Motion time graphs — Edexcel A-level Mathematics (Applied)

Motion time graphsMathematics (Applied)

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Grade C — Questions 1–10

Question 1

The displacement-time graph below shows the motion of a particle.

(a) State the displacement of the particle at t = 4 seconds.

(b) Describe the motion of the particle between t = 4 s and t = 6 s.

Question 2

A cyclist travels 300 m in 20 seconds at constant velocity.

(a) Sketch a displacement-time graph for this journey.

(b) State the gradient of the graph.

Question 3

The velocity-time graph below shows the motion of a car.

(a) State the velocity of the car at t = 10 seconds.

(b) Calculate the acceleration during the first 5 seconds.

Question 4

Explain the difference between:

(a) Distance and displacement

(b) Speed and velocity

Question 5

A particle moves such that its displacement from O is given by the following displacement-time graph.

(a) At what times is the particle at the origin O?

(b) Calculate the velocity between t = 0 and t = 2 s.

(d) Describe the motion of the particle during the 6 seconds.

Question 6

On a velocity-time graph:

(a) What does the gradient represent?

(b) What does the area under the graph represent?

Question 7

A train accelerates uniformly from rest to 30 m/s in 20 seconds.

(a) Sketch a velocity-time graph for this motion.

(b) Calculate the acceleration of the train.

Question 8

The acceleration-time graph below shows the motion of a particle starting from rest.

(a) State the acceleration of the particle.

(b) Calculate the change in velocity during the 8 seconds.

Question 9

A ball is thrown vertically upward. Sketch a velocity-time graph for the ball from the moment it is thrown until it returns to the thrower's hand. Take upward as positive.

Question 10

State whether each of the following is a scalar or a vector quantity:

(a) Distance travelled

(b) Displacement

(d) Velocity

(e) Acceleration

Grade B — Questions 11–20

Question 11

The velocity-time graph shows the motion of two cars, A and B. Both cars start from the same point at t = 0.

(a) At what time do the cars have the same velocity?

(b) Calculate the acceleration of Car A during the first 6 seconds.

(d) Which car is ahead at t = 6 s, and by how much?

Question 12

A particle moves along a straight line. The velocity-time graph for the particle is shown below.

(a) State the initial velocity of the particle.

(b) Calculate the acceleration of the particle.

(d) Calculate the total distance travelled in the 9 seconds.

(e) Calculate the displacement of the particle after 9 seconds.

Question 13

A car accelerates uniformly from 10 m/s to 25 m/s while travelling 140 m.

(a) Sketch a velocity-time graph for this motion.

(b) Calculate the time taken.

Question 14

The displacement-time graph shows the motion of a jogger.

(a) Calculate the velocity during the first 4 minutes.

(b) Describe the motion between t = 4 min and t = 6 min.

(d) Calculate the average velocity over the whole 10 minutes.

Question 15

An acceleration-time graph shows a constant acceleration of 2 m/s² for 5 seconds, followed by zero acceleration for 3 seconds.

(a) Sketch this acceleration-time graph.

(b) If the object starts from rest, sketch the corresponding velocity-time graph.

Question 16

A train starts from rest at station A and accelerates uniformly at 0.8 m/s² for 25 seconds. It then travels at constant velocity for 40 seconds before decelerating uniformly at 1 m/s² until it stops at station B.

(a) Calculate the maximum velocity reached by the train.

(b) Calculate the time taken for the train to decelerate to rest.

(d) Calculate the total distance from station A to station B.

Question 17

The graph shows displacement against time for a particle P.

(a) Estimate the velocity at t = 1 s by drawing a tangent.

(b) At what time is the velocity zero?

Question 18

Convert the following:

(a) 72 km/h to m/s

(b) 15 m/s to km/h

Question 19

A stone is dropped from a height. Taking downward as positive and g = 10 m/s²:

(a) Sketch a velocity-time graph for the first 3 seconds of motion.

(b) Calculate the velocity after 3 seconds.

Question 20

Two particles P and Q move along the x-axis. At time t = 0, P is at the origin and Q is at x = 50 m. P has constant velocity 8 m/s and Q has constant velocity 3 m/s, both moving in the positive x-direction.

(a) Sketch displacement-time graphs for both particles on the same axes.

(b) At what time does P overtake Q?

Grade A — Questions 21–30

Question 21

The velocity-time graph below shows a particle moving along a straight line. The particle starts from the origin.

(a) State the initial velocity and calculate the acceleration during each phase of the motion.

(b) At what time does the particle change direction?

(d) Calculate the total distance travelled in 10 seconds.

(e) Calculate the final displacement from the origin.

Question 22

A car travels along a straight road. The acceleration-time graph is shown. The car has initial velocity 5 m/s.

(a) Calculate the velocity at t = 4 s.

(b) Calculate the velocity at t = 6 s.

(d) Sketch the velocity-time graph for this motion.

Question 23

Two runners A and B race over 100 m. Runner A accelerates uniformly from rest at 4 m/s² for 3 seconds, then continues at constant velocity. Runner B accelerates uniformly from rest at 3 m/s² for 4 seconds, then continues at constant velocity.

(a) Draw velocity-time graphs for both runners on the same axes.

(b) Calculate the maximum velocity of each runner.

(d) Who wins the race and by what margin of time?

Question 24

A car travels along a straight road. Its velocity-time graph is shown below.

(a) Calculate the acceleration during the first 4 seconds.

(b) State the velocity of the car between t = 4 s and t = 8 s.

(d) Calculate the total distance travelled by the car.

(e) Calculate the average velocity over the 12 seconds.

Question 25

A train starts from station A and accelerates uniformly at 0.5 m/s² until it reaches 20 m/s. It then travels at this constant speed until it decelerates uniformly at 1 m/s² to stop at station B. The total journey time is 2 minutes.

(a) Calculate the time spent accelerating.

(b) Calculate the time spent decelerating.

(d) Calculate the distance between stations A and B.

(e) Sketch the velocity-time graph for the journey.

Question 26

A ball is dropped from rest from the top of a building 45 m high. Take g = 10 m/s² and downward as positive.

(a) Calculate the time taken for the ball to reach the ground.

(b) Calculate the velocity of the ball when it hits the ground.

(d) Using the area under your graph, verify the distance fallen.

(e) If a second ball is thrown downward from the same height with initial velocity 10 m/s, calculate the time it takes to reach the ground.

Question 27

A ball is thrown vertically upward from ground level with initial velocity 25 m/s. Take g = 10 m/s² and upward as positive.

(a) Sketch a velocity-time graph for the ball from launch until it returns to ground level.

(b) At what time does the ball reach its maximum height?

(d) At what time does the ball return to ground level?

(e) Sketch the corresponding displacement-time graph.

Question 28

A cyclist and a car start from the same point. The cyclist has constant velocity 8 m/s. The car starts from rest and accelerates uniformly at 2 m/s².

(a) Sketch velocity-time graphs for both on the same axes.

(b) At what time do they have the same velocity?

(d) How far from the start does the overtaking occur?

Question 29

The graph shows the motion of an object.

(a) Find the acceleration during the first 4 seconds.

(b) Find the deceleration during the final 4 seconds.

(d) Sketch the corresponding acceleration-time graph.

Question 30

A particle starts from rest at O and moves along a straight line. Its acceleration is 4 m/s² for the first 3 seconds, then 0 m/s² for the next 2 seconds, then −2 m/s² for the final 6 seconds.

(a) Sketch the acceleration-time graph.

(b) Sketch the velocity-time graph.

(d) Calculate the final velocity.

(e) Calculate the total displacement from O.

Grade A* — Questions 31–40

Question 31

Two cars, P and Q, are travelling along a straight road. At t = 0, car P passes car Q. Car P has constant velocity 15 m/s. Car Q has initial velocity 10 m/s and accelerates uniformly at 1 m/s².

(a) Write expressions for the displacement of each car from Q's initial position at time t.

(b) At what time does Q overtake P?

(d) Sketch displacement-time graphs for both cars on the same axes.

Question 32

Two cyclists P and Q start from the same point at the same time. Their velocity-time graphs are shown below.

(a) Calculate the acceleration of Q during the first 10 seconds.

(b) At what time do P and Q have the same velocity?

(d) At what time does Q overtake P?

(e) How far from the start are they when Q overtakes P?

Question 33

A lift accelerates upward from rest at 2 m/s² for 2 seconds, then moves at constant velocity for 4 seconds, then decelerates at 1 m/s² until it stops.

(a) Find the maximum velocity of the lift.

(b) Find the time taken to decelerate to rest.

(d) Sketch the displacement-time graph for the entire journey.

Question 34

A car is travelling at 25 m/s when the driver sees an obstacle 80 m ahead. The driver's reaction time is 0.4 s (during which the car continues at 25 m/s), after which the driver applies the brakes causing uniform deceleration of 5 m/s².

(a) Calculate the distance travelled during the reaction time.

(b) Calculate the distance required to stop once the brakes are applied.

(d) Sketch the velocity-time graph for the entire motion from when the driver sees the obstacle.

(e) Calculate the total time from seeing the obstacle to stopping.

Question 35

The displacement-time graph for a particle is a parabola passing through the origin. The displacement is given by s = kt² for some constant k.

(a) Given that s = 20 m when t = 2 s, find the value of k.

(b) Find the velocity at t = 2 s by considering the gradient.

(d) Sketch the velocity-time graph.

(e) Sketch the acceleration-time graph.

Question 36

A car starts from rest at point A and accelerates uniformly at 3 m/s² for time T seconds. It then decelerates uniformly at 2 m/s² until it comes to rest at point B. The total distance from A to B is 500 m.

(a) Find the maximum velocity in terms of T.

(b) Find the time taken to decelerate in terms of T.

(d) Hence find the value of T.

(e) Find the total time for the journey.

Question 37

A particle moves along a straight line. It starts from rest at point O and accelerates uniformly at 3 m/s² for 4 seconds. It then immediately decelerates uniformly at 2 m/s² until it comes to rest.

(a) Calculate the maximum velocity reached by the particle.

(b) Calculate the time taken for the particle to decelerate to rest.

(d) Calculate the total distance travelled by the particle.

(e) Calculate the average velocity for the entire journey.

Question 38

A stone is thrown vertically upward from a platform 25 m above the ground with initial velocity 20 m/s. Take g = 10 m/s² and upward as positive.

(a) Calculate the time taken for the stone to reach its maximum height.

(b) Calculate the maximum height of the stone above the ground.

(d) Calculate the total time taken for the stone to hit the ground.

(e) Calculate the velocity with which the stone hits the ground.

Question 39

A train travels from station A to station B. It accelerates uniformly from rest at 0.5 m/s² until it reaches a maximum velocity of 20 m/s. It then travels at constant velocity before decelerating uniformly at 1 m/s² to stop at station B. The total distance from A to B is 1500 m.

(a) Calculate the time spent accelerating and the distance covered during acceleration.

(b) Calculate the time spent decelerating and the distance covered during deceleration.

(d) Calculate the total journey time.

(e) Calculate the average velocity for the journey.

Question 40

The displacement-time graphs for two particles P and Q are shown. P starts from the origin and Q starts from point (0, 30). Q moves with constant velocity 6 m/s.

(a) State the initial displacement of Q from O.

(b) Calculate the velocity of Q.

(c) The curve for P represents constant acceleration. If P has initial velocity 0 and the particles meet at t = 10 s, find the acceleration of P.

(d) At what time do P and Q have the same velocity?

Grade A** — Questions 41–50

Question 41

A particle moves along a straight line. The velocity-time graph consists of two straight line segments meeting at a point. At t = 0, v = u m/s. At t = T, v = 0 m/s (particle momentarily at rest). At t = 2T, v = −u m/s. The particle starts at the origin.

(a) Sketch the velocity-time graph.

(b) Find the acceleration during each phase in terms of u and T.

(d) Find the displacement at t = 2T.

(e) At what time does the particle return to the origin?

Question 42

Two trains A and B travel along parallel tracks. Train A passes train B while both are stationary at a station. Train A accelerates uniformly at 1 m/s² until it reaches 20 m/s, which it then maintains. Train B starts 10 seconds after A and accelerates uniformly at 2 m/s² until it reaches 30 m/s, which it then maintains.

(a) Find the time taken for A to reach 20 m/s and the distance A travels during this time.

(b) Find the time taken for B to reach 30 m/s (measured from when B starts) and the distance B travels during this time.

(d) Find the time at which B overtakes A.

Question 43

A stone is dropped from the top of a tower. One second later, a second stone is thrown vertically downward from the same point with velocity u m/s. The two stones hit the ground at the same time. Taking g = 10 m/s² and the tower height as H metres:

(a) Write an expression for the time taken by the first stone to reach the ground.

(b) Write an expression for the time taken by the second stone to reach the ground.

(d) If H = 80 m, find the value of u.

Question 44

A car travels from A to B, a distance of 600 m. The velocity-time graph for the journey is shown schematically. The car accelerates uniformly from rest for T₁ seconds until reaching velocity V m/s. It then travels at constant velocity for (T₂ − T₁) seconds, before decelerating uniformly to rest for (T₃ − T₂) seconds. Given that the acceleration is 2 m/s², the deceleration is 4 m/s², and the total time T₃ = 40 s:

(a) Express T₁ and (T₃ − T₂) in terms of V.

(b) Form an equation for the total distance in terms of V.

(d) Find the values of T₁, T₂, and T₃.

Question 45

A ball is thrown vertically upward with velocity u m/s from the edge of a cliff of height h metres above sea level. The ball rises, falls past the cliff edge, and hits the sea. Taking downward as positive and using g = 10 m/s²:

(a) Write the displacement of the ball from the cliff edge at time t.

(b) Write the velocity of the ball at time t.

(d) Find the velocity with which the ball hits the sea.

(e) Sketch the velocity-time graph from launch until the ball hits the sea.

Question 46

Particles P and Q move along parallel straight lines. At t = 0, both particles are at the same position. P has constant velocity 12 m/s. Q starts from rest and moves with constant acceleration a m/s².

(a) Find expressions for the displacement of each particle at time t.

(b) Find the time at which P and Q have the same velocity.

(d) Find the value of a if Q catches up with P at t = 12 s.

Question 47

A car starts from rest at point O and accelerates uniformly until it reaches velocity V m/s at point A. It then decelerates uniformly from A until it comes to rest at point B. The distance OA is twice the distance AB.

(a) If the acceleration is a m/s², find the deceleration in terms of a.

(b) If the time from O to A is T seconds, find the time from A to B in terms of T.

(d) Sketch the displacement-time graph for the journey from O to B.

Question 48

A lift moves from the ground floor to the top floor of a building. The lift accelerates from rest at 1.5 m/s² for the first third of the journey, travels at constant velocity for the middle third, and decelerates at 1.5 m/s² for the final third.

(a) Show that the lift reaches the same velocity at the end of the acceleration phase as it has at the start of the deceleration phase.

(b) If the building is 54 m tall, find the maximum velocity of the lift.

(d) Sketch the velocity-time and displacement-time graphs for the journey.

Question 49

Three particles P, Q, and R start from the same point O at the same time and move along a straight line.

P has constant velocity 5 m/s.

Q starts from rest and accelerates at 2 m/s².

R starts with velocity 10 m/s and decelerates at 1 m/s².

(a) At what time(s) do any two particles have the same displacement from O?

(b) At what time do all three particles have the same velocity?

(d) Which particle is furthest from O at t = 10 s?

Question 50

The velocity-time graph for a particle consists of three phases:

Phase 1: Uniform acceleration from rest to velocity V for time T.

Phase 2: Constant velocity V for time 2T.

Phase 3: Uniform deceleration from V to rest for time T.

The total displacement is 240 m.

(a) Show that the acceleration in phase 1 equals the deceleration in phase 3.

(b) Find the total time for the journey in terms of T.

(d) If the acceleration is 3 m/s², find the values of V and T.

(e) Find the total time for the journey.

(f) Sketch the displacement-time graph.

ExamRizz A Level Mathematics | Edexcel | Lesson 7.02 — Kinematics Graphs

Next: Grade B — Questions 11–20