Momentum & Collisions (Cambridge International AS Mathematics): Exam Questions

Three smooth spheres $P$, $Q$ and $R$, with masses 20 kg, 35 kg and $m$ kg respectively, are travelling in the same direction along a straight line with speeds 7 $\text{m s}^{-1}$, $u \text{m s}^{-1}$ and 10 $\text{m s}^{-1}$ respectively.

Given that $P$, $Q$ and $R$ all have the same momentum, find

(i) the value of $m$,

(ii) the value of $u$.

Given that there are no collisions between any of $P$, $Q$ and $R$, state which sphere is travelling at the front, and which is travelling at the back.

A ball of mass 400 grams is travelling at a speed of 2 $\text{m s}^{-1}$. Calculate the momentum of the ball.

A car of mass $m$ kg travelling at a constant speed of 90 $\text{km h}^{-1}$ has momentum 32 500 $\text{kg m s}^{-1}$. Find the value of $m$.

Two particles $P$ and $Q$, of masses 8 kg and 5 kg respectively, lie on a smooth horizontal surface. $Q$ is at rest and $P$ is moving directly towards $Q$ with speed 3 $\text{m s}^{-1}$. The particles collide directly. Immediately after the collision, $P$ continues to move in the same direction with speed 1 $\text{m s}^{-1}$.

Use the principle of conservation of linear momentum to find the speed of $Q$ immediately after the collision.

Find the loss of kinetic energy of the system due to the collision.

Two particles $P$ and $Q$, of masses 10 kg and 16 kg respectively, are moving in the same direction along the same straight line on a smooth horizontal surface when they collide directly. Immediately before they collide, $P$ is moving with speed 13 $\text{m s}^{-1}$ and $Q$ is moving with speed 1 $\text{m s}^{-1}$. Immediately after they collide, $Q$ is moving with speed 10 $\text{m s}^{-1}$.

Explain how you know that the direction of motion of $Q$ does not change as a result of the collision.

(i) Find the speed of $P$ immediately after the collision.

(ii) Explain how you know that the direction of motion of $P$ has been reversed.

Two particles $A$ and $B$, of masses 3 kg and $m$ kg respectively, are moving towards each other along a straight line on a smooth horizontal surface. Immediately before they collide, the speed of $A$ is 6 $\text{m s}^{-1}$ and the speed of $B$ is 5 $\text{m s}^{-1}$. The particles collide directly. Immediately after the collision, $A$ continues to move in the same direction with speed 4 $\text{m s}^{-1}$, and the direction of motion of $B$ is reversed and its speed is 3 $\text{m s}^{-1}$.

Show that $m =$ 0.75.

Find the change in momentum of $A$ due to the collision.

Safa and Adam each roll a ball towards the other along the same straight line on a smooth horizontal table. Adam's ball has mass 0.8 kg. Immediately before the balls collide, Safa's ball has speed 6 $\text{m s}^{-1}$ and Adam's ball has speed 3 $\text{m s}^{-1}$. Immediately after the collision, the direction of motion of Safa's ball is reversed and its speed is 2 $\text{m s}^{-1}$. As a result of the collision, the magnitude of the momentum of Safa's ball decreases by 2.4 $\text{kg m s}^{-1}$.

Show that the mass of Safa's ball is 0.6 kg.

(i) Find the speed of Adam's ball immediately after the collision.

(ii) State whether the direction of motion of Adam's ball is changed by the collision.

Two small smooth spheres $A$ and $B$, of equal radius, are moving in the same direction along the same straight line on a smooth horizontal surface 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 $\text{m s}^{-1}$ and $B$ has speed 1 $\text{m s}^{-1}$. Immediately after the collision, $A$ has speed 1.8 $\text{m s}^{-1}$ and $B$ has speed 5.8 $\text{m s}^{-1}$.

Given that the loss of kinetic energy of $A$ in the collision is 388.8 J, show that the mass of $A$ is 10 kg.

Show that the mass of $B$ is 15 kg.

After the collision, $B$ continues to move with constant speed 5.8 $\text{m s}^{-1}$ along the same straight line and collides directly with a third small smooth sphere $C$, of mass 30 kg and of the same radius. Immediately before this collision $C$ is at rest, and immediately after it $C$ moves with speed 3 $\text{m s}^{-1}$.

(i) Calculate the speed of $B$ immediately after its collision with $C$.

(ii) State whether the collision between $B$ and $C$ changes the direction of motion of $B$.

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

Two small smooth spheres $A$ and $B$, of equal radius and of masses $m$ kg and 1.5 kg respectively, 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 8 $\text{m s}^{-1}$ and $B$ is moving with speed 3 $\text{m s}^{-1}$. Immediately after they collide, $A$ is moving with speed 1 $\text{m s}^{-1}$ and $B$ is moving with speed 4 $\text{m s}^{-1}$ in its original direction.

Find the exact value of $m$ in each of the following cases.

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

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

Two particles $A$ and $B$, of masses 0.6 kg and 1.4 kg respectively, are moving towards each other along a straight line on a smooth horizontal plane. Immediately before they collide, $A$ has speed 8 $\text{m s}^{-1}$ and $B$ has speed 6 $\text{m s}^{-1}$. The particles collide directly and coalesce to form a particle $C$.

Find the speed and direction of motion of $C$ immediately after the collision.

Write down the momentum of $C$ immediately after the collision.

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

Find the value of $m$.

Two particles $P$ and $Q$, of masses $3 m$ kg and $5 m$ kg respectively, are moving towards each other along the same straight line on a smooth horizontal plane. Immediately before they collide, the speed of $P$ is $3 u \text{m s}^{-1}$ and the speed of $Q$ is $2 u \text{m s}^{-1}$. The particles collide directly. 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 by the collision.

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

Find the two possible values of $m$.

Show that the smaller of the two possible losses of kinetic energy of the system due to the collision is 140 J.

Three small smooth spheres $A$, $B$ and $C$, of equal radius and of masses 0.1 kg, 0.3 kg and 0.5 kg respectively, lie in a straight line on a smooth horizontal table, with $B$ between $A$ and $C$. $B$ and $C$ are at rest and $A$ is moving towards $B$ with speed 12 $\text{m s}^{-1}$. $A$ collides directly with $B$ and, immediately after this collision, $A$ continues to move in the same direction with speed 1.5 $\text{m s}^{-1}$.

Show that the speed of $B$ immediately after this collision is 3.5 $\text{m s}^{-1}$.

$B$ then collides directly with $C$. Immediately after this collision, $C$ moves with speed 1.5 $\text{m s}^{-1}$.

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

Given that $A$ and $B$ coalesce in this collision,

(i) show that the speed of the combined spheres immediately after the collision is 1.125 $\text{m s}^{-1}$,

(ii) show that the total loss of kinetic energy due to the three collisions is 6.38 J, correct to 3 significant figures.

Two particles $P$ and $Q$, of masses 0.6 kg and $m$ kg respectively, are moving towards each other along a straight line on a smooth horizontal surface. Immediately before they collide, $P$ has speed 10 $\text{m s}^{-1}$ and $Q$ has speed 15 $\text{m s}^{-1}$. The particles collide directly and coalesce to form a particle $R$. Immediately after the collision, the speed of $R$ is 12 $\text{m s}^{-1}$.

Find the value of $m$.

After the collision, a constant resistance force is the only horizontal force acting on $R$, and $R$ comes to rest after travelling a distance of 120 m.

Find the magnitude of the resistance force.

Lucy and Fran are driving monster trucks towards each other along the same straight line. The combined mass of Lucy and her truck is 5400 kg and the combined mass of Fran and her truck is 4700 kg. Immediately before the trucks collide, Lucy's truck is moving with speed 4 $\text{m s}^{-1}$ and Fran's truck is moving with speed 5 $\text{m s}^{-1}$. The trucks collide directly. The trucks are modelled as particles.

Immediately after the collision, the two trucks move in the same direction with the same speed.

(i) Find this speed.

(ii) State whose direction of motion has been reversed by the collision.

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

It is given instead that, immediately after the collision, the direction of motion of each truck is reversed and the two trucks have equal speeds.

Calculate this speed.

A ball of mass 2 kg is projected with speed 8 $\text{m s}^{-1}$ directly towards a vertical wall along a rough horizontal floor. The ball is initially 20 m from the wall, as shown in the diagram. The coefficient of friction between the ball and the floor is 0.1. The ball is modelled as a particle.

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

The ball rebounds from the wall and moves directly away from it along the same line. The ball comes to rest 3 s after rebounding from the wall.

Find the momentum of the ball immediately after it rebounds from the wall.

Two particles $A$ and $B$, of masses $10 m$ kg and $7 m$ kg respectively, are moving towards each other along the same straight line on a smooth horizontal plane. Immediately before they collide, the speed of $A$ is $5 u \text{m s}^{-1}$ and the speed of $B$ is $λ u \text{m s}^{-1}$, where $λ$ is a constant. The particles collide directly, and the direction of motion of $A$ is reversed. Immediately after the collision, the speed of $A$ is $3 u \text{m s}^{-1}$ and the speed of $B$ is $2 u \text{m s}^{-1}$.

Find the two possible values of $λ$.

A particle $Q$ of mass $m$ kg is at rest on a smooth horizontal plane. A particle $P$ of mass 0.5 kg is moving on the plane with speed 6 $\text{m s}^{-1}$ directly towards $Q$, and the particles collide directly. Immediately after the collision, $P$ and $Q$ have equal speeds, and the magnitude of the momentum of $Q$ is 1.2 $\text{kg m s}^{-1}$.

Find the common speed of the particles immediately after the collision and find the value of $m$.

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

Find the two possible values of the loss of kinetic energy due to the collision.

After the collision, Armina continues to roll with constant speed until she reaches some rough horizontal ground. She comes to rest 2 seconds after reaching the rough ground.

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

Two particles $A$ and $B$, of masses $7 m$ kg and $4 m$ kg respectively, are moving towards each other along the same straight line on a smooth horizontal plane. Immediately before they collide, $A$ has speed $3 u \text{m s}^{-1}$ and $B$ has speed $5 u \text{m s}^{-1}$. The particles collide directly. Immediately after the collision, the direction of motion of $A$ is reversed, $A$ has speed $v_{A} \text{m s}^{-1}$ and $B$ has speed $v_{B} \text{m s}^{-1}$.

Explain why the direction of motion of $B$ must also be reversed by the collision, and show that $v_{B} > \frac{u}{4}$.

Given that $v_{B} = 2 v_{A}$, find $v_{A}$ and $v_{B}$ in terms of $u$.

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

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

After the collision, $Q$ continues to move along the same straight line. A constant resistive force of magnitude 3.2 N acts on $Q$ from the instant of the collision, and $Q$ comes to rest 3 seconds after the collision.

Calculate the value of $u$.

Three small smooth spheres $A$, $B$ and $C$, of equal radius and of masses 0.2 kg, 0.3 kg and 1.5 kg respectively, lie in a straight line on a smooth horizontal surface, with $B$ between $A$ and $C$. $A$ and $B$ are projected directly towards each other with speeds 11 $\text{m s}^{-1}$ and 2 $\text{m s}^{-1}$ respectively, and collide directly. Immediately after the collision, the speed of $B$ is twice the speed of $A$.

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

It is given that the speed of $B$ immediately after the collision is the larger of the two values found in part (a). $B$ then collides directly with $C$, which is at rest. Immediately after this collision, $C$ moves with speed $v \text{m s}^{-1}$.

Given that there is no subsequent collision between $A$ and $B$, find the greatest possible value of $v$.

Three particles $A$, $B$ and $C$, of masses 0.4 kg, 0.2 kg and 0.8 kg respectively, lie at rest in that order in a straight line on a smooth horizontal plane. The distance between $B$ and $C$ is 4 m. $A$ is projected directly towards $B$ with speed 6 $\text{m s}^{-1}$. After $A$ collides with $B$, $A$ continues to move in the same direction with speed 2 $\text{m s}^{-1}$.

Show that the speed of $B$ immediately after the collision is 8 $\text{m s}^{-1}$.

$B$ then moves directly towards $C$ and collides with it. The two particles coalesce.

Find the speed of the combined particle immediately after this collision.

Find the distance from the original position of $C$ at which $A$ collides with the combined particle.

Two particles $P$ and $Q$, of masses $9 m$ kg and $14 m$ kg respectively, are moving towards each other along a straight line on a smooth horizontal surface. Immediately before they collide, $P$ has speed $16 u \text{m s}^{-1}$ and $Q$ has speed $7 u \text{m s}^{-1}$. The particles collide directly. Immediately after the collision, the directions of motion of $P$ and $Q$ are both reversed, $P$ has speed $v_{P} \text{m s}^{-1}$ and $Q$ has speed $v_{Q} \text{m s}^{-1}$.

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

After the collision, both particles decelerate at a constant rate of 2 $\text{m s}^{-2}$. Particle $P$ takes $0 . 3 u$ seconds longer to come to rest than particle $Q$.

Find $v_{P}$ and $v_{Q}$ in terms of $u$.

Harry releases a cricket ball of mass 0.16 kg from rest at a height of 2 m above horizontal ground. The ball falls vertically, hits the ground and rebounds vertically upwards. As a result of hitting the ground, the magnitude of the momentum of the ball decreases by 0.584 $\text{kg m s}^{-1}$. The ball is modelled as a particle and air resistance may be ignored.

Find the greatest height above the ground reached by the ball after it rebounds.

A particle $P$ of mass 5 kg is moving with constant speed 3 $\text{m s}^{-1}$ along a straight line on a smooth horizontal surface. It collides directly with a 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 $\text{m s}^{-1}$. The kinetic energy lost in the collision is 60 J.

(i) Calculate the speed of $Q$ immediately before the collision, and state its direction of motion.

(ii) Calculate the speed of $Q$ immediately after the collision, and state its direction of motion.

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

Show that $P$ reaches $A$ in the subsequent motion.

Determine, showing all necessary working, whether $P$ reaches $A$ before $Q$ reaches $B$.

Three small smooth spheres $R$, $S$ and $T$, of equal radius and of masses 0.4 kg, 0.2 kg and 0.1 kg respectively, lie in a straight line on a smooth horizontal surface, with $S$ between $R$ and $T$. $R$ and $S$ are projected directly towards each other with speeds 3 $\text{m s}^{-1}$ and 1 $\text{m s}^{-1}$ respectively, and collide directly. At the instant that $R$ and $S$ collide, $T$ is projected directly towards $S$ with speed 1 $\text{m s}^{-1}$.

Immediately after the collision between $R$ and $S$, both $R$ and $S$ move towards $T$, with $R$ having speed $v \text{m s}^{-1}$ and $S$ having speed $2 v \text{m s}^{-1}$. $S$ then collides directly with $T$, and $T$ rebounds with speed $x \text{m s}^{-1}$.

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