IB Physics HL A.3 Work, energy and power Questions

SLSL HLHL

Practise IB Physics HL A.3 by interpreting force–displacement evidence, tracking energy transfers, and calculating power, efficiency and fuel requirements.

245 questions in this syllabus scope, with question parts counted together.

Exam points & syllabus

Exam points

Define work using the force component along displacement, or define all symbols in W = Fs cosθ; energy transfer is an accepted alternative.
Match work to the actual force and path: project an oblique pull along motion, and use the distance moved by a frictional contact point.
Use force–distance area or resultant work to find energy change; do not replace it with a time-average force without a justified model.
Separate applied work, gravitational energy change and resistive work when determining a stopping distance or force.
Use the stated path length for resistive work and compare it with the available KE before deciding whether motion can continue.
Track where energy goes: a mechanical-energy decrease or reduced intensity at one location can represent transfer, without total energy disappearing.
Choose the stage before writing an energy balance: free fall before a rope stretches differs from its decelerating elastic-extension stage.
For ideal descent or swinging motion, use the vertical height lost and retain initial KE; show the precision required by the marking scheme.
For engine-off motion away from a planet, explain the speed decrease as KE transferring into GPE; use the stated negligible-loss conditions.
In a spring launch to a higher point, include both final KE and GPE; use the stated conversion fraction when only part of lost GPE becomes KE.
For HL rolling motion without slipping, include translational and rotational KE and use the radius of the surface that actually contacts the track.
For spring recoil from rest, combine equal opposite momenta with the sum of both final kinetic energies; do not assign all energy to one mass.
Distinguish stages around impact: mechanical energy can be conserved before contact while some KE transfers to another body or internal energy in impact.
For a horizontal spring launch, v is proportional to compression; for identical parallel springs under the same total load, total stored energy halves.
For a vertical spring, find equilibrium extension from kx = mg and include changing GPE when interpreting elastic-energy and KE transfers.
Convert energy and mass units before calculating KE or speed; use the actual graph maximum and the extra digits required by a show-that task.
Use the squared-speed difference for bounce energy loss and the squared-speed ratio for the fraction retained, reading speeds immediately around impact.
When using a projectile’s apex data to recover its KE just after a bounce, include both the horizontal KE at the apex and the GPE gained.
For a higher circular orbit, retain the negative KE change ΔEk = −½ΔU even though gravitational potential energy increases.
Use stopping potential to obtain maximum photoelectron KE, or subtract a retarding electrical energy change before finding arrival speed.
Relate KE to momentum using p = √(2mEk); for complete conversion of nonrelativistic electron KE to a photon, frequency scales with speed squared.
For average motor input power, divide useful work by both time and efficiency; use average speed when using a constant force during acceleration.
At constant speed, engine power balances resistive work through P = Fv; a zero resultant does not imply zero energy-transfer rate.
Use the force component parallel to velocity in P = Fv; a contributor’s share is its power divided by the total power of all contributors.
For a given quadratic drag law at steady speed, P ∝ v³; distinguish this from constant-force power trends against time or distance.
Calculate energy from power–time area, subtracting load energy over the same interval when asked for net energy exported.
For dissipation rates, divide the appropriate energy loss by its own interval and distinguish total input power from the system’s KE gain rate.
When a motor accelerates a load upward, include both GPE and KE gains; a hovering drone can still supply KE to moving air.
For conveyor power, distinguish the motor’s Fv from the captured material’s ½ṁv² KE gain rate; momentum flow determines the driving force.
For flow-driven power, convert volume per time into mass per time and use mean gravitational head; use efficiency in the required direction.
For stored-energy duration, divide by combined generator power; for constant-power transfer convert the time interval before using E = Pt.
Use given charge or decay data to obtain energy per time: QV/t for battery power, or λN times joules per decay for nuclear-source power.
Treat ½ρAv³ as incident wind KE flux; use explicit ideal assumptions or supplied efficiency rather than claiming all incident energy is extractable.
In supplied wind-farm models, separate per-rotor output from total demand and allotted land area, applying efficiency and rounding counts upward.
Choose useful output before applying efficiency: divide output by efficiency for input, or multiply input by efficiency to find useful energy or power.
For solar panels, use incident power IA and the relevant intensity conditions; equal output at the same intensity requires efficiency × area to match.
Compare annual output and input using matching energy units, or convert the annual output to average power before calculating efficiency.
From given electrical output VI, divide by efficiency for input power, then divide by supply voltage for input current.
In a Sankey diagram, compare final useful delivered width with original input width, after internal use and transmission losses.
Explain low efficiency with a named transfer or mechanism; an ideal heating estimate understates input because of thermal transfers to surroundings.
Read the fuel energy-density units first: distinguish per kilogram from per litre or m³, and compare fuels only after converting to a common basis.
For fuel consumption or savings, include the stated time, thermal demand, efficiency and number of engine cycles or cylinders before solving for fuel quantity.

Covered syllabus points

A.3.1—Conservation of energy A.3.2—Work as energy transfer A.3.3—Sankey diagrams A.3.4—Work by constant force A.3.5—Work-energy change A.3.6—Mechanical energy A.3.7—Mechanical energy conservation A.3.8—Mechanical energy transformations A.3.9—Kinetic energy A.3.10—Gravitational potential energy A.3.11—Elastic potential energy A.3.12—Power A.3.13—Efficiency A.3.14—Fuel energy density

Exam points

When one component’s energy loss exceeds another’s gain, account for transfer to circuit heating or electromagnetic radiation.
For a falling object slowed by drag, subtract its kinetic-energy gain from its gravitational-energy loss to find energy transferred to the air.
At terminal speed, explain that gravitational energy continues to transfer to the air’s internal energy while the falling object’s KE stays constant.
Explain destructive interference using redistribution: energy absent at dark fringes appears at the intensity maxima.
Trace a wind generator’s energy path from the wind’s kinetic energy through rotating machinery to electrical energy.
For the supplied ideal wind-power bound, explain why actual extraction is lower: departing air retains kinetic energy because its speed is nonzero.

[Maximum number: 1]

Two sources of light produce an interference pattern on a screen.

Light of wavelength 720 nm is incident on two narrow slits that are separated by 0.12 mm . An interference pattern is observed on a screen. $\mathrm{P}_{1}$ and $\mathrm{P}_{2}$ are the points of destructive interference closest to the central maximum M .

Suggest how energy conservation is consistent with the fact that the energy at $\mathrm{P}_{1}$ and $\mathrm{P}_{2}$ is zero.

A beam of light containing all wavelengths in the range $[550 \mathrm{~nm}, 650 \mathrm{~nm}]$ is incident normally on a diffraction grating. The grating has 580 lines per mm . A diffraction pattern is observed on a screen as shown.

The energy missing at $\mathrm{P}_{1}$ and $\mathrm{P}_{2}$ is found at the maxima

Exam points

Define work as force × displacement in the force direction, or use W = Fs cosθ with every symbol defined.

[Maximum number: 2]

This question is about the motion of a ship and observing objects from it.

Outline the meaning of work.

work done = force × distance moved;
(distance moved) in direction of force;
or
energy transferred;
from one location to another;
or
work done $=F s \cos \theta$ ;
with each symbol defined;

[Maximum number: 1]

The Sankey diagram shows the energy input from fuel that is eventually converted to useful domestic energy in the form of light in a filament lamp.

What is true for this Sankey diagram?

The overall efficiency of the process is 10 %.

Generation and transmission losses account for 55 % of the energy input.

Useful energy accounts for half of the transmission losses.

The energy loss in the power station equals the energy that leaves it.

Exam points

Use force direction to distinguish positive, negative and zero work; the radial resultant does no work in uniform circular motion.
When the question specifies an average-force work model, show W = ΔKE before using s = W/Favg; a kinematic-only calculation can lose a mark.
For braking over several rotations, use the travelled distance n2πr, not zero net displacement; friction removes mechanical energy.
For a kite pulling a ship horizontally, project tension with cosθ and convert kilometres to metres before calculating W = Fs cosθ.
At terminal speed, use drag = mg and distance = vt to obtain work transferred to the air, mgvt, despite zero net work on the object.

[Maximum number: 2]

Aibhe and Euan are sitting on opposite sides of a merry-go-round, which is rotating at constant speed around a fixed centre. The diagram below shows the view from above.

Aibhe is moving at speed $1.0 \mathrm{~m} \mathrm{~s}^{-1}$ relative to the ground.

Euan drags his foot along the ground to act as a brake. The merry-go-round comes to a stop after 4.0 rotations. The radius of the merry-go-round is 1.5 m . The average frictional force between his foot and the ground is 45 N . Calculate the work done.

distance travelled by Euan $=4.0 \times 2 \pi \times 1.5(=37.70 \mathrm{~m})$ ;

W\left(=F_{a v} d=45 \times 37.70\right)=1700(\mathrm{~J}) ;

Exam points

Use the increase in kinetic energy, not its final value, when finding the gravitational work needed to double an initial KE.
Include initial kinetic energy in an energy-loss calculation: energy transferred to air = initial KE + GPE lost − final KE.
Find work from force–displacement area; for an acceleration–displacement graph, multiply its area by the object’s mass.
For uphill stopping, set initial KE equal to gravitational-energy gain plus frictional work, using the normal component on the slope.
After the hanging load stops, analyse the box alone: subtract its downslope weight component from the stopping resultant to find friction.
For a smooth ramp followed by rough horizontal stopping, equate mgh to μmgd to obtain μ = h/d.
To find mean stopping friction after trucks couple, divide their combined post-collision kinetic energy by the stopping distance.
When a push is removed, first find the net work during the push, then use the remaining KE to find the further friction-only stopping distance.
For an electron accelerated from rest, equate electric work eV to ½mev²; distinguish the accelerating voltage from a later deflecting voltage.
For a pulled object climbing an incline, subtract its GPE gain from the pulling work before converting the remainder into kinetic energy.
For a spacecraft subject to drag, explain that negative work reduces orbital KE + GPE while energy transfers to thermal forms.
To test whether a vehicle reaches the far end of a bend, compare its initial KE with resistive work over the actual arc length.

[Maximum number: 2]

A girl on a sledge is moving down a snow slope at a uniform speed.

The sledge, without the girl on it, now travels up a snow slope that makes an angle of $6.5^{\circ}$ to the horizontal. At the start of the slope, the speed of the sledge is $4.2 \mathrm{~m} \mathrm{~s}^{-1}$ . The coefficient of dynamic friction of the sledge on the snow is 0.11 .

Calculate the distance along the slope at which the sledge stops moving.

Assume that the coefficient of dynamic friction is constant.

Exam points

For ideal falling or swinging motion, use the vertical height loss in mgh = ΔKE, retaining the precision required by “show that”.
Split a rope-drop problem at the point where stretching starts: the preceding free-fall stage converts GPE to KE without added elastic energy.
For an ideal projectile’s impact speed, combine its full initial KE with GPE lost; the launch angle is unnecessary in the energy method.
Explain an engine-off rocket slowing away from a planet as kinetic energy transferring into gravitational potential energy.
For a cylinder rolling without slipping, include both ½mv² and ½Iω² with ω = v/r; the supplied I = ½mr² gives h = 3v²/(4g).

[Maximum number: 2]

An elastic climbing rope is tested by fixing one end of the rope to the top of a crane. The other end of the rope is connected to a block which is initially at position A. The block is released from rest. The mass of the rope is negligible.

diagram not to scale

The unextended length of the rope is 60.0 m . From position A to position B, the block falls freely.

At position B the rope starts to extend. Calculate the speed of the block at position B.

use of conservation of energy

OR

$\begin{aligned} v^2=u^2+2as \\ v=\sqrt{2\times60.0\times9.81}=34.3\ \mathrm{m\,s^{-1}} \end{aligned}$

Exam points

Separate rope-drop stages: before stretching GPE becomes KE; during deceleration, both GPE and KE transfer into elastic potential energy.
For a spring launch to a higher point, include both final gravitational and kinetic energy; show the substitution before rounding.
Use the stated fraction of GPE lost that becomes KE, taking the vertical height difference rather than the starting height.
For a massless horizontal spring on a frictionless surface, equate ½kx² to ½mv² to obtain v = x√(k/m).
When a spring drives two masses apart from rest, use equal opposite momenta and include both kinetic energies in the energy balance.
Describe a swinging club’s GPE becoming its KE, then identify transfer to ball KE or internal energy during impact.
In a vertical spring’s upward motion, include gravitational energy: decreasing elastic PE and KE can both supply its GPE gain.
For lossless descent, use the height lost to calculate speed; at the top of a loop subtract its diameter from the initial height.
Distinguish the descent before a sticking collision from the collision itself; find the pre-impact speed from energy or momentum.

[Maximum number: 2]

diagram not to scale

The unextended length of the rope is 60.0 m . From position A to position B, the block falls freely.

(a)

For the rope and block, describe the energy changes that take place

[ 2 ]

(i)

between A and B .

[ 1 ]

(ii)

between B and C .

[ 1 ]

Exam points

For constant acceleration from rest, combine v² = 2as with Ek = ½mv² to select a straight-line Ek–distance graph through the origin.
Convert eV to joules and mg or g to kilograms before using v = √(2Ek/m); convert the final speed to the units in the options.
Read the maximum KE from the energy–time graph, convert mJ to J, then show enough digits in a show-that speed calculation.
For a bounce at the same height, find KE lost from ½m(u² − v²); the fraction retained is (v/u)² using speeds just around impact.
For a higher circular orbit, use the KE decrease ΔEk = −½ΔU; retain the negative sign even though gravitational potential energy rises.
To recover KE just after a projectile bounce from its apex, add the apex’s horizontal KE to the GPE gained during its rise.
Use eVs = ½mevmax² for stopping-potential speed, showing complete substitution or the extra significant figures required by “show that”.
At a retarding collecting plate, subtract eV from the emitted electron’s KE before converting eV to joules and calculating arrival speed.
For complete conversion of a nonrelativistic electron’s KE to one photon, use hf = ½mv²: doubling electron speed quadruples frequency.
For the supplied wind KE-flux formula, state its ideal stopping-at-the-blades assumption; constant quantities or generic friction losses earn no credit.
Use p = √(2mEk) for momentum ratios: multiply the mass and KE ratios inside the square root.

[Maximum number: 2]

A toy rocket is made from a plastic bottle that contains some water.

Air is pumped into the vertical bottle until the pressure inside forces water and air out of the bottle. The bottle then travels vertically upwards.

The air-water mixture is called the propellant.
The variation with time of the vertical velocity of the bottle is shown.

The bottle reaches its highest point at time $T_{1}$ on the graph and returns to the ground at time $T_{2}$ . The bottle then bounces. The motion of the bottle after the bounce is shown as a dashed line.

The bottle bounces when it returns to the ground.

Calculate the fraction of the kinetic energy of the bottle that remains after the bounce.

Attempt to evaluate KE ratio as $\left(\frac{v_{\text {final }}}{v_{\text {initial }}}\right)^{2}$

《 $\left(\frac{4.5}{10}\right)^{2}=$ 》 0.20 OR 20 % OR $\frac{1}{5}$

Marking guidance:

Accept ± 0.5 velocity values from graph

Exam points

Estimate a reservoir’s specific gravitational energy using g times its mean height above the outlet, and give the essential unit J kg⁻¹.
Explain reduced uphill range with added load: the same battery energy must supply more mgh for each vertical rise.
For a rolling wheel, equate the centre-of-mass GPE loss to translational plus rotational KE; use the radius of the shaft that contacts the rails.
In a constant gravitational field, compare lifting work using mass × vertical height; doubling mass and quadrupling height multiplies work by eight.
For an ideal falling club head, use its centre-of-mass height loss in mgh = ½mv² to calculate the maximum speed.

[Maximum number: 2]

A girl rides a bicycle that is powered by an electric motor. A battery transfers energy to the electric motor. The emf of the battery is 16 V and it can deliver a charge of 43 kC when discharging completely from a full charge.

On another journey up the slope, the girl carries an additional mass. Explain whether carrying this mass will change the maximum distance that the bicycle can travel along the slope.

Exam points

For a falling elastic rope, use extension beyond its natural length in ½kx² and equate this to the GPE lost by the lowest turning point.
For a frictionless spring launch, ½kx² = ½mv² gives v ∝ x: halving the launch speed requires halving the compression.
With the same total load on two identical parallel springs, each extends half as far; add their two quarter-energies to obtain half the original energy.
Explain the club head’s deformation by elastic energy stored during compression and then transferred to the ball as it leaves.
For a hanging mass at equilibrium, first use kx = mg, then calculate ½kx² using that extension rather than an oscillation amplitude.

[Maximum number: 2]

diagram not to scale

The unextended length of the rope is 60.0 m . From position A to position B, the block falls freely.

The length reached by the rope at C is 77.4 m . Suggest how energy considerations could be used to determine the elastic constant of the rope.

$\mathrm{EPE}=\frac{1}{2}kx^2$ $\mathrm{GPE}\text{ at A}=\mathrm{EPE}\text{ at C}$

OR

$(\mathrm{GPE}+\mathrm{KE})\text{ at B}=\mathrm{EPE}\text{ at C}$

Exam points

Find average motor input power from total work divided by time and efficiency; for uniform acceleration use average, rather than final, speed.
For stored-energy running time, divide the available energy by the combined output of all generators and give the required time unit.
Multiply fuel mass consumed per second by its given specific energy and efficiency to find output power, including W or J s⁻¹.
Convert volume flow to mass flow using density, then use P = ṁg h̄; for a reservoir use the mean height, not just the outlet drop.
For a required hydroelectric output, rearrange Pout = ηṁgh to find water mass flow; divide by efficiency rather than multiply.
For constant-speed motion, balance the driving force against resistance and use P = Fv; zero resultant force does not mean zero engine power.
Calculate average battery power from QV divided by discharge time; retain the precision requested by a show-that task.
For decay power, find the number of nuclei, calculate activity λN, then multiply by energy released per decay in joules.
Distinguish a conveyor’s motor power ṁv² from the collected gravel’s KE gain rate ½ṁv²; use momentum flow to find the belt force.
Use the area under a power–time graph for energy; subtract the load’s energy over the same interval to obtain net energy exported.
For the supplied tidal-basin model, use average head in ρV̇gh̄; a volume flow in m³ s⁻¹ gives power directly, not total energy.
Find average KE supply or dissipation from ½m(v² − u²) divided by the matching motion interval, excluding an earlier push interval.
For upward acceleration, include both lifting work and KE gain in motor power, or use M(g + a) times the average upward speed.
For uniform acceleration with increasing resistance, use P/v = ma + R(v) to distinguish the engine’s power–time graph from a linear increase.
For a heater warming water, subtract mcΔT from its input energy Pt before dividing by heating time to find average heat-loss power.
For air accelerated by hovering blades, calculate power from ½ṁv² using the air speed; the aircraft’s zero speed does not make this power zero.
For energy transferred at constant power, use E = Pt and convert minutes, days or years into seconds before combining with watts.
For the stated wind-flux model, use the rotor’s swept area πr²; interpret ½ρAv³ as incident KE flux, not guaranteed extractable output.
In an ideal wind-flux comparison, use P ∝ l²v³: halving both required power and wind speed doubles the blade length.
For the ideal turbine-count estimate, divide the required power by the stated model’s per-turbine bound and round the count upwards.
Separate total wind-farm output, per-turbine power and incident power using the given efficiency before solving for rotor diameter.
For a wind-farm land estimate, apply the stated efficiency to each rotor’s wind flux, then multiply turbine count by the allotted land per turbine.
For a kite pulling a moving ship, use the horizontal force component in Fv; compare kite power with kite plus engine power.
With the given drag law F = kv² at steady speed, use P = kv³ and a cube-root power ratio to find the new speed.
For constant force from rest, distinguish P ∝ √s against distance from P ∝ t against time; power is the KE change rate.
For average air-resistance dissipation during a fall, subtract final KE from GPE lost and divide by the full fall time read from the graph.

[Maximum number: 2]

The graph shows the variation with time of the height of the International Space Station (ISS) during a 30-day period.

The following data are available:

\begin{aligned} \text { Mass of Earth } & =5.98 \times 10^{24} \mathrm{~kg} \\ \text { Mass of the ISS } & =4.20 \times 10^{5} \mathrm{~kg} \\ \text { Radius of the Earth } & =6.38 \times 10^{6} \mathrm{~m} \end{aligned}

For the 30-day period indicated in the graph the loss of total energy of the ISS is $4.5 \times 10^{9} \mathrm{~J}$ .

Calculate the average rate at which the energy of the ISS is being dissipated.

$\begin{aligned} P=\frac{\Delta E}{\Delta t} \quad P=\text { « } \frac{-4.5 \times 10^{9}}{30 \times 24 \times 3600} \text { » }-1.7 \times 10^{3} W \end{aligned}$

Ignore sign

Exam points

When useful output energy is given, divide it by efficiency to obtain input, then subtract output to find energy dissipated.
For hydroelectric losses, name the location of friction or another specific transfer; a bare “friction” does not earn the explanation mark.
Divide useful electrical output by the gravitational-energy transfer rate to calculate a hydroelectric system’s overall efficiency.
For hill climbing at constant speed, compare useful mgh with supplied Pt; constant kinetic energy does not make efficiency zero.
For photovoltaic efficiency, calculate incident power as intensity × panel area, then divide useful electrical output by this input.
Under the same radiation, equal panel output requires ηA to remain constant; reducing efficiency from 20% to 15% increases area to 4A/3.
For an array of identical panels, multiply intensity × area × efficiency by panel count, converting kW m⁻² to W m⁻² when needed.
For a target solar output, use A = Pout/(ηI) with the intensity specified for the relevant average or peak conditions.
Apply the stated wind-turbine efficiency to incident KE flux before equating the useful power to ṁgh for pumped water.
At equal wind speed, wind-generator output scales with efficiency × radius²; a tripled radius and doubled efficiency give 18 times the output.
When useful heating alone is calculated, identify it as an underestimate of input power and explain thermal transfer to surroundings or components.
For annual efficiency, compare energy totals in the same units, or convert the annual output to average power before dividing by input power.
For a lifting motor, use useful output mgv or mgh/t; multiply input by efficiency for output, and divide output by efficiency for input.
Explain lower actual wind output using residual KE in escaping air, thermal losses in turbine or generator, or wind-related downtime.
In a Sankey diagram, compare the final useful delivered width with the original input width, after internal use and transmission losses.
Explain wasted power with a specific mechanism, such as thermal transfer to surroundings, friction in the generator or electricity used internally.
From given electrical output VI and efficiency, divide by efficiency for input power, then divide by supply voltage to find input current.
Apply generator efficiency before dividing demand by each generator’s output, and round the number of generators up to a whole unit.
For fuel needed to accelerate a vehicle, divide its KE gain by efficiency and by the fuel’s given specific energy, then convert the mass units.
Convert the energy and time units before calculating useful motor power as ηE/t; half a minute is 30 seconds.
Explain why high efficiency matters even when energy is conserved: more input becomes useful work and less is wasted or ejected.

[Maximum number: 5]

This question is about energy sources.

A small island is situated in the Arctic. The islanders require an electricity supply but have no fossil fuels on the island. It is suggested that wind generators should be used in combination with power stations using either oil or nuclear fuel.

(a)

Conventional horizontal-axis wind generators have blades of length 4.7 m .

The average wind speed on the island is $7.0 \mathrm{~ms}^{-1}$ and the average air density is $1.29 \mathrm{~kg} \mathrm{~m}^{-3}$ .

[ 2 ]

(i)

Explain why less energy can actually be generated by the wind generators than the value you deduced in (b)(i).

[ 2 ]

(b)

The energy flow diagram (Sankey diagram) below is for an oil-fired power station that the islanders might use.

[ 3 ]

(i)

Determine the efficiency of the power station.

[ 2 ]

indication that energy supplied to islanders is output and chemical energy input / $\frac{8}{25}$ used;
32 \% / 0.32;

(ii)

Explain why energy is wasted in the power station.

[ 1 ]

Exam points

Before comparing fuel energies per kilogram, convert any energy quoted per gram to the same mass basis and then take the ratio.
Distinguish specific energy from energy per volume: a litre with mass below 1 kg releasing more energy than 1 kg of coal proves greater energy per kg.
Divide energy required per cylinder cycle by the fuel’s energy per litre to find the volume injected in that cycle.
For fuel range, multiply per-cycle volume by cycles per second and cylinder count; use the resulting running time with the given speed.
For fuel needed to heat water, first calculate mcΔT, then divide by the supplied energy per kilogram using consistent units.
For annual fuel consumption, calculate useful Pt, divide by plant efficiency, then divide by fuel energy per kilogram.
For minimum coal saved by replacement electrical output, convert output power to energy over the stated interval before dividing by coal energy per kg.
For a fuel-volume energy estimate, divide the required energy by the supplied energy per m³, converting MJ m⁻³ to J m⁻³ first.

[Maximum number: 2]

At the end of the 30-day period, rockets are fired to bring the ISS back to its initial height. The energy density of liquid hydrogen rocket fuel is $8.5 \times 10^{3} \mathrm{MJ} \mathrm{m}^{-3}$ .

Estimate the volume of fuel needed.

$\begin{aligned} V=\frac{E}{E \text { density }} V=\mu \frac{4.5 \times 10^{9}}{8.5 \times 10^{9}}=>0.53 \mathrm{~m}^{3} \end{aligned}$

Award [2] if $0.53 \ll \mathrm{~m}^{3} \gg$ is seen as the
answer without working

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