IB Physics HL A 3 Work Energy and Power Questions

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

Exam points

Find work from force–displacement data, using W = Fs cos θ for constant force or average force when it varies.
Use net work to relate energy transferred or dissipated to a change in kinetic energy.
Track transfers between energy stores while explaining how the total energy remains conserved.
Apply mechanical-energy conservation to find speed or height when losses are absent, including rotational motion.
Calculate kinetic energy and relate it to momentum using Ek = ½mv² = p²/(2m).
Calculate gravitational or elastic potential-energy changes using mgΔh and ½kx².
Calculate power from energy per time or force and speed, including P = Fv.
Interpret Sankey diagrams and energy/power ratios to calculate efficiency or identify losses.
Use fuel energy density per kilogram or cubic metre to estimate fuel required or energy available.

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

[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

[Maximum number: 11]

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

(a)

Outline the meaning of work.

[ 2 ]

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;

(b)

Some cargo ships use kites working together with the ship's engines to move the vessel.

The tension in the cable that connects the kite to the ship is 250 kN . The kite is pulling the ship at an angle of $39^{\circ}$ to the horizontal. The ship travels at a steady speed of $8.5 \mathrm{~m} \mathrm{~s}^{-1}$ when the ship's engines operate with a power output of 2.7 MW .

[ 6 ]

(i)

Calculate the work done on the ship by the kite when the ship travels a distance of 1.0 km .

[ 2 ]

horizontal force $=250000 \times \cos 39^{\circ}\left(=1.94 \times 10^{5} \mathrm{~N}\right)$ ;
work done $=1.9 \times 10^{8} \mathrm{~J}$ ;

(ii)

Show that, when the ship is travelling at a speed of $8.5 \mathrm{~m} \mathrm{~s}^{-1}$ , the kite provides about 40 % of the total power required by the ship.

[ 4 ]

power provided by kite $=\left(1.94 \times 10^{5} \times 8.5=\right) 1.7 \times 10^{6} \mathrm{~W}$ ;
total power $=(2.7+1.7) \times 10^{6}(\mathrm{~W})\left(=4.4 \times 10^{6} \mathrm{~W}\right)$ ;
fraction provided by kite $=\frac{1.7}{2.7+1.7}$ ;
38 % or 0.38 ;
(must see answer to 2+ sig figs as answer is given)

Marking guidance:

Allow answers in the range of 37 to 39 % due to early rounding.
or
Award [3 max] for a reverse argument such as:
if 2.7 MW is 60 %;
then kite power is $\frac{2}{3} \times 2.7 \mathrm{MW}=1.8 \mathrm{MW}$ ;
shows that kite power is actually 1.7 MW ;
(QED)

(c)

The kite is taken down and no longer produces a force on the ship. The resistive force F that opposes the motion of the ship is related to the speed v of the ship by

F=k v^{2}

where k is a constant.
Show that, if the power output of the engines remains at 2.7 MW , the speed of the ship will decrease to about $7 \mathrm{~ms}^{-1}$ . Assume that k is independent of whether the kite is in use or not.

[ 3 ]

$\quad P=\left(k v^{2}\right) \times v=k v^{3}$ ;
$\frac{v_{1}}{v_{2}}=\left(\sqrt[3]{\left(\frac{P_{1}}{P_{2}}\right)}=\right) \sqrt[3]{\left(\frac{2.7}{4.4}\right)} ;$
final speed of ship $=7.2 \mathrm{~ms}^{-1}$ ; (at least 2 sig figs required).

Approximate answer given, marks are for working only.

[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.

[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: 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}) ;

[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.

[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

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

[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 ]

[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

[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.

[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

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

[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

[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 ]

[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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Covered syllabus points

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