (❌):

• B. $12\ \text{m/s}$ — Initial Velocity Dropped
  Computing $at$ alone and forgetting $u$.
• C. $32\ \text{m/s}$ — Operation Error
  Computing $8\times 4$.
• D. $44\ \text{m/s}$ — Order of Operations
  Computing $(u+a)t$ — addition before multiplication.
• E. $96\ \text{m/s}$ — Operation Error
  Computing $8\times 3\times 4$.

Common Mistake (⚠️):
Adding before multiplying, or using $at$ as the answer and forgetting the initial velocity. The cyclist was already moving at $8\ \text{m/s}$.

Takeaway (📌):
$v = u+at$ when distance is not involved. Multiplication before addition, and never drop the initial velocity.

Correct Answer: C. $8000\ \text{kg/m}^{3}$

Fastest Approach (🚀):
$\dfrac{240}{30} = 8\ \text{g/cm}^{3}$.
$8\times 1000 = 8000\ \text{kg/m}^{3}$.

Matches Option C.

Step-by-Step Breakdown:

1. Compute in the given units

$\rho = \dfrac{m}{V} = \dfrac{240}{30} = 8\ \text{g/cm}^{3}$

2. Convert

$1\ \text{g} = 10^{-3}\ \text{kg}$ and $1\ \text{cm}^{3} = 10^{-6}\ \text{m}^{3}$, so

$1\ \dfrac{\text{g}}{\text{cm}^{3}} = \dfrac{10^{-3}}{10^{-6}} = 10^{3}\ \dfrac{\text{kg}}{\text{m}^{3}}$

Therefore $8\ \text{g/cm}^{3} = 8000\ \text{kg/m}^{3}$.

3. Why the factor is 1000 and not 100

The volume conversion is cubed: $100\ \text{cm} = 1\ \text{m}$ becomes $10^{6}\ \text{cm}^{3} = 1\ \text{m}^{3}$. Applying a linear factor where a cubic one is needed is the standard error, and it lands on $800$ or $80\,000$.

4. Is the answer plausible

Water is $1000\ \text{kg/m}^{3}$, aluminium about $2700$, iron about $7900$ and lead about $11\,300$. A density of $8000$ puts this block close to iron or steel — sensible for a metal, and a useful check on the order of magnitude.

Matches Option C.

Why the Other Options Are Wrong (❌):

• A. $8\ \text{kg/m}^{3}$ — Conversion Omitted
  Leaving the answer in $\text{g/cm}^{3}$ with the wrong unit attached.
• B. $800\ \text{kg/m}^{3}$ — Conversion Error
  Multiplying by $100$ instead of $1000$.
• D. $80\,000\ \text{kg/m}^{3}$ — Conversion Error
  Multiplying by $10\,000$.
• E. $0.125\ \text{kg/m}^{3}$ — Inverted
  Computing $\dfrac{30}{240}$.

Common Mistake (⚠️):
Multiplying by $100$ rather than $1000$, or converting the mass and volume separately and losing track of which way each goes.

Takeaway (📌):
$1\ \text{g/cm}^{3} = 1000\ \text{kg/m}^{3}$, and water is $1000\ \text{kg/m}^{3}$. Memorise that pair and every density conversion follows.

Correct Answer: A. $0.25\ \text{A}$

Fastest Approach (🚀):
$\dfrac{12}{48} = \dfrac14 = 0.25\ \text{A}$.

Matches Option A.

Step-by-Step Breakdown:

1. Rearrange

$V = IR$, so $I = \dfrac{V}{R}$.

2. Substitute

$I = \dfrac{12}{48} = 0.25\ \text{A}$

3. Check the size first

A large resistance opposes current, so a $48\ \Omega$ resistor on only $12\ \text{V}$ must pass a small current. Any answer above one amp is wrong before it is checked, which eliminates three of the five options.

4. What resistance means

One ohm is one volt per amp. A $48\ \Omega$ resistor needs $48\ \text{V}$ to drive one amp through it, so $12\ \text{V}$ drives a quarter of that.

Reasoning from the definition gives the answer without rearranging anything.

5. What follows

$P = VI = 12\times 0.25 = 3\ \text{W}$ is dissipated, and in $60\ \text{s}$ that transfers $180\ \text{J}$.

Matches Option A.

Why the Other Options Are Wrong (❌):

• B. $4\ \text{A}$ — Inverted
  Computing $\dfrac{48}{12}$.
• C. $576\ \text{A}$ — Inverted
  Multiplying voltage by resistance.
• D. $60\ \text{A}$ — Operation Error
  Adding the two values.
• E. $36\ \text{A}$ — Operation Error
  Subtracting the two values.

Common Mistake (⚠️):
Dividing resistance by voltage, or multiplying them. Estimate whether the current should be large or small before choosing an operation.

Takeaway (📌):
$I = \dfrac{V}{R}$. A resistance larger than the voltage always gives a current below one amp.

Correct Answer: B. $255\ \text{m}$

Fastest Approach (🚀):
$340\times 1.5 = 510\ \text{m}$ there and back.
$\dfrac{510}{2} = 255\ \text{m}$.

Matches Option B.

Step-by-Step Breakdown:

1. Find the total distance travelled

$d = vt = 340\times 1.5 = 510\ \text{m}$

2. Halve it

The sound travelled to the cliff and back, so the cliff is
$\dfrac{510}{2} = 255\ \text{m}$ away.

3. Or halve the time first

The one-way trip took $0.75\ \text{s}$, so $d = 340\times 0.75 = 255\ \text{m}$. ✓ Either order works; doing neither gives $510$, which is the most popular wrong answer and sits at Option A.

4. Why the method matters beyond echoes

Exactly the same halving appears in ultrasound scanning, sonar depth-sounding and radar. Any pulse-and-reflection measurement times a round trip, and the factor of two is always there.

Matches Option B.

Why the Other Options Are Wrong (❌):

• A. $510\ \text{m}$ — Round Trip Missed
  Forgetting the return journey.
• C. $227\ \text{m}$ — Operation Error
  Dividing $340$ by $1.5$.
• D. $1020\ \text{m}$ — Direction Reversed
  Doubling rather than halving.
• E. $170\ \text{m}$ — Arithmetic Error
  Dividing the total distance by three.

Common Mistake (⚠️):
Forgetting the return journey and giving $510\ \text{m}$. Ask 'how many times did the sound cover that distance?' before dividing.

Takeaway (📌):
Echo and reflection questions time a round trip. Halve the time or halve the distance — once, and only once.

Correct Answer: E. Beta only

Fastest Approach (🚀):
Paper does nothing $\Rightarrow$ no alpha.
Aluminium stops it $\Rightarrow$ not gamma.

Matches Option E.

Step-by-Step Breakdown:

1. The absorber ladder

alpha: stopped by a sheet of paper, or a few centimetres of air
beta: passes through paper, stopped by a few millimetres of aluminium
gamma: passes through aluminium; needs several centimetres of lead to reduce it appreciably

2. Read the first result

Paper makes no difference, so no alpha is being emitted. Had alpha been present, the count would have dropped at that first step.

3. Read the second result

Aluminium reduces the count almost to background, so essentially everything is being absorbed there. Gamma would have passed straight through, so there is no significant gamma either.

4. The conclusion

What passes paper but is stopped by thin aluminium is beta radiation.

5. Why penetration and ionisation run opposite

Alpha particles are large and doubly charged, so they interact strongly with atoms, ionise heavily, lose energy fast and stop quickly. Gamma rays have no charge and no mass, interact only rarely, ionise weakly and travel far. Beta sits between the two on both counts. The best ioniser is always the worst penetrator, and that single sentence generates the whole table.

Matches Option E.

Why the Other Options Are Wrong (❌):

• A. Alpha only — First Test Misread
  Alpha would have been stopped by the paper.
• B. Gamma only — Second Test Misread
  Gamma would have passed through the aluminium.
• C. Gamma and beta — Second Test Misread
  The gamma component would have survived the aluminium.
• D. Alpha and gamma — Both Tests Misread
  Neither is consistent with these two results.

Common Mistake (⚠️):
Concluding gamma because the radiation passed through paper. Beta also passes through paper — it is the aluminium result that separates the two.

Takeaway (📌):
Paper stops alpha, thin aluminium stops beta, thick lead attenuates gamma. Ionising power runs in the opposite order to penetration.

Correct Answer: E. $8\ \text{kg}$

Fastest Approach (🚀):
$m = \dfrac{48}{6} = 8\ \text{kg}$.

Matches Option E.

Step-by-Step Breakdown:

1. Rearrange

$F = ma$, so $m = \dfrac{F}{a}$.

2. Substitute

$m = \dfrac{48}{6} = 8\ \text{kg}$

3. Let the units check it

$\dfrac{\text{N}}{\text{m/s}^{2}}$. Since $1\ \text{N} = 1\ \text{kg m/s}^{2}$, the $\text{m/s}^{2}$ cancels and kilograms remain. ✓

Multiplying instead would give $\text{N m/s}^{2}$, which is not a mass — so the units rule out Option A without any arithmetic.

4. The triangle

$F$ on top, $m$ and $a$ below. Cover the quantity you want and the arrangement of the other two reads straight off. It is the same device as the mass-moles-$M_r$ triangle in chemistry.

Matches Option E.

Why the Other Options Are Wrong (❌):

• A. $288\ \text{kg}$ — Inverted
  Multiplying rather than dividing.
• B. $42\ \text{kg}$ — Operation Error
  Subtracting the two values.
• C. $54\ \text{kg}$ — Operation Error
  Adding the two values.
• D. $0.125\ \text{kg}$ — Inverted
  Computing $\dfrac{6}{48}$.

Common Mistake (⚠️):
Multiplying force by acceleration. Rearranging on paper before substituting removes the ambiguity entirely.

Takeaway (📌):
$m = \dfrac{F}{a}$. Check the rearrangement with units — they identify the right operation on their own.

Correct Answer: B. $30\,000\ \text{Pa}$

Fastest Approach (🚀):
$\dfrac{600}{0.02} = \dfrac{60\,000}{2} = 30\,000\ \text{Pa}$.

Matches Option B.

Step-by-Step Breakdown:

1. Apply the definition

$p = \dfrac{F}{A} = \dfrac{600}{0.02}$

2. Clear the decimal before dividing

Multiply top and bottom by $100$:
$\dfrac{600}{0.02} = \dfrac{60\,000}{2} = 30\,000\ \text{Pa}$

Dividing by a number smaller than one makes the answer larger — which is worth stating explicitly, because instinct often says otherwise.

3. Read what the result means

$30\ \text{kPa}$ is about a third of atmospheric pressure. One pascal is one newton per square metre, so a pascal is a very small unit and pressures in real problems usually run to thousands.

4. Why area matters so much

Halve the base area and the pressure doubles for the same weight. This is why a drawing pin pierces wood with a light push while the same force through your thumb does not — identical force, vastly different area.

Matches Option B.

Why the Other Options Are Wrong (❌):

• A. $12\ \text{Pa}$ — Inverted
  Multiplying force by area instead of dividing.
• C. $3000\ \text{Pa}$ — Decimal Slip
  A factor of ten lost in the decimal division.
• D. $300\,000\ \text{Pa}$ — Decimal Slip
  A factor of ten gained.
• E. $0.000033\ \text{Pa}$ — Inverted
  Dividing area by force.

Common Mistake (⚠️):
Multiplying by the area rather than dividing, giving $12\ \text{Pa}$. A quick check: the answer must be far bigger than the force, since the area is far less than a square metre.

Takeaway (📌):
$p = \dfrac{F}{A}$ in pascals with the area in $\text{m}^{2}$. Dividing by a decimal less than one always increases the result.

Correct Answer: B. $20\ \Omega$

Fastest Approach (🚀):
$5+7+8 = 20\ \Omega$.

Matches Option B.

Step-by-Step Breakdown:

1. Add them

In series the current passes through every resistor in turn, so the oppositions add:
$R_{\text{total}} = 5+7+8 = 20\ \Omega$

2. The check

A series total is always larger than the largest individual resistance, because each additional resistor adds more opposition. $20 > 8$ ✓

Any answer smaller than $8\ \Omega$ has used the parallel formula by mistake — which is what Options B, D and E are.

3. Contrast with parallel

In parallel, the current has alternative routes, so the total is always smaller than the smallest branch:
$\dfrac{1}{R} = \dfrac15+\dfrac17+\dfrac18$

That gives about $2.06\ \Omega$ — Option B, and the right answer to the other question.

4. Why series and parallel differ this way

Series adds length to the path; parallel adds width. More length means more opposition, more width means less.

Matches Option B.

Why the Other Options Are Wrong (❌):

• A. $2.06\ \Omega$ — Wrong Arrangement
  The parallel combination of the same three resistors.
• C. $280\ \Omega$ — Operation Error
  Multiplying the three values.
• D. $6.67\ \Omega$ — Operation Error
  Averaging the three values.
• E. $1.7\ \Omega$ — Wrong Arrangement
  A mis-computed reciprocal sum.

Common Mistake (⚠️):
Applying the reciprocal formula to a series circuit. Reciprocals belong to parallel, and they always give a smaller total.

Takeaway (📌):
Series adds and exceeds the largest branch. Parallel combines as reciprocals and falls below the smallest.

Correct Answer: E. It bends towards the normal, and the angle of refraction is less than $40^{\circ}$

Fastest Approach (🚀):
Into a denser medium $\Rightarrow$ slower $\Rightarrow$ towards the normal.
Towards the normal $\Rightarrow$ smaller angle.

Matches Option E.

Step-by-Step Breakdown:

1. What happens to the speed

Light travels more slowly in glass than in air. The wavefront meeting the surface at an angle is slowed on one side first, which swings the ray round.

2. Which way it bends

Entering a slower medium bends the ray towards the normal. Leaving into a faster one bends it away. So air into glass bends towards the normal.

3. What that means for the angle

Angles are measured from the normal, so bending towards the normal makes the angle smaller: the angle of refraction is less than $40^{\circ}$. Options B and D disagree only on this point, and D is self-contradictory.

4. Why no total internal reflection here

Total internal reflection requires light travelling from the denser medium to the less dense one, beyond the critical angle. This ray is going the other way, so there is always a refracted ray however large the angle of incidence. Option E is impossible in this direction.

5. What does not change

The frequency is set by the source and is unchanged. The speed falls, so by $v = f\lambda$ the wavelength falls too — the light is not a different colour inside the glass.

Matches Option E.

Why the Other Options Are Wrong (❌):

• A. It bends away from the normal, and the angle of refraction is greater than $40^{\circ}$ — Direction Reversed
  The behaviour on leaving glass for air, not on entering.
• B. It is reflected back into the air with no refracted ray — Wrong Direction
  Total internal reflection needs the ray to start in the denser medium.
• C. It continues without bending, at $40^{\circ}$ — Special Case
  Only true at $0^{\circ}$ incidence, along the normal.
• D. It bends towards the normal, and the angle of refraction is greater than $40^{\circ}$ — Internally Inconsistent
  Self-contradictory: bending towards the normal reduces the angle.

Common Mistake (⚠️):
Measuring the angle from the surface rather than from the normal, which replaces every angle with its complement and reverses which one is 'bigger'.

Takeaway (📌):
Into a denser, slower medium: bend towards the normal, smaller angle. Out into a faster one: away from the normal, larger angle.

Correct Answer: D. $24\ \text{hours}$

Fastest Approach (🚀):
$800\to 400\to 200\to 100\to 50$: four halvings.
$4\times 6 = 24$ hours.

Matches Option D.

Step-by-Step Breakdown:

1. Halve until you arrive

$800 \to 400 \to 200 \to 100 \to 50$

Count the arrows, not the numbers: four halvings.

2. Multiply by the half-life

$4\times 6 = 24$ hours

3. Check with the ratio

$\dfrac{800}{50} = 16 = 2^{4}$, confirming four half-lives. That is the faster route when the numbers are large: divide, then ask what power of two the result is.

4. What half-life actually means

Each half-life halves whatever remains, not the original amount. So the count falls to a half, then a quarter, then an eighth — never reaching zero. It is also a statistical statement about a huge number of nuclei: individual decays are entirely random and unpredictable, and nothing about a nucleus's history changes when it will decay.

Matches Option D.

Why the Other Options Are Wrong (❌):

• A. $16\ \text{hours}$ — Ratio Misused
  Using $\dfrac{800}{50} = 16$ directly as a number of hours.
• B. $96\ \text{hours}$ — Ratio Misused
  Multiplying $16$ by the half-life.
• C. $18\ \text{hours}$ — Miscount
  Counting three halvings.
• E. $30\ \text{hours}$ — Off by One
  Counting five values instead of four steps.

Common Mistake (⚠️):
Counting the numbers in the chain rather than the arrows, giving five half-lives and $30$ hours. Or assuming the activity falls by a fixed amount each half-life rather than by half.

Takeaway (📌):
Halve repeatedly and count the steps. Each half-life halves what is left, so the activity approaches zero without ever reaching it.

Correct Answer: D. $300\ \text{J}$

Fastest Approach (🚀):
$2\times 10\times 15 = 300\ \text{J}$.

Matches Option D.

Step-by-Step Breakdown:

1. Apply the relationship

$E_p = mgh = 2\times 10\times 15 = 300\ \text{J}$

2. Why the path does not matter

Only the vertical height counts. Carrying the mass up a long ramp to the same height transfers the same $300\ \text{J}$ of potential energy — the extra distance along the slope does not appear in the formula.

That is why $h$ is defined as the change in height rather than the distance moved.

3. Where the energy goes next

Released, the mass converts that potential energy back to kinetic energy. Ignoring air resistance, all $300\ \text{J}$ becomes kinetic at the bottom:
$\tfrac12(2)v^{2} = 300 \implies v^{2} = 300 \implies v \approx 17.3\ \text{m/s}$

4. Why $g$ is quoted in N/kg here

Gravitational field strength in $\text{N/kg}$ and acceleration in $\text{m/s}^{2}$ are numerically identical and dimensionally equivalent. Either unit gives joules in this calculation.

Matches Option D.

Why the Other Options Are Wrong (❌):

• A. $30\ \text{J}$ — Factor Omitted
  Computing $mh$ and omitting $g$.
• B. $3000\ \text{J}$ — Decimal Slip
  A factor of ten too large.
• C. $150\ \text{J}$ — Factor Error
  Using $m = 1$, or halving.
• E. $75\ \text{J}$ — Wrong Formula
  Computing $\tfrac12 mgh$, confusing it with kinetic energy.

Common Mistake (⚠️):
Omitting one of the three factors, most often $g$. Multiplying only mass by height gives $30$, which is Option A.

Takeaway (📌):
$E_p = mgh$ with $h$ the vertical height gained. The route taken is irrelevant.

Correct Answer: C. $150\,000\ \text{Pa}$

Fastest Approach (🚀):
$1000\times 10 = 10\,000$, then $\times 15 = 150\,000\ \text{Pa}$.

Matches Option C.

Step-by-Step Breakdown:

1. Apply the formula

$p = \rho gh = 1000\times 10\times 15 = 150\,000\ \text{Pa}$

That is $150\ \text{kPa}$, roughly one and a half atmospheres.

2. What the question excluded

'Due to the water alone' means the atmospheric pressure pushing down on the lake surface — about $100\,000\ \text{Pa}$ — is not included. The total pressure a diver experiences at that depth would be about $250\,000\ \text{Pa}$, which is Option E and is there for anyone who adds it in unasked.

3. What the pressure does not depend on

Not the surface area of the lake, not its total volume, and not the shape of the container. Only the density, $g$, and the vertical depth. A narrow tube and a wide lake filled to the same depth give identical pressure at the bottom.

4. A useful rule of thumb

Every $10\ \text{m}$ of water adds about one atmosphere. At $15\ \text{m}$ that is one and a half atmospheres from the water — matching the calculation.

Matches Option C.

Why the Other Options Are Wrong (❌):

• A. $1500\ \text{Pa}$ — Decimal Slip
  Two factors of ten lost.
• B. $15\,000\ \text{Pa}$ — Decimal Slip
  One factor of ten lost.
• D. $1\,500\,000\ \text{Pa}$ — Decimal Slip
  A factor of ten gained.
• E. $250\,000\ \text{Pa}$ — Misread Question
  Adding atmospheric pressure, which the question excluded.

Common Mistake (⚠️):
Adding atmospheric pressure when the question asked only for the water's contribution, or dropping a factor of ten in the multiplication.

Takeaway (📌):
$p = \rho gh$ depends only on density, gravity and depth. Read carefully whether atmospheric pressure is wanted as well.

Correct Answer: C. $55^{\circ}$

Fastest Approach (🚀):
From the normal: $90-35 = 55^{\circ}$.
Reflection $=$ incidence $= 55^{\circ}$.

Matches Option C.

Step-by-Step Breakdown:

1. Convert to the normal

The normal is perpendicular to the mirror. A ray at $35^{\circ}$ to the surface is therefore at
$90^{\circ}-35^{\circ} = 55^{\circ}$
to the normal. That is the angle of incidence.

2. Apply the law of reflection

angle of reflection $=$ angle of incidence $= 55^{\circ}$

By convention this is also measured from the normal, so $55^{\circ}$ is the answer as it stands.

3. Why the question is phrased that way

Every reflection and refraction angle in physics is measured from the normal. Quoting one from the surface is the standard way of testing whether you know that, and $35^{\circ}$ sits at Option A waiting for anyone who does not convert.

4. The angle between the two rays

The incident and reflected rays are each $55^{\circ}$ from the normal on opposite sides, so the angle between them is $110^{\circ}$ — Option E, and a different question again.

Matches Option C.

Why the Other Options Are Wrong (❌):

• A. $35^{\circ}$ — Wrong Reference Line
  Applying the law to the angle from the surface without converting.
• B. $70^{\circ}$ — Operation Error
  Doubling $35^{\circ}$.
• D. $145^{\circ}$ — Wrong Reference Line
  Computing $180^{\circ}-35^{\circ}$.
• E. $110^{\circ}$ — Wrong Quantity
  The angle between the incident and reflected rays, not the angle of reflection.

Common Mistake (⚠️):
Answering $35^{\circ}$ by applying the law of reflection to an angle measured from the wrong line. The law is right; the angle fed into it is not.

Takeaway (📌):
Always convert to the normal before applying any reflection or refraction law. From the surface, subtract from $90^{\circ}$.

Correct Answer: C. $15\ \text{minutes}$

Fastest Approach (🚀):
$\dfrac{3200}{200} = 16 = 2^{4}$: four halvings.
$\dfrac{60}{4} = 15$ minutes.

Matches Option C.

Step-by-Step Breakdown:

1. Count the halvings

$3200\to 1600\to 800\to 400\to 200$

Four arrows, so four half-lives have elapsed.

2. Or use the ratio

$\dfrac{3200}{200} = 16$, and $16 = 2^{4}$ — the same four halvings, found by one division.

3. Divide the elapsed time

One hour is $60$ minutes, spanning four half-lives:
$\dfrac{60}{4} = 15$ minutes

4. Check it forwards

Starting at $3200$ and halving every $15$ minutes: $1600$ at $15$, $800$ at $30$, $400$ at $45$, $200$ at $60$ ✓

5. The two traps

Counting the values rather than the arrows gives five half-lives and $12$ minutes — Option C. And using the ratio $16$ directly as a number of minutes gives Option E.

Both come from stopping one step early, so running the check forwards is worth the ten seconds.

Matches Option C.

Why the Other Options Are Wrong (❌):

• A. $12\ \text{minutes}$ — Off by One
  Counting five half-lives instead of four.
• B. $4\ \text{minutes}$ — Ratio Misused
  Dividing $60$ by the ratio $16$, roughly.
• D. $30\ \text{minutes}$ — Miscount
  Counting two half-lives.
• E. $16\ \text{minutes}$ — Wrong Quantity
  Quoting the activity ratio as a time.

Common Mistake (⚠️):
Dividing the elapsed time by the activity ratio rather than by the number of halvings. The ratio is $16$; the number of half-lives is $4$.

Takeaway (📌):
Halvings first, then divide the elapsed time by that count. Verify by running the halving forwards to the stated end value.

Correct Answer: B. $18\,000\ \text{kg m/s}$

Fastest Approach (🚀):
$1200\times 15 = 18\,000\ \text{kg m/s}$.

Matches Option B.

Step-by-Step Breakdown:

1. Apply the definition

$p = mv = 1200\times 15 = 18\,000\ \text{kg m/s}$

2. Momentum is a vector

It has the direction of the velocity. That matters the moment two objects interact: momenta in opposite directions carry opposite signs, and conservation only works once a positive direction has been chosen.

For a single object travelling one way, the magnitude is all that is asked for.

3. Why the unit has no special name

Unlike the newton or the joule, momentum's unit is left as $\text{kg m/s}$ — which is convenient, because it states the calculation. Any answer whose units do not come from multiplying a mass by a velocity is wrong.

4. Distinguishing it from kinetic energy

$p = mv$ and $E_k = \tfrac12 mv^{2}$ both describe a moving object, but they are different quantities with different units and different conservation rules. In a collision momentum is always conserved; kinetic energy usually is not.

Matches Option B.

Why the Other Options Are Wrong (❌):

• A. $80\ \text{kg m/s}$ — Inverted
  Computing $\dfrac{1200}{15}$.
• C. $1215\ \text{kg m/s}$ — Operation Error
  Adding rather than multiplying.
• D. $135\,000\ \text{kg m/s}$ — Wrong Quantity
  Computing $\tfrac12 mv^{2}$, which is the kinetic energy in joules.
• E. $1185\ \text{kg m/s}$ — Operation Error
  Subtracting rather than multiplying.

Common Mistake (⚠️):
Adding the two numbers, or computing $\tfrac12 mv$. Momentum is a straightforward product with no factor of a half.

Takeaway (📌):
$p = mv$, in $\text{kg m/s}$, and it is a vector. Kinetic energy is the one with the half and the square.

Correct Answer: C. $6000\ \text{N}$

Fastest Approach (🚀):
$200\,000\times 0.03 = 6000\ \text{N}$.

Matches Option C.

Step-by-Step Breakdown:

1. Rearrange

$p = \dfrac{F}{A}$, so $F = pA$.

2. Substitute

$F = 200\,000\times 0.03 = 6000\ \text{N}$

3. Handle the decimal safely

$0.03 = \dfrac{3}{100}$, so
$\dfrac{200\,000\times 3}{100} = \dfrac{600\,000}{100} = 6000$

Turning the decimal into a fraction avoids counting zeros, which is where these go wrong.

4. Check the direction

The area is far less than one square metre, so the force on it must be far less than the pressure's numerical value. $6000$ is well below $200\,000$ ✓ — which rules out Option C immediately.

5. Why this matters practically

Hydraulic systems use exactly this. The same pressure acting on a larger piston produces a proportionally larger force, which is how a car is lifted by hand — pressure is transmitted undiminished through a fluid, and the area does the multiplying.

Matches Option C.

Why the Other Options Are Wrong (❌):

• A. $6\,666\,667\ \text{N}$ — Inverted
  Dividing rather than multiplying.
• B. $600\ \text{N}$ — Decimal Slip
  A factor of ten lost.
• D. $60\,000\ \text{N}$ — Decimal Slip
  A factor of ten gained.
• E. $0.00015\ \text{N}$ — Inverted
  Dividing area by pressure.

Common Mistake (⚠️):
Dividing pressure by area. That would give a larger number than the pressure, which cannot be a force acting on a small area.

Takeaway (📌):
$F = pA$. Multiplying by an area below one square metre must reduce the number.

Correct Answer: B. Light travelling from a denser to a less dense medium, at an angle greater than the critical angle

Fastest Approach (🚀):
Denser to less dense, and beyond the critical angle.

Matches Option B.

Step-by-Step Breakdown:

1. Why the denser medium first

Leaving a denser medium bends the ray away from the normal, so the refracted angle is larger than the incident one. Increase the incident angle far enough and the refracted angle reaches $90^{\circ}$ — the ray would skim along the boundary. Push past that and there is no refracted ray at all.

Going the other way, into a denser medium, the ray bends towards the normal and can never reach $90^{\circ}$. Total internal reflection is impossible in that direction, whatever the angle.

2. Why beyond the critical angle

The critical angle is defined as the incident angle giving a refracted angle of exactly $90^{\circ}$. Below it, light refracts out normally with a weak partial reflection. Above it, all the light is reflected back — hence 'total'.

3. Both conditions are needed

Denser to less dense but below the critical angle: the light refracts out. Beyond the critical angle but entering a denser medium: the critical angle does not exist in that direction. Only the pair together produces the effect.

4. Where it is used

Optical fibres trap light along the core by total internal reflection at every bounce, which is how internet traffic crosses oceans with almost no loss. The same effect makes a diamond sparkle and lets a periscope work with prisms instead of mirrors.

Matches Option B.

Why the Other Options Are Wrong (❌):

• A. Light travelling from a less dense to a denser medium, at an angle greater than the critical angle — Direction Reversed
  Right angle condition, wrong direction of travel.
• C. Light travelling from a denser to a less dense medium, at an angle less than the critical angle — Angle Condition Reversed
  Right direction, but below the critical angle the light refracts out.
• D. Light striking the boundary along the normal, in either direction — Special Case
  Along the normal the light passes straight through without bending.
• E. Light travelling from a less dense to a denser medium, at any angle — Direction Reversed
  Entering a denser medium never produces total internal reflection.

Common Mistake (⚠️):
Remembering the critical angle condition but reversing the direction of travel. The critical angle only exists for light leaving the denser medium.

Takeaway (📌):
Total internal reflection needs denser to less dense and an angle beyond the critical angle. Both, every time.

Correct Answer: D. In a vacuum all electromagnetic waves travel at the same speed, and gamma rays have the shortest wavelength

Fastest Approach (🚀):
Same speed in a vacuum for all.
Radio longest, gamma shortest.

Matches Option D.

Step-by-Step Breakdown:

1. The common speed

Every electromagnetic wave travels at $c = 3\times 10^{8}\ \text{m/s}$ in a vacuum, from radio waves to gamma rays. That kills Option A immediately: gamma rays are more energetic, not faster.

2. The order of the spectrum

By increasing frequency and decreasing wavelength:
radio, microwave, infrared, visible, ultraviolet, X-ray, gamma

So ultraviolet has a shorter wavelength than infrared (Option B is backwards), and microwaves have a far lower frequency than X-rays (Option E is backwards).

3. They are transverse

Electromagnetic waves are oscillating electric and magnetic fields perpendicular to the direction of travel and to each other. That makes them transverse, so Option C is wrong — and it is also why they can be polarised, which longitudinal waves cannot.

4. What follows from a common speed

Since $c = f\lambda$ is the same for all of them, frequency and wavelength are inversely proportional across the whole spectrum. Highest frequency necessarily means shortest wavelength, which is why gamma rays sit at both extremes at once.

Matches Option D.

Why the Other Options Are Wrong (❌):

• A. Gamma rays travel faster in a vacuum than radio waves — Speed Misconception
  All electromagnetic waves share the same speed in a vacuum.
• B. Ultraviolet has a longer wavelength than infrared — Order Reversed
  Reversed — ultraviolet has the shorter wavelength.
• C. All electromagnetic waves are longitudinal — Wave Type
  Electromagnetic waves are transverse, which is why they can be polarised.
• E. Microwaves have a higher frequency than X-rays — Order Reversed
  Reversed — X-rays have far the higher frequency.

Common Mistake (⚠️):
Assuming higher-energy waves travel faster. In a vacuum they all travel at exactly $c$; the energy is carried by the frequency, not the speed.

Takeaway (📌):
One speed in a vacuum for all, transverse, and ordered radio to gamma by rising frequency and falling wavelength.

Correct Answer: E. $500\ \text{s}$

Fastest Approach (🚀):
$\dfrac{1.5}{3} = 0.5$; $10^{11-8} = 10^{3}$.
$0.5\times 1000 = 500\ \text{s}$.

Matches Option E.

Step-by-Step Breakdown:

1. Rearrange

$\text{speed} = \dfrac{\text{distance}}{\text{time}}$, so $t = \dfrac{d}{v}$.

2. Divide mantissas and indices separately

$\dfrac{1.5}{3} = 0.5$
$\dfrac{10^{11}}{10^{8}} = 10^{11-8} = 10^{3}$

$t = 0.5\times 10^{3} = 500\ \text{s}$

3. Make it meaningful

$500\ \text{s}$ is about $8\tfrac13$ minutes. Sunlight left the Sun over eight minutes before it reaches you — so the Sun you see is always the Sun of eight minutes ago.

4. Why the speed is the same for all of it

Every part of the spectrum travels at $3\times 10^{8}\ \text{m/s}$ in a vacuum, so the visible light, the ultraviolet and the infrared all arrive together. Nothing about the calculation depends on which kind of electromagnetic wave it is.

5. Why Option D is worth noticing

$4.5\times 10^{19}$ is the product rather than the quotient. A travel time longer than the age of the universe should be rejected on sight — an order-of-magnitude check that costs nothing.

Matches Option E.

Why the Other Options Are Wrong (❌):

• A. $0.002\ \text{s}$ — Inverted
  Dividing the wrong way round.
• B. $50\ \text{s}$ — Index Error
  Subtracting the indices incorrectly by one.
• C. $5000\ \text{s}$ — Index Error
  Index error in the other direction.
• D. $4.5\times 10^{19}\ \text{s}$ — Inverted
  Multiplying rather than dividing.

Common Mistake (⚠️):
Multiplying instead of dividing, or adding the indices rather than subtracting them. Handle the mantissas and the powers of ten as two separate divisions.

Takeaway (📌):
$t = \dfrac{d}{v}$. In standard form, divide the mantissas and subtract the indices, then sanity-check the magnitude.

Correct Answer: D. $300\ \text{m/s}$

Fastest Approach (🚀):
$50\times 6 = 300\ \text{m/s}$.

Matches Option D.

Step-by-Step Breakdown:

1. Apply the wave equation

$v = f\lambda = 50\times 6 = 300\ \text{m/s}$

2. Why it works

Frequency is the number of complete waves passing a point each second — here $50$. Each occupies $6\ \text{m}$ of space. So $50\times 6 = 300\ \text{m}$ of wave passes each second, which is the speed.

Read that way, the equation needs no memorising.

3. Check the units

$\text{Hz}\times\text{m} = \text{s}^{-1}\times\text{m} = \text{m/s}$ ✓ — a speed, as required.

4. What each quantity depends on

The medium fixes the speed. The source fixes the frequency. So when a wave crosses into a new medium the speed changes, the frequency cannot, and the wavelength adjusts to keep $v = f\lambda$ true.

Matches Option D.

Why the Other Options Are Wrong (❌):

• A. $0.12\ \text{m/s}$ — Inverted
  Computing $\dfrac{6}{50}$.
• B. $8.3\ \text{m/s}$ — Inverted
  Computing $\dfrac{50}{6}$.
• C. $56\ \text{m/s}$ — Operation Error
  Adding the two values.
• E. $44\ \text{m/s}$ — Operation Error
  Subtracting the two values.

Common Mistake (⚠️):
Dividing rather than multiplying. The units settle it: hertz times metres gives metres per second, and no division does.

Takeaway (📌):
$v = f\lambda$. Frequency counts waves per second; wavelength gives each one's length; the product is distance per second.