Show that an object accelerating with acceleration a from initial velocity u to final velocity v , where 0 < u < v , over a time t is travelling at a velocity of u + v 2 at time t = 1 2 t , that is at the time halfway through the motion the velocity of the object is the mean of the initial and final velocities.

Given initial velocity is u , the final velocity is v , acceleration a acts over the time interval t By definition, velocity can be written as v = s t Using the kinematic equation v 2 - u 2 = 2 a s s = v 2 - u 2 2 a Using identity a 2 - b 2 = ( a + b ) ( a - b ) s = ( v + u ) ( v - u ) 2 a By definition, we can write a = v - u t Displacement at t = t 2 s = ( v + u ) ( v - u ) t 4 ( v - u ) Velocity at t = t 2 v = ( v + u ) ( v - u ) t 4 ( v - u ) × 2 t ⇒ v = v + u 2 Hence proved, at halfway time, the velocity of the object is the mean of initial and final velocity.

EASY

AS and A Level

IMPORTANT

Earn 100

A driver sees the traffic lights change to red $240 m$ away when he is travelling at a velocity of $30 m s^{- 1} .$ To avoid wasting fuel, he does not brake, but lets the car slow down naturally. The traffic lights change to green after $12 s,$ at the same time as the driver arrives at the lights.

What assumptions have been made to answer the question?

Important Questions on Velocity and Acceleration

EASY

AS and A Level

IMPORTANT

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

In a game of curling, competitors slide stones over the ice at a target

38 m

away. A stone is released directly towards the target with velocity

4.8 m s^{- 1}

and decelerates at a constant rate of

0.3 m s^{- 2} .

Find how far from the target the stone comes to rest.

EASY

AS and A Level

IMPORTANT

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

A golf ball is struck

10 m

from a hole and is rolling towards the hole. It has an initial velocity of

2.4 m s^{- 1}

when struck and decelerates at a constant rate of

0.3 m s^{- 2} .

Does the ball reach the hole?

MEDIUM

AS and A Level

IMPORTANT

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

A driverless car registers that the traffic lights change to amber

40 m

ahead. The amber light is a

2 s

warning before turning red. The car is travelling at

17 m s^{- 1}

and can accelerate at

4 m s^{- 2}

or brake safely at

8 m s^{- 2} .

What options does the car have?

EASY

AS and A Level

IMPORTANT

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

The first two equations are $v = u + a t$ and $s = \frac{1}{2} (u + v) t$ . You can use these to derive the other equations. By substituting for $v$ in the second equation, derive $s = u t + \frac{1}{2} a t^{2} .$

EASY

AS and A Level

IMPORTANT

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

The first two equations are $v = u + a t$ and $s = \frac{1}{2} (u + v) t .$ You can use these to derive the other equations.Derive the remaining two equations, $s = v t - \frac{1}{2} a t^{2}$ and $v^{2} = u^{2} + 2 a s,$ from the original two equations.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 1 - Velocity and Acceleration>EXERCISE 1C>Q 15

Show that an object accelerating with acceleration

a

from initial velocity

u

to final velocity

v,

where

0 < u < v,

over a time

t

is travelling at a velocity of

\frac{u + v}{2}

at time

t = \frac{1}{2} t

, that is at the time halfway through the motion the velocity of the object is the mean of the initial and final velocities.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 1 - Velocity and Acceleration>EXERCISE 1C>Q 16

Show that an object accelerating with acceleration

a

from velocity

u

to velocity

v,

where

0 < u < v,

over a displacement

s

is travelling at a speed of

\sqrt{\frac{v^{2} + u^{2}}{2}}

at a distance

\frac{1}{2} s .

Hence, prove that when the object does not change direction the speed at the midpoint of the distance is always greater than the mean of the initial and final speeds. Deduce also that the mean of the initial and final speeds occurs at a point closer to the start of the motion than the end.

MEDIUM

AS and A Level

IMPORTANT

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

Sketch the displacement-time graphs from the information given. In each case, consider north to be the positive direction and home to be the point from which displacement is measured.

Bob leaves his home and heads north at a constant speed of $3 m s^{- 1}$ for $10 s .$

Important Questions on Velocity and Acceleration

Learn and Prepare for any exam you want !