12. Integration

iGCSE (2021 Edition)

Worksheet

1

The velocity v\left(t\right), in metres per second, of an object travelling horizontally along a straight line after t seconds is modelled by: v \left( t \right) = 12 t, \text{ where } t \geq 0The object is initially at the origin.

a

Find the displacement x \left( t \right)= \int v(t) \, dt of the particle at time t.

b

Find the time at which x(t) = 54 \text{ m}.

2

The velocity v\left(t\right), in metres per second, of an object travelling horizontally along a straight line after t seconds is modelled by: v \left( t \right) = 6 t + 10, \text{ where } t \geq 0The object is initially 8 \text{ m} to the right of the origin.

a

Find the displacement s \left( t \right) of the particle at time t.

b

Find the displacement of the object after 5 seconds.

3

The velocity v\left(t\right), in metres per second, of an object travelling horizontally along a straight line after t seconds is modelled by: v \left( t \right) = 12 t^{2} + 30 t + 9, \text{ where } t \geq 0The object is initially 6 \text{ m} to the left of the origin.

a

Find the displacement, s \left( t \right), of the particle at time t.

b

Find the displacement of the object after 5 seconds.

4

The velocity v\left(t\right), in metres per second, of an object travelling horizontally along a straight line after t seconds is modelled by: v \left( t \right) = 12 \sqrt{t}The object is initially 5 \text{ m} to the right of the origin.

a

Find the displacement, x \left( t \right), of the particle at time t.

b

Hence, find the position of the object after 9 seconds.

5

The velocity v\left(t\right), in metres per second, of a particle travelling horizontally along a straight line after t seconds is given by:

v \left( t \right) = 30 \sqrt{t}

a

Find the velocity of the particle after 4 seconds.

b

Sketch a graph of the velocity of the particle over the first 10 seconds.

c

Find the area under the velocity function from t = 0 to t = 4.

d

Describe what the area found in part (c) represents.

6

The velocity v\left(t\right), in metres per second, of a particle travelling horizontally along a straight line after t seconds is given by:

v \left( t \right) = 2 - \dfrac{t}{3}

a

Find the velocity of the particle after 9 seconds.

b

Sketch a graph of the velocity of the particle over the first 12 seconds.

c

Find the area bounded between the velocity function and the horizontal axis from t = 0 to t = 9.

d

Describe what the area found in part (c) represents.

7

The velocity v\left(t\right), in metres per second, of a particle travelling horizontally along a straight line after t seconds is given by:

v \left( t \right) = 6 t + 15

The displacement after 2 seconds is 45 metres to the right of the origin.

a

Calculate the initial velocity of the particle.

b

Find the displacement, x \left( t \right), of the particle at time t.

c

Calculate the displacement of the particle after four seconds.

d

Find the total distance travelled between two and four seconds.

8

The velocity v\left(t\right), in metres per second, of a particle travelling horizontally along a straight line after t seconds is given by: v \left( t \right) = 4 t - 4The particle is instantaneously stationary when it is 1 \text{ m} right of the origin.

a

Find the time when the particle is stationary.

b

Find the displacement, x \left( t \right), of the particle at time t.

c

Find the value of x(t) at:

i

t=0

ii

t=1

iii

t=2

9

v \left( t \right) = 6 t^{2} - 14 t + 3

The displacement after 2 seconds is 4 metres to the left of the origin.

a

Calculate the initial velocity of the particle.

b

Find the displacement, x \left( t \right), of the particle at time t.

c

Calculate the displacement of the particle after four seconds.

d

Assuming the object continues to move in the same direction, find the total distance travelled between two and four seconds.

10

The velocity v\left(t\right), in metres per second, of an object travelling horizontally along a straight line after t seconds is given by: v \left( t \right) = 12 t^{2} - 48 t, \text{ where } t \geq 0The object is initially 5 \text{ m} to the right of the origin.

a

Find the displacement, x \left( t \right), of the particle at time t.

b

Find the times, t, when the object is at rest.

c

Find the displacement at which the object is stationary other than its initial position.

11

v\left(t\right) = 3 t^{\frac{3}{2}} - 2 t

a

Find the initial acceleration of the particle.

b

If the particle is initially at the origin, find an expression for the displacement x(t).

c

Find the exact time t > 0 at which the particle is stationary.

d

How far does the particle travel in the first 2 seconds? Round your answer correct to two decimal places.

12

A pen moves along the x axis ruling a line. The diagram shows the graph of the velocity of the tip of the pen as a function of time.

The velocity, in centimetres per second, after t seconds is given by:

v\left(t\right) = 4 t^{3} - 36 t^{2} + 72 t

When t = 0, the tip of the pen is at x = 5.

a

Find an expression for x\left(t\right), the position of the tip of the pen, as a function of time.

b

What feature will the graph of x\left(t\right) have at the point where t = 3 and how does this relate to the movement of the pen?

c

The pen uses 0.02 milligrams of ink per centimetre travelled. Find the amount of ink used between t = 0 and t = 4.

13

v \left( t \right) = 5 \left(1 - e^{ - t }\right)

a

Calculate the exact total change in displacement over the interval \left[0, 5\right].

b

Hence, calculate the average change in displacement over the interval \left[0, 5\right], correct to three decimal places.

c

Use technology to calculate the average change in the distance over the interval \left[0, 5\right], correct to three decimal places.

d

For the given velocity function, is the average change in displacement equal to the average change in distance for t \geq 0? Explain why or why not.

14

The acceleration a\left(t\right), in \text{m/s}^{2}, of an object travelling horizontally along a straight line after t seconds is modelled by:

a = 2 t - 15, \text{where } t \geq 0

The object is initially moving to the right at 56\text{ m/s}.

a

Find the velocity, v\left(t\right), of the particle at time t.

b

Find for all times at which the particle is at rest.

15

The acceleration a\left(t\right), in \text{ m/s}^{2}, of an object travelling horizontally along a straight line after t seconds is modelled by: a \left( t \right) = 6 t - 27, \text{ where } t \geq 0After 10 seconds, the object is moving at 90 \text{ m/s} in the positive direction.

a

Find the velocity, v\left(t\right), of the particle at time t.

b

Find all the times at which the particle is at rest.

16

The acceleration a\left(t\right), in \text{ m/s}^{2}, of an object travelling horizontally along a straight line after t seconds is modelled by:a\left(t\right) = 2, \text{ where } t \geq 0The object is initially 8 \text{ m} to the right of the origin and moving to the left at 5 \text{ m/s}.

a

Find the velocity, v\left(t\right), of the particle at time t.

b

Find the displacement, x\left(t\right), of the particle at time t.

c

Find the position of the object at 7 seconds.

d

Find the time at which the particle is moving at a speed of 3 \text{ m/s} to the right.

17

A coin is thrown vertically upwards from the top of a building 30\text{ m} high with an initial velocity of 20\text{ m/s}. Using the approximation of acceleration due to gravity of - 10 \text{ m/s}^{2}, find:

a

The time, t, taken for the coin to reach its maximum height.

b

The maximum height reached by the coin.

c

The exact time, t, taken for the coin to reach the ground.

d

The exact velocity of the coin as it hits the ground.

18

An object is thrown vertically upwards with an initial velocity of 15\text{ m/s} and an initial height of 2\text{ m}. Using the approximation of acceleration due to gravity of -9.8 \text{ m/s}^{2}, find:

a

The time, t, taken for the object to reach its maximum height.

b

The maximum height reached by the object, correct to one decimal place.

c

The time, t, taken for the object to reach the ground, correct to one decimal place.

d

The velocity of the object as it hits the ground, correct to one decimal place.

19

The acceleration a\left(t\right), in \text{ cm/s}^{2}, of a particle travelling horizontally along a straight line after t seconds is modelled by:

a\left(t\right) = - e^{ 4 t}The particle is initially at rest.

a

Find the velocity, v\left(t\right), of the particle at time t.

b

Find the change in the particle's position after 5 seconds, to the nearest centimetre.

20

The acceleration a\left(t\right), in \text{ m/s}^{2}, of a particle travelling horizontally along a straight line after t seconds is modelled by:a\left(t \right)= e^{ - t }The particle starts off from the origin with an initial velocity of 2\text{ m/s}.

a

Find the velocity of the particle after 4 seconds, correct to two decimal places.

b

Find the displacement of the particle after 4 seconds, correct to the nearest hundredth of a metre.

21

The acceleration a\left(t\right), in \text{ m/s}^{2}, of a particle travelling horizontally along a straight line after t seconds is modelled by:a \left( t \right) = 2 e^{ 3 t}The initial velocity of a particle is 10\text{ m/s}.

a

Find its velocity, v \left( t \right), after t = 2 seconds. Round your answer to the nearest metre per second.

b

Find the displacement of the particle after 4 seconds, given the initial position is 2 \text{ m} to the left of the origin. Round your answer to the nearest metre.

22

a

Find the acceleration of the particle at t = \dfrac{\pi}{3}.

b

Find the displacement of the particle at t = \dfrac{\pi}{9}, given that when t = 0, the displacement is 2 \text{ m}.

23

v \left( t \right) = 6 \cos 3 t

a

Find the acceleration of the particle at t = \dfrac{\pi}{9}.

b

Find the displacement of the particle at t = \dfrac{\pi}{3}, given that when t = 0, the displacement is 3\text{ m}.

24

a

Find the change in displacement in the first 8 seconds.

b

Given that the distance function is the absolute value of the displacement function, calculate the change in distance in the first 4 seconds.

25

v \left( t \right) = 6 \cos \left(\dfrac{\pi t}{4}\right)

a

Find the change in displacement in the first 9 seconds.

b

Find the change in distance in the first 8 seconds.

26

The acceleration a\left(t\right), in \text{ m/s}^{2}, of a particle travelling horizontally along a straight line after t seconds is modelled by:

a \left( t \right) = 64 \sin 4 ta

Find v \left( t \right), the velocity in \text{m/s} at time t, if the particle is initially traveling at 2 \text{ m/s}.

b

Hence, calculate the speed of the particle at t = \dfrac{\pi}{3}.

c

Find x \left( t \right), the displacement in metres at time t, if the particle is initially located 3 \text{ m} from the origin.

d

Hence, calculate the displacement at t = \dfrac{\pi}{3}.

27

The acceleration a\left(t\right), in \text{ m/s}^{2}, of a particle travelling horizontally along a straight line after t seconds is modelled by:

a \left( t \right) = 50 \cos 5 ta

Find v \left( t \right), the velocity in \text{m/s} at time t, if the particle is initially traveling at 6 \text{ m/s}.

b

Hence, calculate the speed of the particle at t = \dfrac{\pi}{6}.

c

Find x \left( t \right), the displacement in metres at time t, if the particle is initially located at 3 \text{ m} from the origin.

d

Hence, calculate the displacement of the particle at t = \dfrac{\pi}{6}.

28

The acceleration a\left(t\right), in \text{ m/s}^{2}, of a particle travelling horizontally along a straight line after t seconds is modelled by:

a \left( t \right) = 6 \sin \left(\dfrac{\pi t}{3}\right)a

Find the change in velocity between t = 1 and t = 7.

b

Given that the speed function is the absolute value of the velocity function, calculate the total change in speed in the initial 4 seconds.

29

a

Find the change in velocity between t = 1 and t = 3.

b

Calculate the total change in speed in the first 4 seconds.

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Apply differentiation and integration to kinematics problems that involve displacement, velocity and acceleration of a particle moving in a straight line with variable or constant acceleration, and the use of x–t and v–t graphs.