Kinematics [285 marks]

1a.

A particle moves along a straight line so that its velocity,  m s , after seconds is given by , for 0 ≤ ≤ 5.

Find when the particle is at rest.

v −1 tv (t) = 1.4t − 2.7 t

1b. Find the acceleration of the particle when .t = 2

Find the total distance travelled by the particle.

[2 marks]

[2 marks]

1c. Find the total distance travelled by the particle.

2a.

Let , be a periodic function with

The following diagram shows the graph of .

There is a maximum point at A. The minimum value of is −13 .

Find the coordinates of A.

f (x) = 12 cos x − 5 sin x, −π ⩽ x ⩽ 2π f (x) = f (x + 2π)

f

f

[3 marks]

[2 marks]

2b. For the graph of , write down the amplitude.f

2c. For the graph of , write down the period.f

2d. Hence, write in the form .f (x) p cos (x + r)

A ball on a spring is attached to a fixed point O. The ball is then pulled down and released, so

[1 mark]

[1 mark]

[3 marks]

2e.

A ball on a spring is attached to a fixed point O. The ball is then pulled down and released, sothat it moves back and forth vertically.

The distance, d centimetres, of the centre of the ball from O at time t seconds, is given by

Find the maximum speed of the ball.

d (t) = f (t) + 17, 0 ⩽ t ⩽ 5.

Find the first time when the ball’s speed is changing at a rate of 2 cm s .−2

[3 marks]

2f. Find the first time when the ball’s speed is changing at a rate of 2 cm s .−2

A particle P moves along a straight line. The velocity v m s of P after t seconds is given by v (t)−1

[5 marks]

3a.

A particle P moves along a straight line. The velocity v m s of P after t seconds is given by v (t)= 7 cos t − 5t  , for 0 ≤ t ≤ 7.

The following diagram shows the graph of v.

Find the initial velocity of P.

−1

cos t

3b. Find the maximum speed of P.

Write down the number of times that the acceleration of P is 0 m s .−2

[2 marks]

[3 marks]

3c. Write down the number of times that the acceleration of P is 0 m s .−2

3d. Find the acceleration of P when it changes direction.

Find the total distance travelled by P.

[3 marks]

[4 marks]

3e. Find the total distance travelled by P.

4a.

Note: In this question, distance is in metres and time is in seconds.

A particle P moves in a straight line for five seconds. Its acceleration at time is given by , for .

Write down the values of when .

ta = 3t2 − 14t + 8 0 ⩽ t ⩽ 5

t a = 0

4b. Hence or otherwise, find all possible values of for which the velocity of P isdecreasing.

t

= 0 3 m s−1

[3 marks]

[2 marks]

[2 marks]

4c.

When , the velocity of P is .

Find an expression for the velocity of P at time .

t = 0 3 m s−1

t

4d. Find the total distance travelled by P when its velocity is increasing.

m s−1

[6 marks]

[4 marks]

5a.

A particle P moves along a straight line. Its velocity after seconds is given by , for . The following diagram shows the graph of .

Write down the first value of at which P changes direction.

vP m s−1 tvP = √t sin( t)π

2 0 ⩽ t ⩽ 8 vP

t

5b. Find the total distance travelled by P, for .0 ⩽ t ⩽ 8

m s−1

[1 mark]

[2 marks]

5c. A second particle Q also moves along a straight line. Its velocity, after seconds is given by for . After seconds Q has travelled the same totaldistance as P.

Find .

vQ m s−1 t

vQ = √t 0 ⩽ t ⩽ 8 k

k

Note: In this question, distance is in metres and time is in seconds.

[4 marks]

6. Note: In this question, distance is in metres and time is in seconds.

A particle moves along a horizontal line starting at a fixed point A. The velocity of the particle,at time , is given by , for . The following diagram shows the graph of

There are -intercepts at and .

Find the maximum distance of the particle from A during the time and justify youranswer.

v

t v(t) = 2t2−4t

t2−2t+2 0 ⩽ t ⩽ 5 v

t (0, 0) (2, 0)

0 ⩽ t ⩽ 5

cm s−1

[6 marks]

7a.

A particle P starts from a point A and moves along a horizontal straight line. Its velocity after seconds is given by

The following diagram shows the graph of .

Find the initial velocity of .

v cm s−1

t

v(t) = { −2t + 2, for 0 ⩽ t ⩽ 1

3√t + − 7, for 1 ⩽ t ⩽ 124t2

v

P

7b.

P is at rest when and .

Find the value of .

t = 1 t = p

p

When , the acceleration of P is zero.=

[2 marks]

[2 marks]

7c.

When , the acceleration of P is zero.

(i) Find the value of .

(ii) Hence, find the speed of P when .

t = q

q

t = q

7d. (i) Find the total distance travelled by P between and .

(ii) Hence or otherwise, find the displacement of P from A when .

t = 1 t = p

t = p

ms−1

[4 marks]

[6 marks]

8a.

A particle P moves along a straight line so that its velocity, , after seconds, is given by , for . The initial displacement of P from a fixed point O is 4

metres.

Find the displacement of P from O after 5 seconds.

v ms−1 tv = cos 3t − 2 sin t − 0.5 0 ⩽ t ⩽ 5

The following sketch shows the graph of .

[5 marks]

8b.

The following sketch shows the graph of .

Find when P is first at rest.

v

8c. Write down the number of times P changes direction.

8d. Find the acceleration of P after 3 seconds.

Find the maximum speed of P.

[2 marks]

[2 marks]

[2 marks]

8e. Find the maximum speed of P.

9. A particle moves in a straight line. Its velocity after seconds is given by

After seconds, the particle is 2 m from its initial position. Find the possible values of .

v m s−1 t

v = 6t − 6, for 0 ⩽ t ⩽ 2.

p p

ms−1

[3 marks]

[7 marks]

10a.

The velocity of a particle after seconds is given by

, for

The following diagram shows the graph of .

Find the value of when the particle is at rest.

v ms−1 t

v(t) = (0.3t + 0.1)t − 4 0 ≤ t ≤ 5

v

t

10b. Find the value of when the acceleration of the particle is .t 0

ms−1

[3 marks]

[3 marks]

11.

A particle starts from point and moves along a straight line. Its velocity, , after seconds is given by , for . The particle is at rest when .

The following diagram shows the graph of .

Find the distance travelled by the particle for .

A v ms−1 t

v(t) = e cost − 112 0 ≤ t ≤ 4 t = π

2

v

0 ≤ t ≤ π2

12. Ramiro and Lautaro are travelling from Buenos Aires to El Moro.

Ramiro travels in a vehicle whose velocity in is given by , where is inseconds.

Lautaro travels in a vehicle whose displacement from Buenos Aires in metres is given by .

When , both vehicles are at the same point.

Find Ramiro’s displacement from Buenos Aires when .

ms−1 VR = 40 − t2 t

SL = 2t2 + 60

t = 0

t = 10

13a.

A particle moves in a straight line. Its velocity,

, at time

seconds, is given by

Find the velocity of the particle when .

v ms−1

t

v = (t2 − 4)3, for 0 ⩽ t ⩽ 3.

t = 1

13b. Find the value of for which the particle is at rest.t

13c. Find the total distance the particle travels during the first three seconds.

= 6 ( 2 − 4)2

[2 marks]

[8 marks]

[2 marks]

[3 marks]

[3 marks]

13d. Show that the acceleration of the particle is given by .a = 6t(t2 − 4)2

13e. Find all possible values of for which the velocity and acceleration are both positiveor both negative.

t

14a.

A particle moves along a straight line such that its velocity,

, is given by

, for

.

On the grid below, sketch the graph of , for .

v ms−1

v(t) = 10te−1.7t

t ⩾ 0

v 0 ⩽ t ⩽ 4

14b. Find the distance travelled by the particle in the first three seconds.

14c. Find the velocity of the particle when its acceleration is zero.

15. A rocket moving in a straight line has velocity km s and displacement km at time seconds. The velocity is given by . When , .

Find an expression for the displacement of the rocket in terms of .

v –1 st v v(t) = 6e2t + t t = 0 s = 10

t

16a.

The velocity of a particle in ms is given by

, for

.

On the grid below, sketch the graph of .

−1

v = esin t − 10 ≤ t ≤ 5

v

Find the total distance travelled by the particle in the first five seconds.

[3 marks]

[4 marks]

[3 marks]

[2 marks]

[3 marks]

[7 marks]

[3 marks]

16b. Find the total distance travelled by the particle in the first five seconds.

16c. Write down the positive -intercept.t

17a.

A particle’s displacement, in metres, is given by

, for

, where t is the time in seconds.

On the grid below, sketch the graph of .

s(t) = 2t cos t

0 ≤ t ≤ 6

s

17b. Find the maximum velocity of the particle.

18a.

In this question, you are given that

, and

.

The displacement of an object from a fixed point, O is given by

for

.

Find .

cos =π3

12

sin =π3

√32

s(t) = t − sin 2t

0 ≤ t ≤ π

s′(t)

18b. In this interval, there are only two values of t for which the object is not moving. Onevalue is .

Find the other value.

t = π6

Show that between these two values of t .′( ) > 0

[1 mark]

[4 marks]

[4 marks]

[3 marks]

[3 marks]

[4 marks]

18c. Show that between these two values of t .s′(t) > 0

18d. Find the distance travelled between these two values of t .

19a.

A particle moves in a straight line with velocity

, for

, where v is in centimetres per second and t is in seconds.

Find the acceleration of the particle after 2.7 seconds.

v = 12t − 2t3 − 1t ≥ 0

19b. Find the displacement of the particle after 1.3 seconds.

20a.

Let

, where

. The function v is obtained when the graph of f is transformed by

a stretch by a scale factor of

parallel to the y-axis,

followed by a translation by the vector

.

Find , giving your answer in the form .

f(t) = 2t2 + 7t > 0

13

( 2−4

)

v(t) a(t − b)2 + c

20b. A particle moves along a straight line so that its velocity in ms , at time t seconds, isgiven by v . Find the distance the particle travels between and .

−1

t = 5.0 t = 6.8

21a.

The velocity v ms of a particle at time t seconds, is given by

, for

.

Write down the velocity of the particle when .

−1

v = 2t + cos 2t

0 ≤ t ≤ 2

t = 0

21b. When , the acceleration is zero.

(i) Show that .

(ii) Find the exact velocity when .

t = k

k = π4

t = π4

21c. When , and when , .

Sketch a graph of v against t .

t < π4 > 0dv

dtt > π

4 > 0dvdt

Let d be the distance travelled by the particle for .0 ≤ ≤ 1

[3 marks]

[5 marks]

[3 marks]

[3 marks]

[4 marks]

[3 marks]

[1 mark]

[8 marks]

[4 marks]

21d. Let d be the distance travelled by the particle for .

(i) Write down an expression for d .

(ii) Represent d on your sketch.

0 ≤ t ≤ 1

22a.

The following diagram shows part of the graph of a quadratic function f .

The x-intercepts are at

and

, and the y-intercept is at

.

Write down in the form .

(−4, 0)(6, 0)(0, 240)

f(x) f(x) = −10(x − p)(x − q)

22b. Find another expression for in the form .f(x) f(x) = −10(x − h)2 + k

22c. Show that can also be written in the form .f(x) f(x) = 240 + 20x − 10x2

22d. A particle moves along a straight line so that its velocity, , at time t seconds isgiven by , for .

(i) Find the value of t when the speed of the particle is greatest.

(ii) Find the acceleration of the particle when its speed is zero.

v ms−1

v = 240 + 20t − 10t2 0 ≤ t ≤ 6

The velocity v ms of an object after t seconds is given by−1

[3 marks]

[2 marks]

[4 marks]

[2 marks]

[7 marks]

23a.

The velocity v ms of an object after t seconds is given by

, for

.

On the grid below, sketch the graph of v , clearly indicating the maximum point.

−1

v(t) = 15√t − 3t

0 ≤ t ≤ 25

23b. (i) Write down an expression for d .

(ii) Hence, write down the value of d .

24. The acceleration, , of a particle at time t seconds is given by

The particle is at rest when .

Find the velocity of the particle when .

a ms−2

a = + 3 sin 2t, for t ≥ 1.1t

t = 1

t = 5

The following diagram shows the graphs of the displacement, velocity and acceleration of a moving object as

[3 marks]

[4 marks]

[7 marks]

25a.

The following diagram shows the graphs of the displacement, velocity and acceleration of a moving object as

functions of time, t.

Complete the following table by noting which graph A, B or C corresponds to eachfunction.

25b. Write down the value of t when the velocity is greatest.

26a.

In this question s represents displacement in metres and t represents time in seconds.

The velocity v m s of a moving body is given by

where a is a non-zero constant.

(i) If when , find an expression for s in terms of a and t.

(ii) If when , write down an expression for s in terms of a and t.

–1

v = 40 − at

s = 100 t = 0

s = 0 t = 0

26b.

Trains approaching a station start to slow down when they pass a point P. As a train slows down, its velocity is given by

, where

at P. The station is 500 m from P.

A train M slows down so that it comes to a stop at the station.

(i) Find the time it takes train M to come to a stop, giving your answer in terms of a.

(ii) Hence show that .

v = 40 − at

t = 0

a = 85

26c. For a different train N, the value of a is 4.

Show that this train will stop before it reaches the station.

The acceleration,

[4 marks]

[2 marks]

[6 marks]

[6 marks]

[5 marks]

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© International Baccalaureate Organization 2019 International Baccalaureate® - Baccalauréat International® - Bachillerato Internacional®

27a.

The acceleration,

, of a particle at time t seconds is given by

.

Find the acceleration of the particle at .

a ms−2

a = 2t + cos t

t = 0

27b. Find the velocity, v, at time t, given that the initial velocity of the particle is .2 ms−1

27c. Find , giving your answer in the form .∫ 30 vdt p − qcos 3

27d. What information does the answer to part (c) give about the motion of the particle?

28. A particle moves along a straight line so that its velocity, at time t seconds isgiven by . When , the displacement, s, of the particle is 7 metres. Find anexpression for s in terms of t.

v ms−1

v = 6e3t + 4 t = 0

29a.

A toy car travels with velocity v ms for six seconds. This is shown in the graph below.

Write down the car’s velocity at .

−1

t = 3

29b. Find the car’s acceleration at .t = 1.5

29c. Find the total distance travelled.

[2 marks]

[5 marks]

[7 marks]

[2 marks]

[7 marks]

[1 mark]

[2 marks]

[3 marks]