A car starts from rest and accelerates uniformly to a velocity of 24 m/s in 8 seconds.
It then travels at a constant velocity of 24 m/s for a further 20 seconds.
Work out the acceleration of the car during the first 8 seconds.
Work out the total distance travelled by the car in the 28 seconds.
3 marks
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a) 3 m/s2b) 576 m
2
A velocity-time graph shows a cyclist who accelerates uniformly from rest to 12 m/s in 8 seconds, then travels at a constant 12 m/s for 10 seconds.
Work out the acceleration during the first 8 seconds.
Work out the total distance travelled in the 18 seconds.
3 marks
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a) 1.5 m/s²b) 168 m
3
A runner's velocity-time graph is a smooth curve. The table gives her velocity v m/s at time t seconds.
t: 0, 10, 20, 30 v: 0, 4.2, 6.6, 7.8
Use the trapezium rule with 3 strips of equal width to estimate the distance the runner travels in the first 30 seconds.
3 marks
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147 m
4
A train travels at a constant velocity of 20 m/s for 30 seconds. It then brakes uniformly and comes to rest after a further 8 seconds.
Work out the deceleration of the train while it is braking.
Work out the total distance travelled by the train.
4 marks
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a) 2.5 m/s²b) 680 m
5
A cyclist accelerates away from traffic lights. Her speed-time graph is a smooth curve that gets less steep as time increases. The table gives her speed v m/s at time t seconds.
t: 0, 5, 10, 15, 20 v: 0, 3.6, 5.4, 6.4, 7.0
Use the trapezium rule with 4 strips of equal width to estimate the distance travelled in the first 20 seconds.
State, with a reason, whether your answer to part (a) is an underestimate or an overestimate of the actual distance.
4 marks
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a) 94.5 mb) underestimate, because the curve bends above the chords so each trapezium misses some area under the curve
6
A cyclist's velocity increases uniformly from 4 m/s to 10 m/s over 12 seconds.
Work out the acceleration of the cyclist.
Work out the distance travelled in the 12 seconds.
4 marks
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a) 0.5 m/s²b) 84 m
7
The speed-time graph for a car is a smooth curve for 0 ≤ t ≤ 12, where t is the time in seconds and the speed is in m/s.
A tangent drawn by eye to the curve at t = 6 passes through the points (2, 8) and (10, 32).
Use the tangent to estimate the acceleration of the car at t = 6.
Explain why your answer is only an estimate.
4 marks
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a) 3 m/s2b) the acceleration is the gradient of the curve, and a tangent drawn by eye only approximates that gradient
8
A car accelerates uniformly from rest to a velocity of V m/s in 10 seconds. It then travels at a constant velocity of V m/s for 20 seconds, and then decelerates uniformly to rest in 5 seconds. The total distance travelled is 440 m.
Work out the value of V.
Work out the acceleration of the car in the first 10 seconds.
4 marks
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a) 16b) 1.6 m/s²
9
A train travels at a constant velocity of 30 m/s for 40 seconds.
It then decelerates uniformly, coming to rest in a further 25 seconds.
Work out the deceleration of the train.
Work out the total distance the train travels in the 65 seconds.
4 marks
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a) 1.2 m/s2b) 1575 m
10
The velocity, v m/s, of a runner is measured every 2 seconds.
Time, t (seconds)
0
2
4
6
8
Velocity, v (m/s)
0
6
10
12
13
Use the values at t = 2 and t = 6 to estimate the acceleration of the runner at t = 4.
Use the trapezium rule with 4 strips to estimate the distance run in the first 8 seconds.
Is your answer to part (b) an overestimate or an underestimate? Give a reason.
5 marks
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a) 1.5 m/s²b) 69 mc) underestimate, because the curve is concave and lies above the chords that form the tops of the trapezia
11
Car A travels at a constant velocity of 15 m/s. Car B starts from the same place at the same time. Car B accelerates uniformly from rest to 20 m/s in 10 seconds and then continues at a constant 20 m/s.
Work out the distance travelled by each car after 10 seconds.
Work out the time at which car B catches up with car A.
5 marks
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a) A 150 m, B 100 mb) 20 seconds
12
A motorbike accelerates from rest. Its speed-time graph is a smooth curve that gets less steep throughout the first 12 seconds. The table gives the speed v m/s at time t seconds.
t: 0, 4, 8, 12 v: 0, 7, 11, 13
Use 3 strips of equal width to estimate the distance travelled in the first 12 seconds.
Amir says, "The acceleration at t = 8 is 11 ÷ 8 = 1.375 m/s²." Explain why Amir is wrong.
4 marks
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a) 98 mb) Amir has found the gradient of the chord from (0, 0) to (8, 11), which is the average acceleration over the first 8 seconds. The acceleration at t = 8 is the gradient of the tangent to the curve at t = 8, which is smaller here.
13
A lorry starts from rest and accelerates uniformly for 10 seconds until it reaches a velocity of 20 m/s.
It then travels at a constant velocity of 20 m/s for T seconds.
The lorry travels a total distance of 700 m.
Work out the total time of the journey.
5 marks
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40 seconds
14
A ball rolls down a slope. Its velocity-time graph is a smooth curve. The table gives the velocity v m/s at time t seconds.
t: 0, 2, 4, 6 v: 0, 2.2, 5.2, 9.0
A tangent drawn to the curve at t = 4 passes through the points (1, 0.1) and (6, 8.6). Use the tangent to estimate the acceleration of the ball at t = 4.
Use the trapezium rule with 3 strips of equal width to estimate the distance the ball travels in the 6 seconds.
5 marks
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a) 1.7 m/s2 (accept about 1.7 m/s2)b) 23.8 m
15
A car makes a journey in three stages.
Stage 1: it accelerates uniformly from rest to a velocity of V m/s in 12 seconds.
Stage 2: it travels at the constant velocity V m/s for 48 seconds.
Stage 3: it decelerates uniformly from V m/s to rest in 20 seconds.
The total distance of the journey is 960 m.
Work out the value of V.
Work out the acceleration of the car during stage 1.
6 marks
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a) 15b) 1.25 m/s2
16
A drone accelerates from rest. Its velocity-time graph is a smooth curve that gets less steep throughout the first 12 seconds. The table gives the velocity v m/s at time t seconds.
t: 0, 3, 6, 9, 12 v: 0, 4.5, 7.8, 9.9, 10.8
Use the trapezium rule with 4 strips of equal width to estimate the distance travelled by the drone in the 12 seconds.
A tangent drawn to the curve at t = 6 passes through the points (0, 2.4) and (12, 13.2). Use the tangent to estimate the acceleration of the drone at t = 6.
Is your answer to part (a) an underestimate or an overestimate of the actual distance? Give a reason.
7 marks
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a) 82.8 mb) 0.9 m/s2c) underestimate: the velocity increases at a decreasing rate, so the curve bows above each chord and the trapezia miss some area
Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.