Momentum

On a straight horizontal test track, driverless vehicles, with no passengers, are undergoing testing. A car of mass $1600\ \text{kg}$ is pulling a trailer of mass $700\ \text{kg}$ along the track. The brakes are applied, producing a deceleration of $12\ \text{m\ s}^{-2}$. The braking force acts only on the car. In addition to the braking force, there are constant resistance forces of $600\ \text{N}$ on the car and $200\ \text{N}$ on the trailer.

Feb/March 2020

Particles $P$ and $Q$, whose masses are $0.2\,\text{kg}$ and $0.3\,\text{kg}$ respectively, are able to move only along a horizontal straight line on a smooth horizontal plane. $P$ is launched towards $Q$ at speed $0.5\,\text{m s}^{-1}$. Simultaneously, $Q$ is launched towards $P$ at speed $1\,\text{m s}^{-1}$. In the collision that results, $Q$ is brought to rest.

Feb/March 2021

A bead, $A$, with mass $0.1\,\text{kg}$ is placed on a long straight rigid wire inclined at $\sin^{-1}\left(\frac{7}{25}\right)$ to the horizontal. $A$ starts from rest and slides down the wire. The coefficient of friction between $A$ and the wire is $\mu$. After $A$ has moved $0.45\,\text{m}$ down the wire, its speed is $0.6\,\text{m s}^{-1}$.

Feb/March 2022

The diagram represents a smooth track lying in a vertical plane. Section $AB$ is a quarter circle with radius $1.8\,\text{m}$ and centre $O$. Section $BC$ is a horizontal straight segment of length $7.0\,\text{m}$, with $OB$ perpendicular to $BC$. Section $CFE$ is a straight line that is inclined at an angle of $\theta^\circ$ above the horizontal. A particle $P$ of mass $0.5\,\text{kg}$ is released from rest at $A$. Particle $P$ then collides with a particle $Q$ of mass $0.1\,\text{kg}$ that is at rest at $B$. Immediately after the collision, the speed of $P$ is $4\,\text{m s}^{-1}$ in the direction of $BC$. Assume that $P$ is moving horizontally at the moment it collides with $Q$.

Feb/March 2023

The diagram shows two particles $P$ and $Q$ on the line of greatest slope of plane $ABC$. Each particle has mass $m\,\text{kg}$. The plane is inclined at an angle $\theta$ to the horizontal, with $\sin \theta = 0.6$. $AB$ is $0.75\,\text{m}$ long and $BC$ is $3.25\,\text{m}$ long. Section $AB$ of the plane is smooth, whereas section $BC$ is rough. The coefficient of friction between each particle and section $BC$ is $0.25$. Particle $P$ is released from rest at $A$. At the same moment, particle $Q$ is released from rest at $B$.

Feb/March 2024

Particles $P$, $Q$ and $R$ have masses $0.6\,\text{kg}$, $0.4\,\text{kg}$ and $0.8\,\text{kg}$ respectively, and they are initially at rest on a smooth horizontal plane in a straight line. The separation from $P$ to $Q$ is $3\,\text{m}$, and the separation from $Q$ to $R$ is also $3\,\text{m}$. Particle $P$ is then projected directly towards $Q$ with speed $3\,\text{m s}^{-1}$. Once $P$ and $Q$ have collided, $P$ goes on moving in the same direction with speed $1.5\,\text{m s}^{-1}$.

Feb/March 2025

Points $A$ and $B$ are $10\,\text{m}$ apart on one horizontal plane. Particle $P$ begins from rest at $A$ and travels straight towards $B$ with constant acceleration $0.5\,\text{m s}^{-2}$. A second particle $Q$ moves straight towards $A$ at constant speed $0.75\,\text{m s}^{-1}$, and it passes through $B$ at the moment that $P$ starts to move. $T$ s after this moment, particles $P$ and $Q$ collide.

May/June 2014

A particle $P$ with mass $0.3\ \text{kg}$ is placed on a smooth plane that is inclined at $30^{\circ}$ to the horizontal, and it is let go from rest. $P$ travels $2.5\ \text{m}$ down the slope before arriving on a horizontal plane. Its speed does not change as $P$ reaches the horizontal plane. A particle $Q$ of mass $0.2\ \text{kg}$ is at rest on the horizontal plane, $1.5\ \text{m}$ from the lower end of the inclined plane (see diagram). $P$ then collides directly with $Q$.

May/June 2020

Two small smooth spheres $A$ and $B$, with equal radii and masses of $4\,\text{kg}$ and $2\,\text{kg}$ respectively, are on a smooth horizontal plane. At the beginning, $B$ is at rest, while $A$ is travelling towards $B$ at $10\,\text{m s}^{-1}$. After they collide, $A$ keeps moving in the same direction, but at half the speed of $B$.

May/June 2020

Particles $P$ with mass $m\,\text{kg}$ and $Q$ with mass $0.2\,\text{kg}$ are able to move freely on a smooth horizontal plane. $P$ is launched at a speed of $2\,\text{m s}^{-1}$ towards $Q$, which is at rest. After the collision, $P$ and $Q$ travel in opposite directions with speeds of $0.5\,\text{m s}^{-1}$ and $1\,\text{m s}^{-1}$ respectively.

May/June 2020

Three particles $P$, $Q$ and $R$, with masses $0.1\,\text{kg}$, $0.2\,\text{kg}$ and $0.5\,\text{kg}$ respectively, are initially at rest in a straight line on a smooth horizontal plane. Particle $P$ is sent towards $Q$ with speed $5\,\text{m s}^{-1}$. Once $P$ and $Q$ have collided, $P$ rebounds at speed $1\,\text{m s}^{-1}$. $Q$ then collides with $R$. Immediately after the collision with $Q$, $R$ starts moving with speed $V\,\text{m s}^{-1}$.

May/June 2021

Particles $P$, with mass $0.4\,\text{kg}$, and $Q$, with mass $0.5\,\text{kg}$, are free to move on a smooth horizontal plane. $P$ and $Q$ travel straight towards one another at speeds $2.5\,\text{m s}^{-1}$ and $1.5\,\text{m s}^{-1}$ respectively. After the collision between $P$ and $Q$, the speed of $Q$ is twice that of $P$.

May/June 2021

Particles $A$ and $B$, with masses $0.4\,\text{kg}$ and $0.2\,\text{kg}$ respectively, travel down the same line of greatest slope on a smooth plane. The plane is inclined at $30^\circ$ to the horizontal, and $A$ is situated above $B$ on the plane. At the moment of collision, the speeds of $A$ and $B$ are $3\,\text{m s}^{-1}$ and $2\,\text{m s}^{-1}$ respectively. During the collision, the speed of $A$ is lowered to $2.5\,\text{m s}^{-1}$.

May/June 2022

Two small smooth spheres \(A\) and \(B\), each with the same radius and with masses of \(5\,\text{kg}\) and \(3\,\text{kg}\) respectively, rest on a smooth horizontal plane. At the start, \(B\) is at rest, whereas \(A\) moves towards \(B\) at a speed of \(8.5\,\text{m s}^{-1}\). The spheres collide, and afterwards \(A\) continues in the same direction, but at one quarter of \(B\)'s speed.

May/June 2022

Particles $P$ and $Q$, having masses $0.3\,\text{kg}$ and $0.2\,\text{kg}$ respectively, are initially at rest on a smooth horizontal plane. $P$ is sent off at a speed of $4\,\text{m s}^{-1}$ directly towards $Q$. Once $P$ and $Q$ collide, $Q$ starts to move with a speed of $3\,\text{m s}^{-1}$.

May/June 2022

Particles $P$ and $Q$, with masses $m\,\text{kg}$ and $0.3\,\text{kg}$ respectively, are initially stationary on a smooth horizontal plane. $P$ is given a speed of $5\,\text{m s}^{-1}$ straight towards $Q$. After $P$ and $Q$ collide, $P$ continues in the same direction as before, but with speed $2\,\text{m s}^{-1}$.

May/June 2023

Particles $A$ and $B$, whose masses are $3.2\text{ kg}$ and $2.4\text{ kg}$ respectively, rest on a smooth horizontal table. $A$ travels towards $B$ at a speed of $v\text{ m s}^{-1}$ and then strikes $B$, which is travelling towards $A$ at a speed of $6\text{ m s}^{-1}$. After the collision, both particles are at rest.

May/June 2023

Particles $P$ and $Q$, with masses $0.1\,\text{kg}$ and $0.4\,\text{kg}$ respectively, can move freely on a smooth horizontal plane. $P$ is launched at a speed of $4\,\text{m s}^{-1}$ towards stationary $Q$. After $P$ and $Q$ collide, they have equal speeds.

May/June 2023

Particle $P$, with mass $0.2\,\text{kg}$, is launched vertically upwards from level ground at $25\,\text{m s}^{-1}$.

May/June 2024

Particles $A$, $B$ and $C$, with masses $5\text{ kg}$, $1\text{ kg}$ and $2\text{ kg}$ respectively, are initially at rest in that sequence on the straight smooth horizontal track $XYZ$. At the start, $A$ is at $X$, $B$ is at $Y$ and $C$ is at $Z$. $A$ is launched towards $B$ with speed $6\,\text{m s}^{-1}$, and at the same moment $C$ is launched towards $B$ with speed $v\,\text{m s}^{-1}$. In the later motion, $A$ collides with $B$ and they coalesce to form particle $D$. Then $D$ collides with $C$ and they coalesce to form particle $E$, which then moves towards $Z$.

May/June 2024

Particles $P$ and $Q$, with masses $0.2\,\mathrm{kg}$ and $0.5\,\mathrm{kg}$, are initially stationary on a smooth horizontal plane. $P$ is then sent towards $Q$ at a speed of $6\,\mathrm{m\,s^{-1}}$. Once $P$ and $Q$ have collided, the speed of $P$ is $1\,\mathrm{m\,s^{-1}}$.

May/June 2024

For a particle $P$ with mass $m\,\text{kg}$ and speed $u\,\text{m s}^{-1}$, the momentum is $4\,\text{N s}$ and the kinetic energy is $16\,\text{J}$.

May/June 2025

Particles $P$ and $Q$, whose masses are $0.2\text{ kg}$ and $0.1\text{ kg}$ respectively, can move freely along a straight line on a smooth horizontal plane. $P$ is launched towards $Q$ with speed $5\text{ m s}^{-1}$. At the same instant, $Q$ is launched away from $P$ with speed $2\text{ m s}^{-1}$. When $P$ collides with $Q$, the particles coalesce.

May/June 2025

Particles $P$ and $Q$, with masses $0.1\,\text{kg}$ and $0.3\,\text{kg}$ respectively, are initially stationary on a smooth horizontal plane. $P$ is sent directly towards $Q$ at speed $4u\,\text{m s}^{-1}$. At the same moment, $Q$ is sent directly towards $P$ at speed $u\,\text{m s}^{-1}$. Once $P$ and $Q$ have collided, $P$ travels at speed $2\,\text{m s}^{-1}$ and $Q$ travels at speed $4\,\text{m s}^{-1}$.

May/June 2025

In a machine that drives a nail into a wooden block, a hammerhead falls straight down onto the nail’s top. The hammerhead has mass $1.5\,\text{kg}$ and the nail has mass $0.005\,\text{kg}$ (see diagram). The hammerhead strikes the nail at a speed of $32\,\text{m s}^{-1}$ and stays in contact with it after the collision.

May/June 2025

A particle $B$ with mass $5\,\text{kg}$ is initially stationary on a smooth horizontal table. A particle $A$ with mass $2.5\,\text{kg}$ travels along the table at $6\,\text{m s}^{-1}$ and strikes $B$ head-on. During the collision, the two particles join together.

Oct/Nov 2020

Particles $P$ and $Q$, with masses $0.2\,\text{kg}$ and $0.5\,\text{kg}$ respectively, are initially at rest on a smooth horizontal plane. $P$ is launched towards $Q$ at speed $2\,\text{m s}^{-1}$.

Oct/Nov 2020

Two small smooth spheres $A$ and $B$, with equal radii and masses $4\,\text{kg}$ and $m\,\text{kg}$ respectively, are on a smooth horizontal plane. At the start, sphere $B$ is stationary and $A$ is moving towards $B$ at $6\,\text{m s}^{-1}$. Following the collision $A$ has speed $1.5\,\text{m s}^{-1}$ and $B$ has speed $3\,\text{m s}^{-1}$.

Oct/Nov 2020

Two small smooth spheres $A$ and $B$, with equal radii and masses $km\ \text{kg}$ and $m\ \text{kg}$ respectively, where $k > 1$, are able to move freely on a smooth horizontal plane. $A$ moves towards $B$ at speed $6\ \text{m s}^{-1}$, and $B$ moves towards $A$ at speed $2\ \text{m s}^{-1}$. Following the collision, $A$ and $B$ coalesce and then travel with speed $4\ \text{m s}^{-1}$.

Oct/Nov 2021

Particles $P$ and $Q$ have masses $m\,\text{kg}$ and $2m\,\text{kg}$ respectively. Initially, they are both at rest and are $6.4\,\text{m}$ apart on the same line of greatest slope of a rough plane inclined at an angle $\alpha$ to the horizontal, where $\sin\alpha = 0.8$ (see diagram). Particle $P$ is released from rest and moves down the line of greatest slope. At the same time, particle $Q$ is projected up that same line of greatest slope with speed $10\,\text{m s}^{-1}$. The coefficient of friction between each particle and the plane is $0.6$.

Oct/Nov 2021

A metal post is forced straight down into the ground by allowing a heavy object to fall onto it from above. The object has mass $120\text{ kg}$ and the post has mass $40\text{ kg}$ (see diagram). The object strikes the post at speed $8\text{ m s}^{-1}$ and stays in contact with it after the collision.

Oct/Nov 2021

Small smooth spheres $A$ and $B$, with equal radii and masses of $6\text{ kg}$ and $2\text{ kg}$ respectively, are on a smooth horizontal plane. At the start, $A$ is travelling towards $B$ at speed $5\ \text{m\,s}^{-1}$, while $B$ is travelling towards $A$ at speed $3\ \text{m\,s}^{-1}$. After the collision, both $A$ and $B$ travel in the same direction, and the difference between their speeds is $2\ \text{m\,s}^{-1}$.

Oct/Nov 2022

Three particles $A$, $B$ and $C$ have masses $0.3\,\text{kg}$, $0.4\,\text{kg}$ and $m\,\text{kg}$ respectively and are at rest on a smooth horizontal plane in a straight line. The separation of $B$ and $C$ is $2.1\,\text{m}$. $A$ is launched directly at $B$ with speed $2\,\text{m s}^{-1}$. Once $A$ collides with $B$, the speed of $A$ becomes $0.6\,\text{m s}^{-1}$, still travelling in the same direction.

Oct/Nov 2022

Masses of $1.5\ \text{kg}$ and $3\ \text{kg}$ are placed on a plane inclined at angle $\alpha$ to the horizontal, with $\tan \alpha = \frac{3}{4}$. The part of the plane from $A$ to $B$ is smooth, whereas the part from $B$ to $C$ is rough. The $1.5\ \text{kg}$ particle is kept at rest at $A$, and the $3\ \text{kg}$ particle is in limiting equilibrium at $B$. Let $AB = x\ \text{m}$ and $BC = 4\ \text{m}$.

Oct/Nov 2022

A particle $A$, whose mass is $0.5\,\text{kg}$, is launched vertically upwards from level ground at a speed of $25\,\text{m s}^{-1}$.

Oct/Nov 2023

Two particles, $P$ and $Q$, with masses $2m\,\text{kg}$ and $m\,\text{kg}$ respectively, are initially at rest on the same vertical line. Their heights above horizontal ground are $1\,\text{m}$ for $P$ and $2\,\text{m}$ for $Q$. $P$ is thrown vertically upward at speed $2\,\text{m s}^{-1}$, and at that same moment $Q$ is let go from rest.

Oct/Nov 2024

Particles $A$ and $B$, with masses 3 kg and 6 kg respectively, are located on a smooth horizontal plane. At first, $B$ is stationary while $A$ is travelling towards $B$ at speed $8\,\text{m s}^{-1}$. After $A$ and $B$ collide, $A$ has speed $2\,\text{m s}^{-1}$.

Oct/Nov 2024

Particles $P$ and $Q$, with masses 3 kg and 5 kg respectively, start from rest on a smooth horizontal plane. $P$ is then projected directly towards $Q$ at a speed of $4\text{ m s}^{-1}$. After the collision, $P$ moves with speed $1\text{ m s}^{-1}$.

Oct/Nov 2025

Particle $A$, with mass $2.5\text{ kg}$, is let go from rest at the top of a smooth plane inclined at an angle of $\sin^{-1}0.2$ to the horizontal. Two seconds later, $A$ collides with particle $B$, of mass $3\text{ kg}$, which is travelling up the line of greatest slope of the plane. Just before the collision, the speed of $B$ is $3.5\text{ m s}^{-1}$. Immediately after the collision, $B$ moves with velocity $0.5\text{ m s}^{-1}$ down the plane.

Oct/Nov 2025

A particle $P$ with mass $0.1\text{ kg}$ is launched vertically upward at $30\text{ m s}^{-1}$ from level ground. At the same moment, a particle $Q$ with mass $0.4\text{ kg}$ is launched vertically upward at $10\text{ m s}^{-1}$ from a point $15\text{ m}$ above the ground. $P$ and $Q$ travel along the same vertical line. When $P$ and $Q$ collide, they stick together.

Oct/Nov 2025

Particles $P$ and $Q$, with masses $m\text{ kg}$ and $2\text{ kg}$ respectively, are free to move on a smooth horizontal plane. They travel directly towards one another at speeds $u\text{ m s}^{-1}$ and $3u\text{ m s}^{-1}$ respectively. A collision takes place, and the direction of motion of each particle is reversed by the impact. Immediately after the collision, the speed of $P$ is $\frac{u}{3}\text{ m s}^{-1}$.

Oct/Nov 2025

Two particles $A$ and $B$ have masses $km$ and $m$ respectively, where $k$ and $m$ are constants, and are free to move in a straight line on a smooth horizontal plane. Particle $A$ is projected towards $B$ with speed $2u$ and, at the same instant, $B$ is projected towards $A$ with speed $u$. The particles collide. After the collision, $A$ moves with speed $u$ and both particles travel in the same direction as $A$ was originally moving. It is given that $35\%$ of the total kinetic energy is lost in the collision.

Oct/Nov 2025