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.

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.

MEDIUM

AS and A Level

IMPORTANT

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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.

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 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 $10 s .$

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.

Jenny is $30 m$ north of home and walks at a constant speed of $1.5 m s^{- 1}$ until reaching home.

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.

Ryo is sitting still at a point $10 m$ south of his home.

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.

Nina is $300 m$ north of her home. She drives south at a constant speed of $10 m s^{- 1},$ passing her home, until she has travelled a total of $500 m .$

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 1 - Velocity and Acceleration>EXERCISE 1D>Q 2

Sketch the displacement-time graphs from the information given. In each case consider upwards to be the positive direction and ground level 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 firework takes off from ground level, accelerating upwards for $10 s$ with constant acceleration $4 m s^{- 2} .$

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 1 - Velocity and Acceleration>EXERCISE 1D>Q 2

A ball is thrown upwards from a point $1 m$ above the ground with initial speed $5 m s^{- 1} .$ It accelerates downwards at a constant rate of $10 m s^{- 2}$ until it stops moving upwards, when it is caught by someone standing on a ladder.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 1 - Velocity and Acceleration>EXERCISE 1D>Q 2

A rocket is falling at $10 m s^{- 1}$ at a height of $100 m$ above the ground when its engines turn on to provide a constant acceleration of $2 m s^{- 2}$ upwards. The engines remain on until the rocket has reached a height of $175 m$ above ground level.

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+v2 at time t=12t, that is at the time halfway through the motion the velocity of the object is the mean of the initial and final velocities.

Important Questions on Velocity and Acceleration

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