Sketch the displacement-time graphs from the information given. In each case consider forwards to be the positive direction and the traffic lights to be the point from which displacement is measured. Remember to include the values for time and displacement at any points where the motion changes. A scooter accelerates from rest 100 m before the traffic lights at a constant rate of 1 . 5 m s - 2 until it reaches 6 m s - 1 . It then travels at this speed until it reaches a point 50 m beyond the traffic lights. At that point the scooter starts to slow at a constant rate of 1 m s - 2 until it stops.

First, we find displacement for the accelerating part as follows using the basic formulas of the kinematics v = u + a t 6 = 0 + 1 . 5 t t = 4 s Now, put the time value in the acceleration formula as given below s = u t + 1 2 a t 2 s = 0 + 1 2 × 1 . 5 × 4 2 = 12 m So, the remaining distance to approach the origin is - 100 + 12 = - 88 Now we need to find time for constant speed: 50 - - 88 = 138 s = u t + 1 2 a t 2 138 = 6 t + 0 t = 23 s For the decelerating stage: v = u + a t 0 = 6 - t t = 6 s And for the displacement s = u t + 1 2 a t 2 s = 6 × 6 - 1 2 × 1 × 6 2 = 18 m So here total time is 4 + 23 + 6 = 33 s And total displacement is 18 + 50 = 68 m So, now the graph for given problems is as given below

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

HARD

AS and A Level

IMPORTANT

Earn 100

Sketch the displacement-time graphs from the information given. In each case consider forwards to be the positive direction and the traffic lights to be the point from which displacement is measured. Remember to include the values for time and displacement at any points where the motion changes.

A motorbike passes the traffic lights at a constant speed of $10 m s^{- 1} .$ After $6 s$ it starts to slow at a constant rate of $2 m s^{- 2}$ until it comes to rest.

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 3

Sketch the displacement-time graphs from the information given. In each case consider forwards to be the positive direction and the traffic lights to be the point from which displacement is measured. Remember to include the values for time and displacement at any points where the motion changes.

A truck is moving at a constant speed of $8 m s^{- 1}$ and is approaching the traffic lights $60 m$ away. When it is $20 m$ away it accelerates at a constant rate of $2 m s^{- 2}$ to get past the lights before they change colour.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 1 - Velocity and Acceleration>EXERCISE 1D>Q 3

A scooter accelerates from rest $100 m$ before the traffic lights at a constant rate of $1.5 m s^{- 2}$ until it reaches $6 m s^{- 1} .$ It then travels at this speed until it reaches a point $50 m$ beyond the traffic lights. At that point the scooter starts to slow at a constant rate of $1 m s^{- 2}$ until it stops.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 1 - Velocity and Acceleration>EXERCISE 1D>Q 4

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.

Question Image

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.

Important Questions on Velocity and Acceleration

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