Momentum & Collisions (Cambridge (CIE) A Level Maths: Mechanics): Exam Questions

Two objects $A$ and $B$ have masses 0.6 kg and 1.4 kg respectively. They are moving towards each other in opposite directions in a straight line on a smooth horizontal plane. The objects collide directly and coalesce to form an object $C$. Immediately before the collision, $A$ is moving with speed 8 m s^-1 and $B$ is moving with speed 6 m s^-1.

By modelling the objects as particles, find the speed and direction of motion of $C$ after the collision.

At the instance that object $C$ is moving with speed $um{s}^{-1}$, the magnitude of its momentum is 13 N s.

Find the value of $u$.

Two marbles $M$ and $N$ have masses 3 kg and $m$ kg respectively. The marbles are rolled towards each other in a straight line on a smooth horizontal table and they collide directly bringing $M$ to rest. Immediately before the collision, the speeds of $M$ and $N$ are $3m{s}^{-1}$ and $5m{s}^{-1}$ respectively. Immediately after the collision, $N$ moves with speed $4m{s}^{-1}$. The marbles are modelled as particles.

Find the value of $m$.

Two particles, $P$ and $Q$, with masses $3m$ and $5m$ respectively are moving towards each other in opposite directions along the same straight line on a smooth horizontal plane. They collide directly. Immediately before the collision, the speed of $P$ is $3u$ and the speed of $Q$ is $2u$. Immediately after the collision, the speed of $P$ is halved and its direction of motion is reversed.

Find the speed of $Q$ immediately after the collision.

State whether the direction of motion of $Q$ is changed immediately after the collision.

Two small balls $A$and $B$ have masses $m$ kg and 1.5 kg respectively. They are moving in the same direction along the same straight line on a smooth horizontal table when they collide directly. Immediately before they collide, $A$is moving with speed $8m{s}^{-1}$ and $B$ is moving with speed $3m{s}^{-1}$. Immediately after they collide, $A$ is moving with speed $1m{s}^{-1}$ and $B$ is moving with speed $4m{s}^{-1}$. The direction of motion of $B$ is unchanged after the collision.

Find the value of $m$ in the case that:

(i) the direction of motion of $A$ is unchanged after the collision.

(ii) the direction of motion of $A$ is reversed after the collision.

Amber works at a circus and routinely performs the human cannonball act. The act involves Amber, mass 65 kg, being ejected horizontally out of a specially designed cannon, mass 300 kg, into a net 100 metres away. Both the cannon and Amber start from rest before Amber is shot out at a speed of 108 km h^-1.

(i) Find the speed of the cannon after ejecting Amber.

(ii) Show that the cannon recoils after ejecting Amber.

Amber is travelling with a horizontal speed of 26 m s^-1 when she hits the net. The horizontal resistance force is assumed to be constant.

Find the time it takes Amber to reach the net after being ejected from the canon.

Calculate the magnitude of the resistance force.

Two spheres $A$ and $B$ are of equal radius and have masses $5kg$ and $mkg$ 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 $12m{s}^{-1}$ and $2m{s}^{-1}$ respectively. Immediately after the collision, the speed of $A$ is $2m{s}^{-1}$, the speed of $B$ is $4m{s}^{-1}$ and the direction of motion of $B$ is unchanged.

Find the two possible values of $m$.

Show that the minimum total kinetic energy lost in the collision is $140\mathrm{J}.$

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 $0.1kg,0.3kg$ and $0.5kg$ respectively and $A$ moves with speed $12m{s}^{-1}$ while $B$ and $C$ are at rest. $A$ and $B$ collide directly which does not change the direction of motion of $A$ and subsequently $A$ moves with speed $1.5m{s}^{-1}.$

Show that the speed of $B$ immediately after it collides with $A$ is $3.5m{s}^{-1}.$

In the subsequent motion, $B$ collides directly with $C$. Immediately after this collision, $C$ moves with speed $1.5m{s}^{-1}.$

Show that there will be another collision between $A$ and $B$.

Given that A and B coalesce in this collision,

show that the speed of the combined spheres is $1.125m{s}^{-1},$

show that the total loss of kinetic energy due to the three collisions is $6.38\mathrm{J}$ correct to 3 significant figures.

Two objects $P$ and $Q$ have masses 0.6 kg and $m$ kg respectively. They are moving towards each other in opposite directions in a straight line on a smooth horizontal surface. The objects collide directly and coalesce to form an object $R$. Immediately before the collision, $P$ is travelling with speed $10m{s}^{-1}$ and $Q$ is travelling with speed $15m{s}^{-1}.$ The speed of $R$ after the collision is $12m{s}^{-1}.$

Find the value of $m$.

There is a constant resistance force acting on $R$ which causes $R$ to come to rest after travelling 120 m.

Find the magnitude of the resistance force acting on $R$.

Two particles, $A$ and $B$, with masses $7m$ and $4m$ respectively are moving towards each other in opposite directions along the same straight line on a smooth horizontal plane when they collide directly. Immediately before the collision, the speed of $A$is $3u$and the speed of $B$ is $5u$. Immediately after the collision, the speed of $A$is $v_A$ and the speed of $B$ is $v_B$.

The direction of motion of $A$is reversed after the collision.

Show that $v_B>\frac{u}{4}$.

State whether the direction of motion of $B$ is reversed after the collision.

Given that $v_B$ is double $v_A,$ find the values of $v_A$ and $v_B.$

Lucy and Fran are driving monster trucks along the same straight line in opposite directions towards each other. The combined mass of Lucy and her truck is 5400 kg and the combined mass of Fran and her truck is 4700 kg. They collide directly. Immediately before the collision, Lucy’s truck is moving with speed $4m{s}^{-1}$ and Fran’s truck is moving with speed $5m{s}^{-1}$. Immediately after the collision, Lucy and Fran’s monster trucks are moving at the same speed.

(i) Given that Lucy and Fran’s monster trucks are moving in the same direction after the collision, find the speed at which they both travel.

(ii) State which person’s direction of motion has been reversed.

(iii) Calculate the change in momentum of Fran’s monster truck due to the collision.

Given instead that Lucy and Fran’s monster trucks bounce off each other and move in opposite directions with equal speeds after the collision, calculate the speed at which they both travel.

A ball of mass 2 kg is projected along a rough horizontal floor in a straight line with speed $8m{s}^{-1}.$ Initially the ball is 20 m away from the wall as shown in the diagram above. The coefficient of friction between the ball and the floor is 0.1.

Find the momentum of the ball immediately before it hits the wall.

The ball rebounds from the wall and travels for $3s$ before coming to rest.

Find the momentum of the ball immediately after it hits the wall.

Two particles $P$ and $Q$, with masses 0.4 kg and 2.4 kg respectively, are moving in the same direction along a straight line on a smooth horizontal plane. At the instant when $P$ has speed $10m{s}^{-1}$ and $Q$ has speed $um{s}^{-1}$, the two particles collide directly. Immediately after the collision, the two particles are moving in opposite directions and the speed of $P$ is half of the speed of $Q$.

Show that the speed of $Q$ immediately after the collision is $\frac{2}{11}(10+6u)m{s}^{-1}$

After the collision $Q$ continues travelling in the same straight line for 3 seconds before being brought to rest by a constant resistive force of magnitude 3.2 N.

Calculate the value of $u$.

Paul, a teenager of mass $50kg$, is skating in a straight line on a skateboard of mass $5kg$. In the instant that Paul and the skateboard are travelling at $3m{s}^{-1}$, Paul jumps forward off the skateboard. Paul’s speed is $4m{s}^{-1}$ immediately after he jumps off the skateboard.

(i) Calculate the speed of the skateboard immediately after Paul jumps off.

(ii) State whether the skateboard continues to move in the same direction as Paul.

Three small balls $A,B$ and $C$ of equal radius have masses $0.2kg,$ $0.3kg$ and $1.5kg$ 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 and collide directly when they are travelling with speeds $11m{s}^{-1}$ and $2m{s}^{-1}$ respectively. Immediately after the collision, the speed of $B$ is twice the speed of $A.$

Find the two possible values for the speed of $B$ immediately after the collision.

Immediately after the collision the speed of $B$ is the larger of the two values found in part (a). In the subsequent motion, $B$ collides with $C$ which is at rest. After the collision, $C$ moves with speed $vm{s}^{-1}.$

Given that there are no further collisions between $A$ and $B$, find the maximum value of $v.$

Two objects A and B have masses $10m$ and $7m$ respectively. The objects are moving towards each other in opposite directions along the same straight line on a smooth horizontal plane. The objects collide directly which causes the direction of motion of A to be reversed.

Immediately before the objects collide, the speed of A is $5u$ and the speed of B is $\lambda u$, where $\lambda$ is a constant.

Immediately after the objects collide, the speed of A is $3u$ and the speed of B is $2u$.

Find the two possible values of $\lambda$.

Two particles P and Q move towards each other in opposite directions along a straight line on a smooth horizontal surface. The mass of P is $9m$ and the mass of Q is $14m$. They collide directly. Immediately before the collision, P moves with speed $16u$ and Q moves with speed $7u$. Immediately after the collision the directions of P and Q are reversed and P has speed $v_p$ and Q has speed $v_Q$.

Show that $v_Q>\frac{23}{7}u$.

After the collision, both particles decelerate at a constant rate of $2\mathrm{m}$ $s^{-2}$. Particle P takes $0.3u$ seconds longer to come to rest than particle Q.

Find the values of $v_p$ and $v_Q$.

Harry drops a cricket ball, mass 0.16 kg, from a height 2 m above the ground. The only force acting on the cricket ball as it falls is the force due to gravity. The ball falls in a straight vertical line, hits the ground and bounces back upwards still following the same straight line. Immediately after the ball hits the ground, the magnitude of its momentum decreases by $0.584\mathrm{kg}\mathrm{m}{\mathrm{s}}^{-1}$.

Find the greatest height above the ground reached by the cricket ball after it bounces off the ground.

An empty car of mass $m$ kg is stopped in the middle of a road. A lorry of mass 3800 kg is driving at a constant speed of $48\mathrm{km}{\mathrm{h}}^{-1}$ when it collides with the car. The magnitude of the momentum of the car immediately after the collision is $12500\mathrm{kg}\mathrm{m}{\mathrm{s}}^{-1}$.

Given that the speeds of the lorry and the car are equal immediately after the collision, find their common speed and find the value of $m$.

In this question, take $g=10\mathrm{m}{\mathrm{s}}^{-2}$.

AB is a line of greatest slope, of length 4 m, on a smooth plane inclined at $40°$ to the horizontal. Particle P of mass 0.4 kg is projected down the slope from the point A and particle Q of mass 0.33 kg is projected up the slope from the point B. When P has speed $1\mathrm{m}{\mathrm{s}}^{-1}$ and Q is moving up the slope with speed $5\mathrm{m}{\mathrm{s}}^{-1}$, the particles collide directly at a point 1 m down the slope from A, as shown in the diagram above. Immediately after the collision, the direction of motion of Q is reversed and it has speed $1\mathrm{m}{\mathrm{s}}^{-1}$.

Show that P reaches the top of the slope, A, in the subsequent motion.

Determine, with calculations, whether P reaches A before Q reaches B.

Armina and Dillon are two armadillos with masses 5 kg and $m$ kg respectively. They curl into balls and roll towards each other in opposite directions along a straight line on smooth ground. They collide directly which causes Armina's direction of motion to be reversed. Immediately before the collision the speed of Armina is $3\mathrm{m}{\mathrm{s}}^{-1}$ and the speed of Dillon is $6\mathrm{m}{\mathrm{s}}^{-1}$. Immediately after the collision, the speed of Armina is $2.6\mathrm{m}{\mathrm{s}}^{-1}$ and the speed of Dillon is $2\mathrm{m}{\mathrm{s}}^{-1}$. Armina and Dillon are modelled as particles.

Find the two possible values for the loss of kinetic energy in the collision.

After the collision, Armina continues to roll with a constant speed until she reaches some rough ground. Once Armina enters the rough ground she continues to roll for 2 seconds before coming to rest

Find the coefficient of friction between Armina and the rough ground.

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

(i) 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.

R, S and T are three small marbles of equal radius and masses 0.4 kg, 0.2 kg and 0.1 kg 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 $3\mathrm{m}{\mathrm{s}}^{-1}$ and $1\mathrm{m}{\mathrm{s}}^{-1}$ respectively and collide directly. At the instant that R and S collide, T is projected towards S with a speed of $1\mathrm{m}{\mathrm{s}}^{-1}$. In the subsequent motion, R travels towards T with a speed of $v\mathrm{m}{\mathrm{s}}^{-1}$ and S travels towards T with a speed of $2v\mathrm{m}{\mathrm{s}}^{-1}$. S and T collide directly and T rebounds with speed $x\mathrm{m}{\mathrm{s}}^{-1}.$

Given that there are no further collisions, find the maximum value of $x$.