4.1 Collisions

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3 marks

A particle of mass 2.5 kg is moving with velocity $(3 i + 5 j) m s^{- 1}$ when it receives an impulse $(20 i - 15 j)$ N s.

Find the velocity of the particle immediately after receiving the impulse.

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2 marks

Find the magnitude of the impulse.

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1 mark

The velocity $v$ m s^-1 of a particle $P$ , after $t$ seconds, is given by

$v = (3 t - 4) i + (t^{2} + 1) j .$

Find the velocity of $P$ when $t = 2$ .

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3 marks

The mass of particle $P$ is 1.25 kg. When $t = 2$ , the particle $P$ receives an impulse $(3 i - 7 j)$ N s.

Show that $P$ is moving with velocity $(4.4 i - 0.6 j)$ m s^-1 immediately after it receives the impulse.

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2 marks

Find the speed of $P$ immediately after it receives the impulse

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6 marks

Two particles $P$ and $Q$ are moving in a straight line on a smooth horizontal plane with velocities $u_{1}$ and $u_{2}$ respectively. The two particles collide directly. Immediately after the collision, the velocities of $P$ and $Q$ are $v_{1}$ and $v_{2}$ respectively.

The coefficient of restitution between $P$ and $Q$ is given by

$e = \frac{Speed of separation}{Speed of approach} = \frac{v_{2} - v_{1}}{u_{1} - u_{2}} .$

Find the coefficient of restitution for each of the scenarios:

Immediately before the collision,

P

and

Q

are moving in the same direction with speeds 5 m s^-1 and 2 m s^-1 respectively. Immediately after the collision,

P

and

Q

are moving in the same direction with speeds 3 m s^-1 and 4 m s^-1 respectively.

ii)

Immediately before the collision,

P

and

Q

are moving in opposite directions with speeds 5 m s^-1 and 2 m s^-1 respectively. Immediately after the collision,

P

and

Q

are moving in opposite directions with speeds 3 m s^-1 and 4 m s^-1 respectively.

iii)

Immediately before the collision,

P

and

Q

are moving in opposite directions with speeds 5 m s^-1 and 2 m s^-1 respectively. Immediately after the collision,

P

and

Q

move with speeds 3 m s^-1 and 4 m s^-1 respectively. The direction of motion of

P

is unchanged but the direction of motion of

Q

is reversed.

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4 marks

Two small, smooth spheres of equal size, $M$ and $N$ , have masses 2 kg and 3kg respectively. They are travelling in the same direction along the same straight line when they collide directly. Immediately before the collision, the speed of $M$ is 8 m s^-1 and the speed of $N$ is 2 m s^-1. Immediately after the collision, the speed of N is 5 m s^-1.

Show that the speed of

M

immediately after the collision is 3.5 m s^-1.

ii)

State whether the direction of motion of

M

is unchanged in the collision.

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2 marks

Show that the coefficient of restitution between the particles $M$ and $N$ is $\frac{1}{4}$ .

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4 marks

Show that

M

loses 51.75 J of kinetic energy in the collision,

ii)

N

gains 31.5 J of kinetic energy in the collision.

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2 marks

A small ball of mass 0.6 kg is moving with constant speed 5 m s^-1 in a straight line on a smooth horizontal plane. It collides directly with a smooth vertical wall which is perpendicular to the direction of motion. The ball rebounds with speed 3 m s^-1 along the same straight line.

Calculate the coefficient of restitution between the ball and the wall.

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2 marks

Calculate the loss in kinetic energy of the ball in the collision.

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2 marks

Two spheres $A and B$ , with equal radii, are moving in the same direction along the same straight line on a smooth horizontal ground when they collide directly. The directions of motion of $A and B$ are unchanged by the collision.

Immediately before the collision, $A$ has speed 9 m s^-1 and $B$ has speed 1 m s^-1.

Immediately after the collision, $A$ has speed 1.8 m s^-1 and $B$ has speed
5.8 m s^-1.

Calculate the coefficient of restitution between $A$ and $B$ .

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3 marks

After the collision, $B$ continues to move with constant speed 5.8 m s^-1 along the same straight line when it collides directly with another sphere $C$ of equal radius. The coefficient of restitution between $B$ and $C$ is $\frac{3}{4}$ . Before the collision $C$ is at rest and after the collision $C$ is moving with speed 2 m s^-1.

Calculate the speed of

B

immediately after the collision with

C

ii)

State whether the collision between

B

and

C

changes the direction of motion of

B

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1 mark

Explain why there will be another collision between $A$ and $B$ .

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3 marks

Two small balls of equal radius, $M$ and $N$ , have masses 1 kg and 4 kg respectively. $M$ and $N$ are moving in the same direction along a straight line on a smooth horizontal surface with speeds 5 m s^-1 and 1 m s^-1 respectively when they collide directly. The coefficient of restitution between $M$ and $N$ is $\frac{3}{4}$ . Taking the initial direction of motion as the positive direction, the velocities of $M$ and $N$ after the collision are $v_{M}$ m s^-1 and $v_{N}$ m s^-1 respectively.

Show that $v_{N} - v_{M} = 3$ and use the conservation of linear momentum to find a second equation involving $v_{M}$ and $v_{N}$ .

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3 marks

Show that after the collision between $M$ and $N$ ,

the direction of motion of

M

is reversed and it is moving with speed

\frac{3}{5}

m s^-1.

ii)

N

is moving with speed

\frac{12}{5}

m s^-1.

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1 mark

After $M$ and $N$ collide, $N$ continues to move with constant speed $\frac{12}{5}$ m s^-1 until it hits a smooth vertical wall which is perpendicular to the direction of motion of $N$ . The coefficient of restitution between $N$ and the wall is $e$ .

Show that the speed of $N$ after it collides with the wall is $\frac{12}{5} e$ m s^-1.

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2 marks

Given that there is another collision between $M$ and $N$ , show that $e > \frac{1}{4}$ .

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7 marks

A ball of mass 0.5 kg is moving with velocity $(- 5 i - 7 j)$ m s^-1 when it is kicked by a child. The magnitude of the impulse exerted by the child on the ball is 6.5 N s. Immediately after the child kicks the ball, the velocity of the ball is $(7 i + k j)$ m s^-1.

Find the two possible values of $k$ .

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6 marks

An eagle of mass 5 kg is flying in a horizontal plane with velocity $(6 i)$ m s^-1 when it is spotted by a birdwatcher. The acceleration of the eagle, $a$ m s^-2, is given by

$a = (0.2 t + 0.3) i + (0.1 t^{2}) j,$

where $t$ is the time in seconds since the eagle was spotted by the birdwatcher.

Find the speed of the eagle two seconds after it was spotted by the birdwatcher.

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3 marks

Two seconds after the birdwatcher spotted the eagle, the eagle receives an impulse $(10 i - 11 j)$ N s due to a gust of wind.

Find the velocity of the eagle immediately after the receiving the impulse due to the gust of wind.

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3 marks

Calculate the kinetic energy gained by the eagle by the gust of wind.

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6 marks

q3-hard-4-1-collisions-edexcel-a-level-mechanics

An object $M$ of mass 8 kg is moving in a straight line with speed 20 m s^-1on a smooth horizontal plane. $M$ receives an impulse of magnitude 18 N s which has negative vector components. The impulse forms an acute angle $α °$ , where $\tan α = \frac{7}{24}$ , below the initial direction of motion, as shown in the diagram above.

Find the speed of

M

immediately after the impulse is applied.

ii)

Find the direction of motion of

M

immediately after the impulse is applied.

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7 marks

Two particles $P and Q$ with masses 0.5 kg and 0.9 kg respectively are moving towards each other in opposite directions along a straight line on a smooth horizontal surface. They collide directly. Immediately before the collision, $P$ has speed 5 m s^-1 and $Q$ has speed 7 m s^-1. The coefficient of restitution between $P and Q$ is $\frac{3}{8}$ .

Find the speed of $P$ and the speed of $Q$ immediately after the collision. Clearly state the direction of motion of each particle.

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3 marks

Calculate the kinetic energy lost in the collision.

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1 mark

Give one form of energy into which the lost kinetic energy could have been transformed.

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5 marks

Two smooth spheres $S and T$ with the same radii have masses $4 m$ and $7 m$ respectively. The spheres are moving in opposite directions along a straight line on a smooth horizontal table when they collide directly. Immediately before the collision, the speed of $S$ is $5 u$ and the speed of $T$ is $2 u$ . The collision causes the directions of motion to be reversed for both $S$ and $T$ . The coefficient of restitution between $S and T$ is $e$ .

By modelling the spheres as particles, show that the speed of $T$ immediately after the collision is $\frac{2}{11} (3 + 14 e) u .$

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4 marks

Find the range of possible values of $e$ .

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2 marks

Given that the speed of $T$ immediately after the collision is $\frac{34}{11} u$ , explain why there is no change in the total kinetic energy before and after the collision.

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5 marks

A small marble of mass 300 g is projected vertically upwards with speed 10 m s^-1 from a height 1.5 m above a smooth horizontal ground. The marble rebounds from the ground and travels upwards for half a second before instantaneously coming to rest.

Find the coefficient of restitution between the marble and the ground.

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6 marks

Three small balls $A, B and C$ of equal radius have masses $2 m, 3 m and 7 m$ respectively. The three balls lie in the same straight line on a smooth horizontal ground with $B$ in the middle of $A and C$ . $A and B$ are projected towards each other in opposite directions with speeds $5 u$ and $3 u$ respectively. $A and B$ collide directly which causes the direction of motion of $B$ to reverse. The coefficient of restitution between $A and B$ is $e$ .

Show that the speed of

B

immediately after the collision with

A

\frac{1}{5} (1 + k e) u

, where

k

is a constant to be found.

ii)

Hence state the maximum speed of

B

immediately after the collision with

A

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5 marks

After $A and B$ collide, $B$ moves with constant speed towards $C$ which is at rest. $B and C$ collide directly and the coefficient of restitution between $B and C$ is $\frac{5}{9}$ .

Show that the speed of $B$ immediately after the collision with $C$ is $\frac{4}{225} (1 + 16 e) u$ and that the direction of motion of $B$ is reversed.

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3 marks

Given that $e = \frac{1}{20}$ , determine whether there will be another collision between $A and B$ .

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6 marks

Two particles $S and T$ have masses $3 m and 7 m$ respectively. $S$ is moving with speed $6 u$ along a straight line on a smooth horizontal surface towards $T$ . $T$ is moving along the same straight line towards $S$ with speed $u$ . $S$ and $T$ collide directly. The coefficient between S and $T$ is $e$ .

Show that the speed of

T

immediately after the collision is

\frac{1}{10} (11 + 21 e) u

ii)

State whether it is possible for the collision to cause

T

to come to rest.

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6 marks

After the collision, both particles continue with constant speed. In the subsequent motion $T$ collides with a fixed vertical wall which is perpendicular to the direction of motion of $T$ . The coefficient of restitution between $T$ and the wall is $f$ .

Given that there is a second collision between $S and T$ ,

find the range of possible values of

e

in the case that

f = 0

ii)

find the range of possible values of

f

in the case that

e = 0.5

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4 marks

Kelsie is playing squash. The squash ball, of mass 0.025 kg, is moving with velocity $(- 2 i + 3 j)$ m s^-1 when Kelsie strikes it with her squash racket. Immediately after being struck, the ball has velocity $(23 i + 15 j)$ m s^-1.

Find the magnitude of the impulse exerted by the racket on the ball.

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2 marks

Find the angle between the vector $i$ and the impulse exerted by the racket. Give your answer to 1 decimal place.

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5 marks

A mouse of mass 0.8 kg is running on a smooth horizontal surface and its position vector, $r$ metres, at time $t$ seconds is given by

${r = (2 t - 0.1 t^{3}) i + (5 t - 7) j}^{}$

Calculate the speed of the mouse when $t = 4$ .

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3 marks

When $t = 4,$ the mouse is pushed by a cat. The mouse receives an impulse $(8 i - 3 j) N$ s from the cat.

Find the velocity of the mouse immediately after it has been pushed by the cat.

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4 marks

q3-medium-4-1-collisions-edexcel-a-level-mechanics

A particle $P$ of mass 6 kg is moving with speed 12 m s^-1 in a horizontal plane when it receives an impulse. Immediately after receiving the impulse, $P$ is moving with speed 25 m s^-1 at an angle of 45° to the horizontal as shown in the diagram above.

Find the magnitude of the impulse.

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2 marks

Find the acute angle between the direction of the impulse and the horizontal.

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3 marks

Two spheres $A$ and $B$ are of equal radius and have masses 5 kg and $m$ kg respectively. $A$ and $B$ are moving in the same direction in a straight line on a smooth horizontal surface when they collide directly. Immediately before the collision, $A$ and $B$ are moving with speeds 12 m s^-1 and 2 m s^-1 respectively. Immediately after the collision, the speed of $B$ is 4 m s-1 and the direction of motion is unchanged. The coefficient of restitution between $A$ and $B$ is $\frac{3}{5}$ .

Show that the speed of

A

immediately after the collision is 2 m s^-1.

ii)

State whether the direction of motion of

A

is changed in the collision.

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3 marks

Find the value of $m$ .

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3 marks

Show that the total kinetic energy lost in the collision is 140 J.

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6 marks

A smooth sphere $P$ of mass $2 m$ is moving along a straight line on a smooth horizontal surface when it collides with another smooth sphere $Q$ of mass $3 m$ . The spheres collide directly. Immediately before the collision, $P$ has speed $u$ and $Q$ is at rest. Immediately after the collision, the direction of motion of $P$ is unchanged. The coefficient of restitution between $P$ and $Q$ is $e$ . The spheres can be modelled as particles with the same radii.

Show that the speed of

P

immediately after the collision is

\frac{1}{5} (2 - 3 e) u

ii)

Show that the speed of

Q

immediately after the collision is

\frac{2}{5} (1 + e) u

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2 marks

Find the range of possible values of $e$ .

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5 marks

A small ball of mass 400 g is rolling along a straight line on a smooth horizontal ground towards a smooth vertical wall perpendicular to its direction of motion. The magnitude of the impulse exerted on the ball by the wall is 9.3 N s. The coefficient of restitution between the ball and the wall is 0.55.

Find the speeds of the ball immediately before and after colliding with the wall.

ii)

Find the loss in kinetic energy in the collision.

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6 marks

Three uniform spheres $A, B and C$ of equal radius move in the same direction on the same straight line on a smooth horizontal table with $B$ in the middle of $A$ and $C$ . The masses of $A, B$ and $C$ are $m, 3 m and 5 m$ respectively and $A$ moves with speeds $12 u$ while $B and C$ are at rest. $A and B$ collide directly which does not change the direction of motion of $A$ . The coefficient of restitution between $A and B$ is $e$ .

Show that the speed of $B$ immediately after it collides with $A$ is $3 (1 + e) u$ .

Show that the speed of $A$ immediately after it collides with $B$ is $3 (1 - 3 e) u$ .

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5 marks

In the subsequent motion, $B$ collides directly with $C$ . The coefficient of restitution between $B$ and $C$ is $\frac{1}{2}$ .

Given that $e = \frac{1}{6}$ ,

show that the speed of

B

immediately after colliding with

C

\frac{7}{32} u

ii)

Explain why there will be a second collision between

A and B

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6 marks

Two particles $P and Q$ , with masses $5 m$ and $2 m$ respectively, are moving towards each other in opposite directions along a straight line on a smooth horizontal plane with speeds $4 u$ and $u$ respectively. $P and Q$ collide directly and immediately afterwards the direction of motion of $Q$ is reversed and it moves with speed $3 u$ .

Find the speed of

P

immediately after the collision.

ii)

Calculate the coefficient of restitution between

P and Q

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4 marks

After colliding with $P, Q$ hits a fixed vertical wall that is perpendicular to the direction of motion of $Q$ . The magnitude of the impulse on the wall on $Q$ is $\frac{34}{5} m u .$

Calculate the coefficient of restitution between $Q$ and the wall.

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1 mark

State, with a reason, whether there will be another collision between $P and Q$ .

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1 mark

Two particles $A and B$ have masses 2 kg and 3 kg respectively. $A and B$ are moving along the same straight line on a smooth horizontal plane. The velocity of $A$ is $(3 i - 2 j)$ m s^-1 and the velocity of $B$ is $(λ i + 10 j)$ m s^-1.

Write down the value of $λ$ .

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3 marks

$A and B$ collide directly. The velocity of $A$ immediately after the collision is $(- 6 i + 4 j)$ m s^-1.

Find the velocity of $B$ immediately after the collision.

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4 marks

Calculate the magnitude of the impulse exerted on $A$ by $B$ in the collision.

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6 marks

A particle of mass 4 kg is moving with velocity $(- 4 i + 10 j)$ m s^-1 when it receives an impulse $Q$ N s. Immediately after receiving the impulse, the particle moves with velocity $(2 k i + (k - 2) j)$ m s^-1, where $k$ is a constant.

Given that the magnitude of $Q$ is $8 \sqrt{145}$ , show that one possible value of $k$ is 10 and find the other possible value.

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3 marks

Show that when

k = 10

the directions of motion of the particle, before and after receiving the impulse, are perpendicular.

ii)

For

k = 10

, find the impulse in the form

Q = a i + b j .

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9 marks

A force $F$ N acts on a particle $P$ of mass 5 kg. At time $t$ seconds, $F = (2 t^{3}) i + (2 - 10 t) j$ . When $t = 1$ , the velocity of $P$ is $(i - 3 j)$ m s^-1. When $t = k$ , $P$ receives an impulse $(20 i + 41 j)$ N s and immediately after this impulse the velocity of $P$ is $(13 i - 2 j)$ m s^-1.

Find the value of $k$ .

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7 marks

q3-vhard-4-1-collisions-edexcel-a-level-mechanics

Particle $P$ , of mass 1.2 kg, is moving in a straight line with a speed 3 m s^-1 on a smooth horizontal surface. $P$ receives an impulse of magnitude $I$ N s which immediately causes $P$ to move in a different direction with speed 7 m s^-1. The angle between the initial direction of motion of $P$ and the direction of the impulse is 30° as shown in the diagram above.

Find the value of $I$ .

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3 marks

Find the acute angle formed by the directions of $P$ before and after receiving the impulse.

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12 marks

A particle $P$ of mass 5 kg is travelling along a straight line on a smooth horizontal surface with constant speed 3 m s^-1. It collides directly with particle $Q$ which is moving along the same straight line. Immediately after the collision, the direction of motion of $P$ is reversed and its speed is 1 m s^-1. The kinetic energy lost in the collision is 60 J and the coefficient of restitution between $P$ and $Q$ is $\frac{2}{5}$ .

Calculate the speed of

Q

immediately before the collision. Clearly state the direction of motion.

ii)

Calculate the speed of

Q

immediately after the collision. Clearly state the direction of motion.

iii)

Calculate the mass of particle

Q

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8 marks

Two balls $A and B$ of masses $8 m$ and $5 m$ are rolled towards each other in opposite directions along a straight line on a smooth horizontal ground. The balls are the same size and they collide directly. Immediately before they collide, $A$ moves with speed $3 u$ and $B$ moves with speed $4 u$ . Immediately after they collide, both particles move in the same direction and the speed of $B$ is more than double the speed of $A$ . The coefficient of restitution between $A and B$ is $e$ .

Find the range of possible values of $e$ .

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4 marks

Given that $e = \frac{1}{14},$ show that the loss of kinetic energy in the collision is $75 m u ²$ .

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6 marks

John projects a bouncy ball vertically downwards with speed 2 m s^-1 from a height 5 m above a smooth horizontal ground. The coefficient of restitution between the ball and the ground is $\frac{1}{3}$ . Once the ball hits the ground, it rebounds and follows the same straight line. The ball is modelled as a particle and it is assumed that there are no external forces acting on the ball.

Show that the total distance travelled by the ball is approximately 6.3 m.

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11 marks

$R, S and T$ are three small marbles of equal radius and masses $4 m, 2 m$ and $m$ respectively. The three marbles lie in the same straight line on a smooth horizontal floor with $S$ in the middle. $R and S$ are projected towards each other with speeds $3 u and u$ respectively. $R and S$ collide directly and the coefficient of restitution between $R and S$ is e. At the instant that $R and S$ collide, $T$ is projected towards $S$ with a speed of $u$ . $S and T$ collide directly and this collision does not change the total kinetic energy between $S and T$ .

Given that there is a second collision between $R and S$ , find the range of possible values of $e$ .

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8 marks

Two particles $P and Q$ have masses $k m$ and $30 m$ respectively, where $k$ is a constant. $P and Q$ are moving in the same direction along a straight line on a smooth horizontal plane when they collide directly. Immediately before the collision, $Q$ has speed $2 u$ . Immediately after the collision, the direction of motion of $P$ is reversed and it moves with speed $2 u$ . The speed of $Q$ is $4 u$ . The kinetic energy lost in the collision is $60 m u ²$ .

ii)

Show that

k = 5

ii)

Show that the coefficient of restitution between

P and Q

\frac{3}{4}

How did you do?

5 marks

At the instant $P and Q$ collide, they are a distance $x$ from a fixed vertical wall perpendicular to the direction of the motion of the particles. After colliding with $P, Q$ continues with constant speed $4 u$ until it hits the wall and rebounds. A second collision between $P and Q$ occurs a distance $4 x$ from the wall.

Find the coefficient of restitution between $Q$ and the wall.

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Edexcel International A Level Maths: Mechanics 2

Topic Questions

4.1 Collisions