The sketch shows a displacement-time graph of a car slowing down with constant acceleration before coming to rest at a set of traffic lights. The equation of the displacement-time graph can be written in the form s = p ( t - q ) 2 + r . Using the two points marked and the fact that the car is stationary at t = 10 , find p , q and r .

As given in the question s = p ( t - q ) 2 + r Stationary at t = 10 s so maximum at t = 10 s ( t - q ) 2 = 0 a t t = 10 10 - q = 0 q = 10 At the same point s = 0 0 = 0 + r r = 0 t = 0 gives s = - 50 So, - 50 = p ( 0 - 10 ) 2 + 0 - 50 = 100 p p = - 0 . 5

The sketch shows a displacement-time graph of a car slowing down with constant acceleration before coming to rest at a set of traffic lights. By comparison with the equation s = s 0 + u t + 1 2 a t 2 , Find the initial speed and acceleration of the car.

As given in the question s = p ( t - q ) 2 + r Stationary at t = 10 s so maximum at t = 10 s ( t - q ) 2 = 0 a t t = 10 10 - q = 0 q = 10 At the same point s = 0 0 = 0 + r r = 0 t = 0 gives s = - 50 So, - 50 = p ( 0 - 10 ) 2 + 0 - 50 = 100 p p = - 0 . 5 So, here new displacement equation be like s = - 0 . 5 t - 10 2 + 0 s = - 0 . 5 t 2 + 10 t - 50 s = - 50 + 10 t - 0 . 5 t 2 Compare it with the above-given equation s = s o + u t + 1 2 a t 2 u = 10 a n d a = - 1

MEDIUM

AS and A Level

IMPORTANT

Earn 100

The sketch shows a displacement-time graph of the position of a train passing a station. The displacement is measured from the entrance of the station to the front of the train. Find the equation of the displacement-time graph and hence the time at which the front of the train reaches the entrance of the station.

Important Questions on Velocity and Acceleration

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 1 - Velocity and Acceleration>EXERCISE 1D>Q 5

The sketch shows a displacement-time graph of a car slowing down with constant acceleration before coming to rest at a set of traffic lights.

Question Image

The equation of the displacement-time graph can be written in the form $s = p (t - q)^{2} + r .$ Using the two points marked and the fact that the car is stationary at $t = 10,$ find $p, q$ and $r .$

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 1 - Velocity and Acceleration>EXERCISE 1D>Q 5

The sketch shows a displacement-time graph of a car slowing down with constant acceleration before coming to rest at a set of traffic lights.

Question Image

By comparison with the equation $s = s_{0} + u t + \frac{1}{2} a t^{2},$ Find the initial speed and acceleration of the car.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 1 - Velocity and Acceleration>EXERCISE 1D>Q 8

A cyclist is stationary when a second cyclist passes, travelling at a constant speed of

8 m s^{- 1} .

The first cyclist then accelerates for

5 s

At a constant rate of

2 m s^{- 2}

Before continuing at constant speed until overtaking the second cyclist. By sketching both graphs, find the equations of the two straight-line sections of the graphs and, hence, find how long it is before the first cyclist overtakes the second.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 1 - Velocity and Acceleration>EXERCISE 1D>Q 10

A leader in a race has

500 m

to go and is running at a constant speed of

4 m s^{- 1}

, but with

100 m

to go increases her speed by a constant acceleration of

0.1 m s^{- 1}

. The second runner is

100 m

behind the leader when the leader has

500 m

to go, and running at

3.8 m s^{- 1}

when she starts to accelerate at a constant rate. Find the minimum acceleration she needs in order to win the race.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 1 - Velocity and Acceleration>EXERCISE 1D>Q 11

A van driver wants to pull out from rest onto a road where cars are moving at a constant speed of

20 m s^{- 1} .

When there is a large enough gap between cars, the van driver pulls out immediately after one car passes. She then accelerates at a constant rate of

4 m s^{- 2}

until moving at

20 m s^{- 1} .

To do this safely the car behind must always be at least

10 m

away. Find the minimum length of the gap between the cars for the van driver to pull out.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 1 - Velocity and Acceleration>EXERCISE 1D>Q 12

A police motorcyclist is stationary when a car passes, driving dangerously at a constant speed of

40 m s^{- 1} .

At the instant the car passes, the motorcyclist gives chase, accelerating at

2.5 m s^{- 2}

until reaching a speed of

50 m s^{- 1}

before continuing at a constant speed. Show that the motorcyclist has not caught the car by the time he reaches top speed. Find how long after the car initially passed him the motorcyclist catches up to the car.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 1 - Velocity and Acceleration>EXERCISE 1D>Q 13

The front of a big wave is approaching a beach at a constant speed of

11.6 {ms}^{- 1} .

When it is

30 m

away from a boy on the beach, the wave starts decelerating at a constant rate of

1.6 m s^{- 2}

and the boy walks away from the sea at a constant speed of

2 m s^{- 1} .

Show that the wave will not reach the boy and find the minimum distance between the boy and the wave.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 1 - Velocity and Acceleration>EXERCISE 1D>Q 14

Swimmers going down a waterslide

30 m

long push themselves off with an initial speed of between

1 m s^{- 1}

and

2 m s^{- 1} .

They accelerate down with constant acceleration

0.8 m s^{- 2}

for the first

20 m

before more water is added and the acceleration is

1 m s^{- 2}

for the last

10 m

of the slide. For safety there must be at least

5 s

between swimmers arriving at the bottom of the slide. Find the minimum whole number of seconds between swimmers being allowed to start the slide.

Important Questions on Velocity and Acceleration

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