Space, Time, and Motion — Free IB Physics HL Practice Questions

1FoundationMCQLorentz transformation equations1 markPaper 1~2 min

An observer on Earth measures the length of a rocket moving at

0.80c

to be

60 \text{ m}

. What is the proper length of the rocket?

Solutions

2MasteryMCQTorque and rotational motion1 markPaper 1~2 min

A uniform rod of length

L

and mass

M

is pivoted at one end and held stationary at

30°

above the horizontal by a horizontal force applied at the free end. What is the magnitude of the torque due to the weight of the rod about the pivot?

Solutions

3FoundationMCQLorentz transformation equations1 markPaper 1~2 min

A spacecraft moves at

0.60c

relative to Earth. An observer on Earth measures the length of the spacecraft to be

24\text{ m}

. What is the proper length of the spacecraft?

Solutions

4FoundationMCQLorentz transformation equations1 markPaper 1~2 min

A muon is created in the upper atmosphere and travels vertically toward the ground at

v = 0.998c

relative to a ground observer. The proper lifetime of the muon is

2.20\ \mu\text{s}

. What is the muon's lifetime as measured by the ground observer?

Solutions

5FoundationMCQConservation of momentum1 markPaper 1~2 min

An astronaut of mass

70\text{ kg}

is initially at rest relative to their spacecraft in deep space. The astronaut throws a toolkit of mass

2.0\text{ kg}

directly away from the spacecraft a speed of

6.0\text{ m s}^{-1}

relative to the spacecraft. What is the speed of the astronaut relative to the spacecraft immediately after throwing the toolkit?

Solutions

6MasteryMCQConservation of momentum1 markPaper 1~2 min

A firefighter of mass

75 \text{ kg}

jumps from a stationary platform and lands on a stationary safety mattress of mass

60 \text{ kg}

. The firefighter arrives at the mattress with a downward speed of

4.0 \text{ m s}^{-1}

and immediately moves together with it. What is the speed of the firefighter and mattress immediately after landing?

Solutions

7MasteryMCQNewton’s laws of motion1 markPaper 1~2 min

A cyclist and bicycle of combined mass

80 \text{ kg}

travel at

8.0 \text{ m s}^{-1}

on a level road. The cyclist stops pedalling and coasts to rest in

20 \text{ s}

. Friction is the only horizontal force acting during coasting. What is the magnitude of the friction force on the cyclist and bicycle during this phase?

Solutions

8FoundationMCQMomentum and impulse1 markPaper 1~2 min

A tennis ball of mass

0.058 \text{ kg}

travels toward a player at

30 \text{ m s}^{-1}

. The player strikes the ball, which then moves in the opposite direction at

25 \text{ m s}^{-1}

. The contact time between racquet and ball is

0.060 \text{ s}

. What is the magnitude of the average force exerted by the racquet on the ball?

Solutions

9FoundationMCQMomentum and impulse1 markPaper 1~2 min

Two ice skaters are initially at rest on a frictionless ice rink. They push off from each other. Skater A has a mass of

50 \text{ kg}

and moves to the right at

1.2 \text{ m s}^{-1}

immediately after the push. Skater B has a mass of

40 \text{ kg}

. What is the velocity of skater B immediately after the push?

Solutions

10MasteryMCQNewton’s laws of motion1 markPaper 1~2 min

A box of mass

5.0 \text{ kg}

is pushed across a rough horizontal floor by a constant horizontal force of

30 \text{ N}

. The box accelerates at

2.0 \text{ m s}^{-2}

in the direction of the applied force. What is the magnitude of the frictional force acting on the box?

Solutions

11FoundationMCQKinetic and potential energy1 markPaper 1~2 min

A block of mass

0.40 \text{ kg}

starts from rest and slides down a frictionless incline of vertical height

3.0 \text{ m}

. What is the speed of the block at the bottom of the incline?

Solutions

12FoundationMCQKinetic and potential energy1 markPaper 1~2 min

A spring with spring constant

500 \text{ N m}^{-1}

is compressed by

0.20 \text{ m}

from its natural length. A ball of mass

0.10 \text{ kg}

is placed against the compressed spring on a frictionless horizontal surface and then released. What is the maximum speed of the ball?

Solutions

13FoundationMCQKinetic and potential energy1 markPaper 1~2 min

A ball of mass

0.30\ \text{kg}

is swung in a vertical circle on a string of length

0.80\ \text{m}

. At the top of the circle the ball moves at

4.0\ \text{m s}^{-1}

. What is the kinetic energy of the ball at the bottom of the circle?

Solutions

14FoundationMCQKinetic and potential energy1 markPaper 1~2 min

An arrow of mass

0.050 \text{ kg}

is fired vertically upward. The bowstring exerts an average force of

150 \text{ N}

over a draw length of

0.60 \text{ m}

. What is the maximum height reached by the arrow above its launch point? (Ignore air resistance.)

Solutions

15FoundationMCQScalars and vectors1 markPaper 1~2 min

A student walks

3.0 \text{ km}

east from home to a library, then turns and walks

4.0 \text{ km}

north to a park. What is the magnitude of the student's displacement from home to the park?

Solutions

16MasteryMCQGraphical analysis of motion1 markPaper 1~2 min

A ball is thrown vertically upward from a balcony. Its velocity–time graph is a straight line from

(0 \text{ s},\ 15 \text{ m s}^{-1})

to

(1.5 \text{ s},\ 0 \text{ m s}^{-1})

, where positive values represent the upward direction. What is the displacement of the ball during this 1.5 s interval?

Solutions

17MasteryMCQGraphical analysis of motion1 markPaper 1~2 min

A train moves along a straight track. Its displacement–time graph consists of three straight-line segments: slope

+20 \text{ m s}^{-1}

from

t = 0

to

t = 10 \text{ s}

, slope

0

from

t = 10 \text{ s}

to

t = 20 \text{ s}

, and slope

-15 \text{ m s}^{-1}

from

t = 20 \text{ s}

to

t = 30 \text{ s}

. What is the displacement of the train at

t = 30 \text{ s}

relative to its position at

t = 0

?

Solutions

18FoundationMCQScalars and vectors1 markPaper 1~2 min

A hiker walks

5.0 \text{ km}

due south, then turns and walks

12 \text{ km}

due east. What is the magnitude of the hiker's resultant displacement?

Solutions

19FoundationMCQGraphical analysis of motion1 markPaper 1~2 min

A toy train moves along a straight track. Its displacement–time graph is a straight line passing through

(0\ \text{m},\ 0\ \text{s})

and

(8\ \text{m},\ 4\ \text{s})

. What is the velocity of the train?

Solutions

20ChallengeSAQ-LMoment of inertia and angular acceleration9 marksPaper 2~14 min

A flywheel used in a stamping press is a solid steel disc of radius 0.40 m and mass 120 kg. It rotates at 600 revolutions per minute (rpm). The press engages a clutch that applies a constant frictional torque of 240 N m, bringing the flywheel to rest in 12.5 s. Moment of inertia of a solid disc about its central axis:

I = \frac{1}{2}MR^2

Moment of inertia of a thin ring (all mass at rim) about its central axis:

I = MR^2

(a)

Calculate the moment of inertia of the flywheel modelled as a solid disc. [1 mark]

(b)

Show that the magnitude of the angular deceleration of the flywheel is approximately

5.0 \text{ rad s}^{-2}

. [3 marks]

(c)

Calculate the number of complete revolutions the flywheel makes during the braking period. [2 marks]

(d)

An engineer proposes redesigning the flywheel so that the same total mass of 120 kg is concentrated entirely at the rim at radius 0.40 m, keeping the same operating speed and applied braking torque. Calculate the ratio of the rotational kinetic energy stored in the rim design to that stored in the solid disc design, and hence evaluate whether this redesign improves the energy available for stamping. [3 marks]

Solutions

21MasterySAQ-SMoment of inertia and angular acceleration8 marksPaper 2~12 min

A maintenance engineer is testing a large industrial grinding wheel. The wheel is a uniform disc of mass 24 kg and radius 0.40 m, mounted on a frictionless axle. The wheel is initially at rest. A constant tangential force of 45 N is applied to the rim of the wheel.

(a)

State what the moment of inertia of an object represents in rotational dynamics. [1 mark]

(b)

Show that the moment of inertia of the grinding wheel is approximately

1.9 \text{ kg m}^2

. [2 marks]

(c)

Calculate the angular acceleration of the grinding wheel. [2 marks]

(d)

The engineer doubles the applied force and simultaneously replaces the wheel with one of identical mass and radius but shaped as a thin ring (hoop) rather than a solid disc. Determine the ratio

\dfrac{\alpha_{\text{new}}}{\alpha_{\text{original}}}

and explain whether the angular acceleration increases, decreases, or stays the same. [3 marks]

Solutions

22ChallengeSAQ-LMoment of inertia and angular acceleration7 marksPaper 2~11 min

A wind turbine has three blades, each of length 35 m and mass 1800 kg. Each blade is modelled as a uniform thin rod rotating about one end. The moment of inertia of a thin rod about one end is:

I = \frac{1}{3}ML^2

The turbine starts from rest and reaches an operating angular speed of

2.0 \text{ rad s}^{-1}

in 45 s under constant torque.

(a)

Calculate the total moment of inertia of the three blades about the rotation axis. [2 marks]

(b)

Determine the angular acceleration of the turbine during start-up. [1 mark]

(c)

Calculate the constant torque applied to the turbine. [2 marks]

(d)

Real turbine blades taper from a wide base near the hub to a narrow tip. Deduce whether the uniform rod model overestimates or underestimates the moment of inertia of a real blade, and justify your answer. [2 marks]

Solutions

23MasterySAQ-SMoment of inertia and angular acceleration7 marksPaper 2~11 min

Each blade of a wind turbine can be modelled as a uniform rod of mass 750 kg and length 25 m, rotating about one end (the hub). The turbine has three identical blades.

(a)

Explain why a large moment of inertia of the blades makes startup of the turbine slower when the wind applies a constant torque. [2 marks]

(b)

Calculate the moment of inertia of one blade about the hub. [2 marks]

(c)

The wind exerts a net torque of

1.20 \times 10^4 \ \text{N m}

on the turbine. Determine the angular acceleration of the three-blade rotor at startup. [3 marks]

Solutions

24MasterySAQ-SForce, mass, and acceleration7 marksPaper 2~11 min

Data

F = ma

,

\cos 40° = 0.766

A box of mass 5.0 kg is pulled along a rough horizontal surface by a force of 25 N applied at 40° above the horizontal. The surface exerts a constant friction force of 8.0 N on the box.

(a)

Explain why the vertical component of the applied force does not contribute to the horizontal acceleration of the box. [2 marks]

(b)

Calculate the horizontal acceleration of the box. [3 marks]

(c)

The angle of the applied force is increased above 40° while the magnitude of the applied force remains 25 N. The friction force remains 8.0 N. Deduce whether the horizontal acceleration of the box increases or decreases. [2 marks]

Solutions

25MasterySAQ-SForce, mass, and acceleration7 marksPaper 2~11 min

A velocity-time graph for a 3.0 kg object moving in a straight line consists of a straight line from

(0, 0)

to

(4, 12)

and then a horizontal line from

(4, 12)

to

(8, 12)

, where velocity is in

\text{m s}^{-1}

and time is in s.

(a)

Determine the net force acting on the object during the first 4 seconds. [2 marks]

(b)

Calculate the impulse delivered to the object between

t = 0

and

t = 8

s. [2 marks]

(c)

The object then decelerates uniformly to rest over a further 6 seconds. Deduce whether the magnitude of the net force during this deceleration phase is greater than, equal to, or less than the net force found in (a). [3 marks]

Solutions

26ChallengeSAQ-LNewton’s laws of motion7 marksPaper 2~11 min

A student investigates the motion of a small rocket of total mass 0.80 kg on a horizontal frictionless track. The rocket ejects water backwards at a constant rate of 0.20 kg s

^{-1}

with a speed of 5.0 m s

^{-1}

relative to the rocket. The rocket starts from rest.

(a)

State the principle that relates thrust force to the rate of change of momentum of the ejected fluid. [1 mark]

(b)

Determine the thrust force exerted on the rocket by the ejected water. [2 marks]

(c)

Determine the acceleration of the rocket at

t = 1.0\ \text{s}

after launch. [2 marks]

(d)

The thrust force remains constant throughout the motion. Evaluate how the acceleration of the rocket changes over time, justifying your answer with reference to Newton's second law. [2 marks]

Solutions

27MasterySAQ-SForce, mass, and acceleration7 marksPaper 2~11 min

A 2.0 kg object is placed on a frictionless inclined plane that makes an angle of 30° with the horizontal. The object is released from rest.

(a)

(i) State the two components into which the weight of the object can be resolved on the incline. [1]
(ii) Explain why only one of these components causes the object to accelerate along the incline. [2 marks]

(b)

Calculate the acceleration of the object down the incline. [2 marks]

(c)

A student claims that doubling the mass of the object will double the net force and therefore double the acceleration down the incline. Evaluate this claim. [2 marks]

Solutions

28FoundationSAQ-SPower and efficiency5 marksPaper 2~8 min

Data

g = 9.81 \text{ m s}^{-2}

A pump is used to lift water from a well 25 m deep at a rate of 200 kg per minute. The water is discharged at ground level with a speed of

4.0 \text{ m s}^{-1}

. The pump has an overall efficiency of 70%.

(a)

State the useful energy conversions that take place as the pump operates. [1 mark]

(b)

Identify one source of energy loss in the pump system and state the form in which this energy is dissipated. [1 mark]

(c)

Calculate the electrical power input required to operate the pump. [3 marks]

Solutions

29MasterySAQ-SConservation of mechanical energy5 marksPaper 2~8 min

A ball of mass 0.150 kg is thrown vertically upward from ground level with an initial speed of

12.0 \text{ m s}^{-1}

. Air resistance is negligible. Take

g = 9.81 \text{ m s}^{-2}

.

(a)

Calculate the total mechanical energy of the ball at the moment of release. [1 mark]

(b)

Describe how the kinetic energy and the gravitational potential energy of the ball change as it travels from the point of release to its maximum height. [2 marks]

(c)

Determine the maximum height reached by the ball. [1 mark]

(d)

The experiment is repeated with a ball of identical mass but the initial speed is increased to

17.0 \text{ m s}^{-1}

. Show that the ratio of the new maximum height to the original maximum height depends only on the ratio of the initial speeds, and calculate this ratio. [1 mark]

Solutions

30ChallengeSAQ-LConservation of mechanical energy8 marksPaper 2~12 min

Data

E_k = \frac{1}{2}mv^2

,

\quad E_p = mgh

,

\quad F_c = \frac{mv^2}{r}

,

\quad g = 9.81 \text{ m s}^{-2}

,

\quad \cos 60° = 0.5

A pendulum consists of a bob of mass 0.200 kg attached to a light string of length 1.50 m. The bob is pulled aside until the string makes an angle of 60° with the vertical, then released from rest.

(a)

Determine the speed of the bob at the lowest point of its swing. [3 marks]

(b)

Determine the tension in the string at the lowest point. [2 marks]

(c)

The string is replaced by a massless rigid rod of the same length. The bob is released from the same position (60° from vertical) and given a small additional push at the lowest point so that it completes a full vertical circle. (i) State the minimum speed the bob must have at the top of the circle for the rod to remain taut. [1]
(ii) Explain why a string could not keep the bob on a circular path at the top of the circle under the same conditions, but a rigid rod can. [2 marks]

Solutions

31FoundationSAQ-SPower and efficiency5 marksPaper 2~8 min

A solar panel of area

2.5 \text{ m}^2

receives solar radiation of intensity

800 \text{ W m}^{-2}

. The panel converts 18% of the incident solar energy into electrical energy. The electrical energy is used to charge a battery with a charging efficiency of 90%.

(a)

State what is meant by the term efficiency as applied to an energy conversion device. [1 mark]

(b)

Calculate the electrical power output of the solar panel. [2 marks]

(c)

Determine the rate at which energy is stored in the battery. [2 marks]

Solutions

32FoundationSAQ-SPower and efficiency5 marksPaper 2~8 min

A 1200 kg car accelerates from rest to

20 \ \text{m s}^{-1}

along a straight horizontal road. The engine provides a constant power output of 50 kW. Air resistance and friction together provide a constant resistive force of 400 N.

(a)

Explain how the driving force from the engine changes as the car accelerates from rest. [2 marks]

(b)

Calculate the distance travelled by the car as it accelerates from rest to

20 \ \text{m s}^{-1}

. [3 marks]

Solutions

33MasterySAQ-SConservation of mechanical energy8 marksPaper 2~12 min

A roller coaster car of mass 500 kg starts from rest at point P, which is 25.0 m above the ground. The car travels along a track to point Q, which is 10.0 m above the ground. Take

g = 9.81 \text{ m s}^{-2}

.

(a)

State the principle of conservation of mechanical energy. [1 mark]

(b)

The track between P and Q is frictionless. Calculate the speed of the car at Q. [2 marks]

(c)

The track between P and Q is now roughened. The speed of the car at Q is measured to be

14.0 \text{ m s}^{-1}

. Determine the energy transferred to thermal energy by friction between P and Q. [2 marks]

(d)

A second, identical roller coaster operates on a track where the energy transferred to thermal energy between P and Q is always 30% of the initial gravitational potential energy relative to Q. Evaluate whether the car reaches Q with sufficient speed to continue over a hill of height 12.0 m above the ground immediately after Q. [3 marks]

Solutions

34ChallengeSAQ-LDisplacement, velocity, and acceleration7 marksPaper 2~11 min

A car travels along a straight road. Starting from rest, it accelerates uniformly at

3.0 \text{ m s}^{-2}

for

8.0 \text{ s}

. It then travels at constant velocity for

12 \text{ s}

. Finally, it decelerates uniformly to rest in

5.0 \text{ s}

.

(a)

Determine the maximum velocity reached by the car. [1 mark]

(b)

Determine the total displacement of the car during the entire journey. [3 marks]

(c)

Explain why the average velocity for the entire journey is less than the maximum velocity. [1 mark]

(d)

Determine the average velocity for the entire journey. [2 marks]

Solutions

35MasterySAQ-SGraphical analysis of motion5 marksPaper 2~8 min

A car moves along a straight road. The displacement-time graph of the car is shown below.

(a)

State the velocity of the car during the time interval

t = 4\ \text{s}

to

t = 8\ \text{s}

. [1 mark]

(b)

Calculate the average speed of the car over the entire 12 seconds of motion. [2 marks]

(c)

The average velocity of the car over the 12 seconds is

0\ \text{m s}^{-1}

. Explain, with reference to the graph, why the average speed and the magnitude of the average velocity are not equal in this case, and state the condition under which they would be equal. [2 marks]

Solutions

36ChallengeSAQ-LGraphical analysis of motion7 marksPaper 2~11 min

A cyclist travels along a straight, level road. The velocity-time graph of her journey consists of three distinct phases. - Phase A (

t = 0

to

t = 10

s): velocity increases uniformly from

0

to

8.0\ \text{m s}^{-1}

- Phase B (

t = 10

s to

t = 25

s): velocity is constant at

8.0\ \text{m s}^{-1}

- Phase C (

t = 25

s to

t = 35

s): velocity decreases uniformly from

8.0\ \text{m s}^{-1}

to

0

(a)

State the acceleration of the cyclist during Phase B. [1 mark]

(b)

Determine the total distance travelled over the entire 35-second journey. [3 marks]

(c)

State the displacement of the cyclist at

t = 10\ \text{s}

,

t = 25\ \text{s}

, and

t = 35\ \text{s}

. [1 mark]

(d)

Describe the shape of the displacement-time graph during each of the three phases, justifying your answer with reference to the velocity in each phase. [2 marks]

Solutions

37MasterySAQ-SGraphical analysis of motion7 marksPaper 2~11 min

A skydiver jumps from a stationary balloon and falls vertically downward. The velocity-time graph of the skydiver is shown below.

(a)

(i) State the type of motion of the skydiver during the first 5 seconds. [1]
(ii) Calculate the acceleration of the skydiver during the first 5 seconds. [1 mark]

(b)

Calculate the distance fallen by the skydiver during the first 5 seconds. [2 marks]

(c)

The magnitude of the skydiver's acceleration during the first 5 seconds is less than

g = 9.81 \text{ m s}^{-2}

. Explain why this is the case. [1 mark]

(d)

The skydiver has a mass of 75 kg. Determine the magnitude of the air resistance force acting on the skydiver during the first 5 seconds. [2 marks]

Solutions

38MasterySAQ-SGraphical analysis of motion5 marksPaper 2~8 min

Data

g = 9.81 \text{ m s}^{-2}

A ball is thrown vertically upward from the ground. The displacement–time graph of the ball is shown below.

(a)

Describe the velocity of the ball during the interval

t = 0

to

t = 2.0

s, including both its direction and how its magnitude changes. [2 marks]

(b)

Calculate the initial speed of the ball. [3 marks]

Solutions

39ChallengeLAQTime dilation and length contraction10 marksPaper 3~15 min

A high-energy physics experiment at CERN involves a beam of muons travelling at

0.994c

relative to the laboratory. In the laboratory frame, the muon beam travels a distance of

2.40\,\text{km}

from the production point to the detector. The mean proper lifetime of a muon is

2.20 \times 10^{-6}\,\text{s}

.

(a)

Calculate the Lorentz factor

\gamma

for the muons. [1 mark]

(b)

Calculate the mean lifetime of the muons as measured in the laboratory frame. [2 marks]

(c)

Determine the maximum distance a muon can travel in the laboratory frame before decaying, and state whether the muons reach the detector. [3 marks]

(d)

Calculate the contracted length of the

2.40\,\text{km}

path as measured in the muon rest frame, and determine the time for the detector to reach the muon in that frame. [3] (e) Explain why the results of (c) and (d) are consistent with the principle of relativity. [1 mark]

Solutions

40ChallengeLAQTime dilation and length contraction10 marksPaper 3~15 min

A particle accelerator produces pions travelling at

0.999c

relative to the laboratory. The proper half-life of a pion is

1.77 \times 10^{-8}\,\text{s}

. The pions travel a distance of

750\,\text{m}

from the production point to a detector.

(a)

Calculate the Lorentz factor

\gamma

for the pions. [1 mark]

(b)

Using time dilation, calculate the fraction of pions that survive the journey to the detector as measured in the laboratory frame. [4 marks]

(c)

Show that length contraction in the pion rest frame gives the same fraction of surviving pions. [3 marks]

(d)

The travel time measured in the laboratory frame and the proper time experienced by the pions are different. Discuss why the fraction of surviving pions must nevertheless be the same in both frames, identifying which quantities are frame-dependent and which are invariant. Data for use in this question:

\gamma = \frac{1}{\sqrt{1 - \dfrac{v^2}{c^2}}}, \quad \Delta t = \gamma\,\Delta t_0, \quad L = \frac{L_0}{\gamma}

N = N_0\,e^{-\lambda t}, \quad \lambda = \frac{\ln 2}{t_{1/2}}, \quad c = 3.00 \times 10^8\,\text{m\,s}^{-1}

[2 marks]

Solutions

Space, Time, and Motion — Free Physics HL Practice Questions