Theme A: Space, time and motion

Choose a positive direction and keep signs on displacement, velocity, acceleration and momentum. On a velocity-time graph the gradient is acceleration and the area is displacement.

A lesson for this theme

Multiple choice, Paper 1A style

Question 1

A runner completes one lap of a 400 m track in 80 s and finishes where she started. What are her average speed and the magnitude of her average velocity?

Question 2

The graph shows how the velocity v of a cyclist moving in a straight line varies with time t. What is the cyclist's average velocity over the 12 s?

Question 3

The graph shows the displacement s of a toy car from its starting point against time t. What is the car's velocity between t = 6 s and t = 10 s?

Question 4

A car travelling at 25 m s⁻¹ brakes with uniform deceleration and stops in a distance of 40 m. What is the magnitude of its deceleration?

Question 5

A stone is dropped from rest down a deep well. Air resistance is negligible. How far does it fall in the first 3.0 s?

Question 6

A ball is thrown vertically upwards at 15 m s⁻¹. Air resistance is negligible. What is the maximum height it reaches above the point of release?

Question 7

A ball is thrown vertically upwards and caught again at the same height. Air resistance is negligible. Which statement describes the ball's acceleration?

Question 8

A ball is kicked horizontally at 12 m s⁻¹ from the top of a vertical cliff 20 m high, as shown. Air resistance is negligible. How far from the foot of the cliff does the ball land?

Question 9

A ball is launched from level ground at 20 m s⁻¹ at 30° above the horizontal. Air resistance is negligible. What is the maximum height of the ball?

Question 10

A ball is projected at an angle above level ground. How does air resistance change its path compared with the same launch in a vacuum?

Question 11

The graph shows how the vertical speed v of a skydiver varies with time t. She opens her parachute at t = 25 s. During which interval is the resultant force on her upwards?

Question 12

Table of the two measurements: height h = 1.50 m with absolute uncertainty 0.01 m; fall time t = 0.55 s with absolute uncertainty 0.02 s.
Quantity	Value	Absolute uncertainty
h / m	1.50	0.01
t / s	0.55	0.02

A student drops a ball from rest through a measured height h = 1.50 ± 0.01 m and times its fall as t = 0.55 ± 0.02 s. She calculates g = 2h/t². What is the percentage uncertainty in her value of g?

Question 13

A car starts from rest and accelerates uniformly. It travels 2.0 m in the first 1.0 s. How far does it travel in the first 3.0 s?

Question 14

A trolley moves in a straight line with an initial velocity of 4.0 m s⁻¹. The graph shows its acceleration a against time t. What is its velocity at t = 5.0 s?

Question 15

A box of mass m rests without moving on a rough slope at angle θ to the horizontal. The coefficient of static friction is μₛ. What is the size of the frictional force on the box?

Question 16

A 3.0 kg block on a smooth horizontal table is joined by a light string over a frictionless pulley to a hanging 2.0 kg block, as shown. What is the acceleration of the blocks?

Question 17

A book rests on a table. One force on the book is the gravitational pull of the Earth on the book. Which force forms a Newton's third law pair with it?

Question 18

A 60 kg person stands on the floor of a lift that is accelerating upwards at 1.5 m s⁻². What is the normal force of the floor on the person?

Question 19

The graph shows the force F needed to stretch a spring against its extension Δx. What is the spring constant?

Question 20

A small sphere of radius 2.0 mm moves at 0.050 m s⁻¹ through oil of viscosity 0.80 Pa s. What is the viscous drag on the sphere?

Question 21

A block of wood of density 600 kg m⁻³ floats in water of density 1000 kg m⁻³. What fraction of the block's volume is below the surface?

Question 22

Table for the two trolleys before the collision. Trolley P: mass 2.0 kg, velocity 3.0 m s⁻¹. Trolley Q: mass 1.0 kg, velocity 0. After the collision they move together.
Trolley	m / kg	v before / m s⁻¹
P	2.0	3.0
Q	1.0	0

A 2.0 kg trolley moving at 3.0 m s⁻¹ collides with a stationary 1.0 kg trolley, and they move off joined together. How much kinetic energy is transferred to other forms in the collision?

Question 23

A 0.40 kg football at rest is kicked. The graph shows the force F of the boot on the ball against time t. What is the speed of the ball as it leaves the boot?

Question 24

A 1200 kg car travels round a flat circular bend of radius 50 m at a constant 15 m s⁻¹. What is the size of the frictional force from the road that keeps it on the bend?

Question 25

A stone tied to a string is whirled in a horizontal circle at constant speed. Which statement about the stone is correct?

Question 26

A seat on a fairground ride moves in a horizontal circle of radius 4.0 m, completing one revolution every 2.5 s. What is the centripetal acceleration of the seat?

Question 27

An object at rest explodes into two pieces of mass 2.0 kg and 3.0 kg that move apart in opposite directions. What is the ratio (kinetic energy of the 2.0 kg piece) ÷ (kinetic energy of the 3.0 kg piece)?

Question 28

A 3.0 kg block rests on a level bench. A horizontal pulling force is increased slowly from zero. The graph shows the friction force on the block against the pulling force. What is the coefficient of static friction?

Question 29

A sledge is pulled 15 m across level snow by a rope at 35° above the horizontal. The tension in the rope is 80 N. How much work does the rope do on the sledge?

Question 30

An object of mass 4.0 kg has a momentum of 12 kg m s⁻¹. What is its kinetic energy?

Question 31

Two objects X and Y have the same kinetic energy. The mass of X is 4 times the mass of Y. What is (momentum of X) ÷ (momentum of Y)?

Question 32

A roller-coaster car starts from rest at a height of 12.0 m and runs down a frictionless track. What is its speed when it is 4.0 m above the ground?

Question 33

An electric motor takes 500 W from the supply and raises a 20 kg load vertically at a constant 1.5 m s⁻¹. What is the efficiency of the motor?

Question 34

The graph shows the force F needed to stretch a spring against its extension Δx. How much work is done in stretching the spring from an extension of 0.10 m to 0.20 m?

Question 35

A car travels at a constant 25 m s⁻¹ on a level road. The total resistive force on it is 600 N. What useful power does the engine deliver to the wheels?

Question 36

An engine of efficiency 30% burns fuel of energy density 46 MJ kg⁻¹. What mass of fuel must it burn to do 69 MJ of useful work?

Question 37

Which of the following is equivalent to the joule?

Question 38

A ball is dropped and falls through the air. The energy bar charts show its energies at release and just before it hits the ground. Each block is 0.50 J. How much work is done against air resistance during the fall?

Question 39

A 2.0 kg box slides 5.0 m down a frictionless slope inclined at 30° to the horizontal. How much work does gravity do on the box?

Question 40

The graph shows the kinetic energy Eₖ of a braking car against the distance s it travels. What is the size of the braking force, assumed constant?

Question 41 · HL

A force of 40 N is applied to the end of a spanner 0.25 m from the centre of a nut. The force acts at 60° to the line of the spanner. What is the torque about the centre of the nut?

Question 42 · HL

A uniform plank of length 4.0 m and mass 20 kg rests on a pivot 1.5 m from its left end. A 30 kg child sits at the left end. How far to the right of the pivot must a 40 kg child sit for the plank to balance?

Question 43 · HL

A solid cylinder and a thin-walled hollow cylinder have the same mass and the same radius. They are released together from rest at the top of a slope and roll down without slipping. Which statement is correct?

Question 44 · HL

A wheel of moment of inertia 0.50 kg m² starts from rest. A constant resultant torque of 2.0 N m acts on it for 3.0 s. What is its angular velocity at the end of the 3.0 s?

Question 45 · HL

A disc spinning at 30 rad s⁻¹ slows down with uniform angular deceleration and stops after turning through 10 revolutions. What is the size of its angular deceleration?

Question 46 · HL

Table for the skater. Arms out: moment of inertia 4.0 kg m², rotation rate 2.0 revolutions per second. Arms in: moment of inertia 1.6 kg m², rotation rate unknown.
Position	I / kg m²	f / rev s⁻¹
Arms out	4.0	2.0
Arms in	1.6	?

An ice skater spins at 2.0 revolutions per second with a moment of inertia of 4.0 kg m². She pulls her arms in, reducing her moment of inertia to 1.6 kg m². Friction is negligible. What is her new rate of rotation?

Question 47 · HL

A spinning ice skater pulls her arms in, so her moment of inertia decreases and her angular velocity increases. Friction is negligible. What happens to her rotational kinetic energy, and why?

Question 48 · HL

A solid sphere, with moment of inertia I = ⅖MR², rolls without slipping along a level floor. What fraction of its total kinetic energy is rotational?

Question 49 · HL

The graph shows the angular velocity ω of a turntable against time t. How many revolutions does it make in the first 10 s?

Question 50 · HL

Which conditions must hold for an extended rigid body to be in equilibrium?

Question 51 · HL

A uniform disc of mass 2.0 kg and radius 0.30 m rotates at 40 rad s⁻¹ about its central axis. The moment of inertia of a uniform disc is ½MR². What is its angular momentum?

Question 52 · HL

A student applies different torques τ to a wheel and measures its angular acceleration α. The data are plotted. There is a constant frictional torque at the axle. What is the moment of inertia of the wheel?

Question 53 · HL

Which statement is one of the two postulates of special relativity?

Question 54 · HL

A train moves forwards at 30 m s⁻¹ relative to the ground. A passenger throws a ball towards the back of the train at 10 m s⁻¹ relative to the train. Using Galilean relativity, what is the velocity of the ball relative to the ground?

Question 55 · HL

In frame S an event happens at x = 500 m and t = 4.0 s. Frame S′ moves at 50 m s⁻¹ in the +x direction relative to S, and the origins coincided at t = 0. Using a Galilean transformation, what is the position of the event in S′?

Question 56 · HL

A clock moves at 0.80c relative to an observer. The clock records a time interval of 3.0 s. What is the time interval measured by the observer?

Question 57 · HL

A spaceship has a proper length of 120 m. It moves at 0.60c relative to a space station. What is its length measured by an observer on the station?

Question 58 · HL

Muons made high in the atmosphere move at 0.995c relative to the Earth. A muon's mean lifetime, measured at rest, is 2.2 μs. How far does a muon travel, in the Earth's frame, during one mean lifetime measured in that frame?

Question 59 · HL

Two lightning strikes hit the front and back ends of a fast train. An observer on the platform, standing level with the middle of the train, finds the two strikes simultaneous. Which statement describes what an observer at the middle of the train concludes?

Question 60 · HL

A spaceship moves at 0.60c relative to the Earth. It launches a probe forwards at 0.50c relative to the ship. What is the speed of the probe relative to the Earth?

Question 61 · HL

On a spacetime diagram for frame S, ct is plotted vertically and x horizontally. Frame S′ moves at 0.50c relative to S. What angle does the ct′ axis make with the ct axis?

Question 62 · HL

The spacetime diagram shows the worldline of a rocket in frame S between two events, its launch and its arrival. In the rocket's own frame both events happen at the same place. What time interval does the rocket's clock record between them?

Question 63 · HL

Two inertial observers move relative to each other and both measure the same pair of events. Which quantity do they always agree on?

Question 64 · HL

In frame S an event happens at x = 6.0 × 10⁸ m and t = 4.0 s. Frame S′ moves at 0.60c in the +x direction relative to S, and the origins coincided at t = t′ = 0. What is the time of the event in S′?

Structured questions, Paper 1B and Paper 2 style

Measuring g with a falling ballPaper 1B style · SL · 7 marks

A student releases a steel ball from rest and an electronic timer records the time t it takes to fall through a measured height h to a pad on the floor. She repeats this for five heights. The table shows her mean results. The uncertainty in each h is ±0.002 m and in each t is ±0.005 s.

Table of the five trials. Height h in m (uncertainty ±0.002 m) and mean fall time t in s (uncertainty ±0.005 s): h = 0.200, t = 0.207; h = 0.400, t = 0.289; h = 0.600, t = 0.353; h = 0.800, t = 0.406; h = 1.000, t = 0.454.
h / m ± 0.002	t / s ± 0.005
0.200	0.207
0.400	0.289
0.600	0.353
0.800	0.406
1.000	0.454

(a) Which graph should she plot to obtain a straight line from which g can be found? [1]

(b) Using the first and last rows of the table, determine the gradient of a graph of h against t², in m s⁻². [2]

m s⁻²

(d) Calculate the percentage uncertainty in t² for the 0.200 m height. [2]

(e) Her best-fit line of h against t² does not pass through the origin: it meets the h axis at about h = −0.010 m. What does this suggest? [1]

A trolley on a ramp and a light gatePaper 1B style · SL · 6 marks

A trolley is released from rest at a height h on a ramp. At the bottom a card of width 0.050 ± 0.001 m on the trolley passes through a light gate, which records the time t for which the beam is blocked. The table shows the results; each t has an uncertainty of ±0.1 ms.

Table of results. Release height h in m and light-gate time t in ms (±0.1 ms): h = 0.10, t = 37.6; h = 0.20, t = 26.6; h = 0.30, t = 21.7; h = 0.40, t = 18.8; h = 0.50, t = 16.8.
h / m	t / ms ± 0.1
0.10	37.6
0.20	26.6
0.30	21.7
0.40	18.8
0.50	16.8

(a) Calculate the speed of the trolley at the light gate for the 0.30 m release height, in m s⁻¹. [1]

m s⁻¹

(b) If no energy were dissipated, mgh = ½mv². Which graph would be a straight line through the origin with a gradient of 2g? [1]

(c) The gradient of the student's graph of v² against h is 17.6 m s⁻². Calculate the fraction of the trolley's loss in gravitational potential energy that becomes kinetic energy. [2]

(d) Calculate the percentage uncertainty in the speed for the 0.50 m release height, where t = 16.8 ms. [2]

A ball kicked towards a wallPaper 2 style · SL · 5 marks

A football is kicked from level ground at 18.0 m s⁻¹ at 40.0° above the horizontal towards a vertical wall 20.0 m away. Air resistance is negligible.

(a) Calculate the horizontal component of the ball's initial velocity, in m s⁻¹. [1]

m s⁻¹

(b) The horizontal component is 13.8 m s⁻¹. Calculate the time the ball takes to reach the wall, in s. [1]

(c) The vertical component of the initial velocity is 11.6 m s⁻¹ and the ball reaches the wall after 1.45 s. Calculate the height of the ball when it reaches the wall, in m. [2]

(d) Is the ball rising or falling when it reaches the wall? [1]

Two trolleys couple togetherPaper 2 style · SL · 7 marks

Trolley P, of mass 0.80 kg, moves at 1.5 m s⁻¹ along a level track towards trolley Q, of mass 1.2 kg, which is at rest. They collide and move off joined together. Friction is negligible.

(a) Calculate the speed of the joined trolleys just after the collision, in m s⁻¹. [2]

m s⁻¹

(b) The joined trolleys move off at 0.60 m s⁻¹. Calculate the kinetic energy transferred to other forms in the collision, in J. [2]

(c) The collision lasts 0.15 s. Calculate the average force exerted on trolley Q during the collision, in N. Use a final speed of 0.60 m s⁻¹. [2]

(d) What is the average force that Q exerts on P during the collision? [1]

A car on a flat bendPaper 2 style · SL · 5 marks

A car of mass 1100 kg goes round a flat circular bend of radius 60 m. The coefficient of static friction between the tyres and the dry road is 0.70.

(a) Calculate the greatest speed at which the car can go round the bend without skidding, in m s⁻¹. [2]

m s⁻¹

(b) Calculate the frictional force on the car when it goes round the bend at 15 m s⁻¹, in N. [2]

A block launched by a springPaper 2 style · SL · 6 marks

A block of mass 0.40 kg is pushed against a spring of spring constant 250 N m⁻¹, compressing it by 0.12 m. When released, the spring pushes the block along a smooth section of floor; the block then slides onto a rough section where the coefficient of dynamic friction is 0.25, and stops.

(a) Calculate the elastic potential energy stored in the compressed spring, in J. [1]

(b) The spring stores 1.8 J. Calculate the speed of the block as it leaves the spring, in m s⁻¹. [2]

m s⁻¹

(c) The block enters the rough section with 1.8 J of kinetic energy. Calculate the distance it slides on the rough section before stopping, in m. [2]

(d) Where is the 1.8 J of energy once the block has stopped? [1]

A child steps onto a roundaboutPaper 2 style · HL · 8 marks

A playground roundabout has a moment of inertia of 250 kg m² about its axis and turns freely at 1.20 rad s⁻¹. A child of mass 30.0 kg, initially at rest, steps onto its edge, 2.00 m from the axis. Treat the child as a point mass and ignore friction at the axle.

(a) Calculate the angular momentum of the roundabout before the child steps on, in kg m² s⁻¹. [1]

kg m² s⁻¹

(b) Calculate the angular velocity of the roundabout once the child is on it, in rad s⁻¹. [2]

rad s⁻¹

(c) The new angular velocity is 0.811 rad s⁻¹. Calculate the rotational kinetic energy transferred to other forms when the child steps on, in J. [2]

(d) Why is rotational kinetic energy not conserved although angular momentum is? [1]

(e) Once the child is on, a brake brings the roundabout to rest in 5.00 s. Calculate the average braking torque, in N m. Use a total moment of inertia of 370 kg m² and an angular velocity of 0.811 rad s⁻¹. [2]

N m