Energy, work and power

A cyclist of mass $85\,\text{kg}$ and a bicycle of mass $20\,\text{kg}$ travel along a horizontal road while opposing a total resistance force of $40\,\text{N}$.

Feb/March 2016

A resistive force of magnitude $1350\,\text{N}$ acts continuously on a car with mass $1200\,\text{kg}$.

Feb/March 2016

A particle with mass $0.4\,\text{kg}$ is launched at a speed of $12\,\text{m s}^{-1}$ up the line of greatest slope on a smooth plane inclined at $30^\circ$ to the horizontal.

Feb/March 2017

A particle with mass $0.6\,\text{kg}$ rests on a rough plane that makes an angle of $21^\circ$ with the horizontal. A force of magnitude $P$, acting parallel to the line of greatest slope of the plane, holds the particle in equilibrium, as shown in the diagram. The coefficient of friction between the particle and the plane is $0.3$.

Feb/March 2017

A girl with mass $40\,\text{kg}$ moves down a slide in a water park. She begins at point $A$ and travels to point $B$, which lies $7.2\,\text{m}$ vertically beneath the level of $A$, as the diagram shows.

Feb/March 2018

A car with mass $1200\,\text{kg}$ can move at a maximum constant speed of $60\,\text{m s}^{-1}$ on a straight horizontal road. When the car is moving at $v\,\text{m s}^{-1}$, the resistive force has magnitude $35v\,\text{N}$.

Feb/March 2018

A $1500\text{ kg}$ car is towing a $300\text{ kg}$ trailer on a straight level road at a steady speed of $20\text{ m s}^{-1}$. The car and trailer are represented as two particles joined by a light rigid horizontal rod. The car’s engine has a power output of $6000\text{ W}$. The resistances to motion are constant, with $R\text{ N}$ acting on the car and $80\text{ N}$ acting on the trailer.

Feb/March 2019

The figure gives the vertical cross-section $PQR$ of a slide. Segment $PQ$ is a straight section of length $8\,\text{m}$, making an angle $\alpha$ with the horizontal, where $\sin\alpha = 0.8$. The straight section $PQ$ is tangent to the curved section $QR$, and $R$ is $h\,\text{m}$ above the level of $P$. The straight section $PQ$ is rough, whereas the curved section $QR$ is smooth. A particle of mass $0.25\,\text{kg}$ is projected from $P$ towards $Q$ with speed $15\,\text{m s}^{-1}$ and comes to rest at $R$. The coefficient of friction between the particle and $PQ$ is $0.5$.

Feb/March 2019

A lorry with mass $16000\,\text{kg}$ moves along a straight horizontal road. Its engine is operating at constant power. In $10\,\text{s}$, the work done by the driving force is $750000\,\text{J}$.

Feb/March 2020

The figure gives the vertical cross-section of a surface. $A$, $B$ and $C$ are three points on the cross-section. $B$ is at a level $h\,\text{m}$ above $A$. $C$ is at a level $0.5\,\text{m}$ below $A$. A particle with mass $0.2\,\text{kg}$ is projected up the slope from $A$ with initial speed $5\,\text{m}\,\text{s}^{-1}$. As it moves from $A$ to $C$, the particle remains in contact with the surface.

Feb/March 2020

A car with mass $1400\,\text{kg}$ moves at constant speed up a straight hill inclined at $\alpha$ to the horizontal, where $\sin \alpha = 0.1$. A constant resistive force of magnitude $600\,\text{N}$ acts on it. The engine power of the car is $22\,500\,\text{W}$.

Feb/March 2021

Particles $P$ and $Q$ have masses $0.5\,\text{kg}$ and $m\,\text{kg}$ respectively, and they are connected by a light inextensible string. The string runs over a fixed smooth pulley mounted at the top of two inclined planes. At the start, both particles are at rest, with $P$ on a smooth plane at $30^\circ$ to the horizontal and $Q$ on a plane at $45^\circ$ to the horizontal. The string is taut, and the particles may travel along the lines of greatest slope of the two planes. A force of magnitude $0.8\,\text{N}$ is applied to $P$ down the plane, making $P$ move down the plane (see diagram).

Feb/March 2021

A crane is used to lift a block of mass $600\,\text{kg}$ straight up at constant speed through a height of $15\,\text{m}$. A resistive force acts on the block, and the crane does $10000\,\text{J}$ of work to overcome it.

Feb/March 2022

The combined mass of the cyclist and her bicycle is $70\,\text{kg}$. She is climbing a straight hill at constant power of $180\,\text{W}$, with the hill inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = 0.05$. At the instant when her speed is $6\,\text{m s}^{-1}$, her acceleration is $-0.2\,\text{m s}^{-2}$. A constant resistive force of magnitude $F\,\text{N}$ acts against the motion.

Feb/March 2022

A toy railway locomotive, with mass $0.8\text{ kg}$, is pulling a truck of mass $0.4\text{ kg}$ along a straight horizontal track at a steady speed of $2\text{ m s}^{-1}$. The locomotive experiences a constant resistive force of magnitude $0.2\text{ N}$, whereas the truck has no resistive force acting on it. The locomotive and the truck are joined by a light rigid horizontal coupling.

Feb/March 2023

A car with mass $1800\,\text{kg}$ is pulling a trailer with mass $300\,\text{kg}$ along a straight road that slopes at an angle $\alpha$ to the horizontal, with $\sin \alpha = 0.05$. A tow-bar, which is light, rigid and parallel to the road, links the car and trailer. A resistance force of $800\,\text{N}$ acts on the car, while the trailer experiences a resistance force of $F\,\text{N}$. The car's engine provides a driving force of $3000\,\text{N}$.

Feb/March 2024

The aeroplane is travelling at a steady speed.

Feb/March 2025

The car has mass $1150 \text{ kg}$ and moves uphill along a straight slope that makes an angle of $1.2^{\circ}$ with the horizontal. The force resisting the motion is $975 \text{ N}$.

May/June 2010

$P$ and $Q$ are two fixed points on the line of greatest slope of an inclined plane. Point $Q$ is $0.45 \text{ m}$ above the level of $P$. A particle with mass $0.3 \text{ kg}$ moves in the upward direction along $PQ$.

May/June 2010

A car with mass $1150\,\text{kg}$ moves along a straight hill that is tilted at $1.2^\circ$ above the horizontal. The resistive force acting on the car is $975\,\text{N}$.

May/June 2010

$P$ and $Q$ are two fixed points lying on the line of greatest slope of an inclined plane. The point $Q$ is $0.45\,\text{m}$ higher than the level of $P$. A particle with mass $0.3\,\text{kg}$ travels upwards along $PQ$.

May/June 2010

A load is moved along a horizontal straight track from $A$ to $B$ by a force of magnitude $P\,\text{N}$ acting at $30^\circ$ above the horizontal. The distance $AB$ is $80\,\text{m}$. As the load travels from $A$ to the midpoint $M$ of $AB$, its speed stays constant at $1.2\,\text{m s}^{-1}$.

May/June 2010

A car with mass $700\,\text{kg}$ moves along a straight horizontal road. The resistance to motion is constant at $600\,\text{N}$.

May/June 2011

A crane lifts a load with mass $1250\,\text{kg}$ from rest on level ground to rest at a height of $1.54\,\text{m}$ above the ground. The work done in overcoming resistance to motion is $5750\,\text{J}$.

May/June 2011

Masses $A$ and $B$, with masses $1.2\,\text{kg}$ and $2.0\,\text{kg}$ respectively, are fixed to the two ends of a light inextensible string that passes over a fixed smooth pulley. $A$ is kept at rest while $B$ hangs freely, and both vertical sections of the string are straight. When $A$ is released, it moves upwards. In the later motion, it does not reach the pulley.

May/June 2011

A load is dragged over horizontal ground for a distance of $76\,\text{m}$ by means of a rope. The rope makes an angle of $5^\circ$ above the horizontal, and the rope tension is $65\,\text{N}$.

May/June 2011

An object of mass $8\,\text{kg}$ moves down the line of greatest slope on an inclined plane. At the top of the plane, its initial speed is $3\,\text{m s}^{-1}$, and at the bottom its speed is $8\,\text{m s}^{-1}$. The work done against the resistance to motion of the object is $120\,\text{J}$.

May/June 2011

A block is dragged across a level floor for $50\,\text{m}$ by a rope making an angle of $\alpha^\circ$ with the floor. The rope tension is $180\,\text{N}$, and the work done by the tension is $8200\,\text{J}$.

May/June 2011

Particle $P$ is launched from the upper end of a smooth ramp with speed $u\,\text{m s}^{-1}$ and moves along the steepest descent. The ramp is $6.4\,\text{m}$ long and makes an angle of $30^\circ$ to the horizontal. At the same moment that $P$ is launched, a second particle $Q$ is let go from rest at a point $3.2\,\text{m}$ vertically above the foot of the ramp (see diagram). If $P$ and $Q$ arrive at the bottom of the ramp at the same time,

May/June 2011

A lorry with mass $15\,000\,\text{kg}$ travels up a hill that is $500\,\text{m}$ long at constant speed. The slope is at an angle of $2.5^\circ$ to the horizontal. The resistance to the lorry’s motion stays constant at $800\,\text{N}$. On the way back, the lorry arrives at the top of the hill with speed $20\,\text{m s}^{-1}$ and moves downhill with a constant driving force of $2000\,\text{N}$. The resistance to the lorry’s motion is again constant at $800\,\text{N}$.

May/June 2011

A car with mass $880\,\text{kg}$ is moving on a level straight road, and its engine is delivering power at a steady rate of $P\,\text{W}$. The resistive force opposing the motion is $700\,\text{N}$. When the car’s speed is $16\,\text{m s}^{-1}$, its acceleration is $0.625\,\text{m s}^{-2}$.

May/June 2012

A load with mass $160\,\text{kg}$ is lifted vertically upward from rest at a fixed point $O$ on the ground by means of a winding drum. When the load reaches point $A$, which is $20\,\text{m}$ above $O$, its speed is $1.25\,\text{m s}^{-1}$ (see diagram). Find, for the motion from $O$ to $A$,

May/June 2012

A car with mass $1250\,\text{kg}$ moves from the foot to the top of a straight hill of length $400\,\text{m}$, which makes an angle $\alpha$ to the horizontal where $\sin\alpha = 0.125$. The resistive force acting on the car is $800\,\text{N}$. Determine the work done by the car’s engine in each case below.

May/June 2012

A ring passes over a fixed horizontal bar. One end of a light inextensible string is fastened to the ring and is used to draw the ring along the bar at a steady speed of $0.5\,\text{m s}^{-1}$. The string stays at a constant angle of $24^{\circ}$ to the bar, and the tension in the string is $6\,\text{N}$ (see diagram).

May/June 2012

A car with mass $1230\,\text{kg}$ speeds up from $4\,\text{m s}^{-1}$ to $21\,\text{m s}^{-1}$ over $24.5\,\text{s}$. The table beneath gives matching values of time $t\,\text{s}$ and speed $v\,\text{m s}^{-1}$.

May/June 2012

A lorry with mass $16000\,\text{kg}$ travels along a straight hillside that makes an angle $\alpha^{\circ}$ with the horizontal. The slope has length $500\,\text{m}$.

May/June 2012

A car with mass $1250\text{ kg}$ moves from the foot to the summit of a straight hill of length $600\text{ m}$, inclined at $2.5^\circ$ to the horizontal. The constant resistance to the car's motion is $400\text{ N}$. The work done by the driving force is $450\text{ kJ}$. At the bottom of the hill, the car's speed is $30\text{ m s}^{-1}$.

May/June 2013

A and B lie 50\,\text{m} apart along a straight path inclined at an angle $\theta$ to the horizontal, with $\sin\theta = 0.05$, and A is above B. A block of mass $16\,\text{kg}$ is pulled down the path from A to B. It starts from rest at A and arrives at B with speed $10\,\text{m s}^{-1}$. The work done by the pulling force on the block is $1150\,\text{J}$.

May/June 2013

A car with mass $1000\,\text{kg}$ is moving along a straight horizontal road. Its engine delivers constant power $P\,\text{kW}$. The resistive force on the car is $600\,\text{N}$. When the car’s speed is $25\,\text{m s}^{-1}$, its acceleration is $0.2\,\text{m s}^{-2}$.

May/June 2013

A straight ice track with a length of $50\,\text{m}$ is tilted at $14^\circ$ to the horizontal. A man sets off from the top of the track on a sledge, with speed $8\,\text{m s}^{-1}$. He goes down the sledge to the bottom of the track. The coefficient of friction between the sledge and the track is $0.02$.

May/June 2013

Particles $A$, with mass $1.6\,\text{kg}$, and $B$, with mass $2\,\text{kg}$, are connected to the two ends of a light inextensible string. The string runs over a small smooth pulley fixed at the top of a smooth plane inclined at angle $\theta$, where $\sin \theta = 0.8$. Particle $A$ is initially held at rest at the lower end of the plane, while $B$ is suspended at a height of $3.24\,\text{m}$ above the level of the bottom of the plane (see diagram). $A$ is released from rest and the particles begin to move.

May/June 2013

A car with mass $1100\,\text{kg}$ leaves O from rest and travels along the road OAB. OA is a straight stretch of length $1760\,\text{m}$, and it slopes to the horizontal, with A $160\,\text{m}$ above the level of O. AB is a straight horizontal stretch. As the car moves, the driving force is $1800\,\text{N}$ and the resistance to the car’s motion is $700\,\text{N}$. When the car has covered $x\,\text{m}$ from O, its speed is $v\,\text{m s}^{-1}$.

May/June 2014

A car with mass $600\text{ kg}$ is moving on a straight, level road. The resistive force opposing the car’s motion remains constant at $R\text{ N}$.

May/June 2014

A light inextensible rope has block $A$, of mass $5\text{ kg}$, at one end and block $B$, of mass $16\text{ kg}$, at the other. The rope runs over a smooth pulley fixed at the top of a rough plane inclined at an angle of $30^{\circ}$ to the horizontal. Block $A$ is kept at rest at the bottom of the plane, while block $B$ hangs below the pulley (see diagram). The coefficient of friction between $A$ and the plane is $\frac{1}{\sqrt{3}}$. Block $A$ is released from rest and the system begins to move. After each block has moved a distance of $x\text{ m}$, each has speed $v\,\text{m s}^{-1}$.

May/June 2014

A car with mass $1250\text{ kg}$ moves up a straight hill at an angle $\alpha$ to the horizontal, where $\sin \alpha = 0.02$. The engine supplies $23\text{ kW}$. The resistance to motion is constant and equals $600\text{ N}$. Find the car’s speed at the moment when its acceleration is $0.5\text{ m s}^{-2}$.

May/June 2014

A lorry with mass $16000\text{ kg}$ moves at steady speed from the base, $O$, to the summit, $A$, of a straight hill. The length $OA$ is $1200\text{ m}$ and $A$ lies $18\text{ m}$ above the level of $O$. The lorry’s driving force is constant and equal to $4500\text{ N}$. When it reaches $A$ the lorry goes on along a straight horizontal road, working against a constant resistance of $2000\text{ N}$. The driving force is then no longer constant, and the speed rises from $9\text{ m s}^{-1}$ at $A$ to $21\text{ m s}^{-1}$ at point $B$ on the road. The distance $AB$ is $2400\text{ m}$.

May/June 2014

A block $B$ with mass $2.7\,\text{kg}$ is being pulled at a steady speed in a straight path across a rough horizontal floor. The pulling force is $25\,\text{N}$ and is applied at an angle of $\theta$ above the horizontal. The normal component of the contact force on $B$ is $20\,\text{N}$.

May/June 2015

A block weighing $6.1\,\text{N}$ moves down a slope that makes angle $\tan^{-1}\left(\frac{11}{60}\right)$ with the horizontal. The coefficient of friction between the block and the slope is $\frac{1}{4}$. As the block goes through point $A$, its speed is $2\,\text{m s}^{-1}$.

May/June 2015

A lorry with mass $14\,000\,\text{kg}$ travels along a road, beginning at rest at point $O$. It gets to point $A$, and then continues to point $B$, where it has a speed of $24\,\text{m s}^{-1}$. The stretch $OA$ is straight, horizontal and $400\,\text{m}$ long. The stretch $AB$ is straight, slopes downwards at an angle of $\theta^\circ$ to the horizontal, and has length $300\,\text{m}$.

May/June 2015

A cyclist together with her bicycle has a combined mass of $84\,\text{kg}$. She delivers power at a steady rate of $P\,\text{W}$ while travelling along a straight road that is sloping at an angle $\theta$ to the horizontal, where $\sin\theta = 0.1$. When she is going uphill, the cyclist’s acceleration is $1.25\,\text{m s}^{-2}$ at the moment when her speed is $3\,\text{m s}^{-1}$. When she is going downhill, the cyclist’s acceleration is $1.25\,\text{m s}^{-2}$ at the moment when her speed is $10\,\text{m s}^{-1}$. The resistive force opposing the cyclist’s motion, whether she is travelling uphill or downhill, is $R\,\text{N}$.

May/June 2015

A block is fastened to one end of a light inextensible string. The string is inclined at $60^\circ$ above the horizontal and pulls the block along a horizontal floor in a straight line with acceleration $0.5\,\text{m s}^{-2}$. The tension in the string is $8\,\text{N}$. The block starts moving with speed $0.3\,\text{m s}^{-1}$. During the first $5\,\text{s}$ of the motion, find

May/June 2015

The combined mass of the cyclist and the cycle is $80\,\text{kg}$. There is no resistance to motion.

May/June 2015

A plane makes an angle of $\sin^{-1}\!\left(\tfrac{1}{8}\right)$ with the horizontal. $A$ and $B$ are two points on the same line of steepest descent, with $A$ lying above $B$. The length $AB$ is $12\,\text{m}$. A small object $P$ of mass $8\,\text{kg}$ is released from rest at $A$ and slides down the plane, reaching $B$ with speed $4.5\,\text{m s}^{-1}$. For the motion of $P$ from $A$ to $B$, determine

May/June 2015

A block is being pulled across a level floor by a rope that is horizontal. The rope tension is $500\,\text{N}$, and the block travels at a steady speed of $2.75\,\text{m s}^{-1}$.

May/June 2015

Particles $A$ and $B$, with masses of $0.35\,\text{kg}$ and $0.15\,\text{kg}$ respectively, are joined by a light inextensible string at their ends. $A$ starts from rest on a smooth horizontal surface, while the string passes over a small smooth pulley fixed at the edge of the surface. $B$ hangs vertically below the pulley, at a height of $h\,\text{m}$ above the floor (see diagram). $A$ is released and the particles move. $B$ reaches the floor and $A$ then arrives at the pulley with a speed of $3\,\text{m s}^{-1}$.

May/June 2015

A box with mass $25\,\text{kg}$ is dragged a distance of $36\,\text{m}$ up a rough plane that is inclined at $20^\circ$ to the horizontal, moving at constant speed. It travels along the line of greatest slope while a steady frictional force of $40\,\text{N}$ acts against it. The pulling force is parallel to the line of greatest slope.

May/June 2016

A car with mass $1000\,\text{kg}$ moves in a straight line on a level road while experiencing total resistive forces of $300\,\text{N}$.

May/June 2016

A particle of mass $8\,\text{kg}$ is launched at a speed of $5\,\text{m s}^{-1}$ up the line of greatest slope on a rough plane inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = \frac{5}{13}$. Its motion is opposed by a constant friction force of magnitude $15\,\text{N}$. After moving a distance $x\,\text{m}$ up the plane, the particle is brought to instantaneous rest.

May/June 2016

A car with mass $1100\,\text{kg}$ is travelling along a road while a steady resisting force of $1550\,\text{N}$ acts opposite to the motion.

May/June 2016

A particle with mass $8\,\text{kg}$ is drawn, at constant speed, a distance of $20\,\text{m}$ along a rough plane that is inclined at an angle of $30^\circ$ to the horizontal by a force acting in the line of greatest slope.

May/June 2016

A constant force of magnitude $650\,\text{N}$ opposes the motion of a car with mass $1400\,\text{kg}$.

May/June 2016

A particle with mass $0.6\text{ kg}$ is released from rest at a point $8\text{ m}$ above the ground. Just before it reaches the ground, its speed is $10\text{ m s}^{-1}$.

May/June 2017

A car with mass $800\,\text{kg}$ is travelling uphill on a slope making an angle of $\theta^\circ$ with the horizontal, where $\sin\theta = 0.15$. Its initial speed is $8\,\text{m s}^{-1}$. Twelve seconds later, it has gone $120\,\text{m}$ up the slope and its speed is $14\,\text{m s}^{-1}$.

May/June 2017

A block is attached to one end of a light inextensible string. The string is inclined at an angle of $\theta^\circ$ to the horizontal. The tension in the string is $20\,\text{N}$. The string moves the block across a horizontal surface at a steady speed of $1.5\,\text{m s}^{-1}$ for $12\,\text{s}$. The work done by the tension in the string is $50\,\text{J}$.

May/June 2017

The diagram depicts a wire $ABCD$ made up of a straight section $AB$ with length $5\,\text{m}$ and a section $BCD$ formed as a semicircle of radius $6\,\text{m}$ with centre $O$. The diameter $BD$ of the semicircle lies horizontally, while $AB$ is vertical. A small ring is placed on the wire and can move along it. It is released from rest at $A$. The section $AB$ of the wire is rough, so the ring has constant acceleration of $2.5\,\text{m s}^{-2}$ from $A$ to $B$.

May/June 2017

A car with mass $1200\,\text{kg}$ is travelling along a straight road while a steady resisting force of $850\,\text{N}$ acts against its motion.

May/June 2017

A particle with mass $0.12\,\text{kg}$ rests on a plane that is tilted at $40^\circ$ to the horizontal. A force of size $P\,\text{N}$ acts up the plane at $30^\circ$ above a line of greatest slope, as shown in the diagram, and the particle is in equilibrium. The coefficient of friction between the particle and the plane is $0.32$.

May/June 2017

A man propels a wheelbarrow of mass $25\,\text{kg}$ over a level road using a constant force of magnitude $35\,\text{N}$ applied at $20^\circ$ below the horizontal. A steady resistive force of $15\,\text{N}$ acts against the motion. Starting from rest, the wheelbarrow travels $12\,\text{m}$.

May/June 2017

A car with mass $1200\text{ kg}$ is moving along a level road.

May/June 2017

The mass of the car is $1250\,\text{kg}$.

May/June 2018

A train with mass $240\,000\,\text{kg}$ is moving up a slope that makes an angle of $4^\circ$ to the horizontal. A constant resistive force of magnitude $18\,000\,\text{N}$ acts on the train. At the moment when its speed is $15\,\text{m s}^{-1}$, the train’s deceleration is $0.2\,\text{m s}^{-2}$.

May/June 2018

Particles $A$ and $B$, whose masses are $0.8\,\text{kg}$ and $1.6\,\text{kg}$, are joined by a light inextensible string. $A$ is on a smooth plane inclined at angle $\theta$ to the horizontal, with $\sin \theta = \frac{3}{5}$. A small smooth pulley $P$ is fixed at the top of the plane, and $B$ hangs freely (see diagram). The part $AP$ of the string is parallel to the line of greatest slope of the plane. The particles are let go from rest while both parts of the string remain taut.

May/June 2018

A car with mass $1400\,\text{kg}$ moving at speed $v\,\text{m}\,\text{s}^{-1}$ is subject to a resistive force of magnitude $40v\,\text{N}$. On a straight level road, the car’s highest possible constant speed is $56\,\text{m}\,\text{s}^{-1}$.

May/June 2018

The lorry’s mass is $12000\,\text{kg}$.

May/June 2019

A particle with mass $13\,\text{kg}$ lies on a rough plane inclined at an angle of $\theta$ to the horizontal, where $\tan \theta = \frac{5}{12}$. The coefficient of friction between the particle and the plane is $0.3$. A force of magnitude $T\,\text{N}$, acting parallel to the line of greatest slope, pulls the particle $2.5\,\text{m}$ up the plane at a constant speed.

May/June 2019

A car with mass $1400\,\text{kg}$ is moving uphill on a slope that makes an angle of $4^{\circ}$ to the horizontal. A constant resistive force of magnitude $1550\,\text{N}$ acts on the car.

May/June 2019

A particle with mass $18\,\text{kg}$ is placed on a plane inclined at an angle of $30^\circ$ to the horizontal. It is projected along the line of greatest slope of the plane with speed $20\,\text{m s}^{-1}$.

May/June 2019

A car with mass $1800\,\text{kg}$ is pulling a trailer of mass $400\,\text{kg}$ on a straight horizontal road. The car and trailer are joined by a light rigid tow-bar. The car has acceleration $1.5\,\text{m}\,\text{s}^{-2}$. Constant resistance forces act, with $250\,\text{N}$ on the car and $100\,\text{N}$ on the trailer.

May/June 2020

A child with mass $35\,\text{kg}$ is on a rope swing. Model the child as a particle $P$, and model the rope as a light inextensible string of length $4\,\text{m}$. At the start, $P$ is being held at an angle of $45^\circ$ to the vertical (see diagram).

May/June 2020

A car with mass $1250\,\text{kg}$ is travelling along a straight road.

May/June 2020

A minibus with mass $4000\,\text{kg}$ is moving along a straight horizontal road. The resistive force is $900\,\text{N}$.

May/June 2020

A block $B$ with mass $4\,\text{kg}$ is driven upwards along the line of greatest slope on a smooth plane that is tilted at $30^\circ$ to the horizontal by a force applied to $B$, and this force acts in the same direction as the motion of $B$. The block goes through points $P$ and $Q$ with speeds $12\,\text{m s}^{-1}$ and $8\,\text{m s}^{-1}$, respectively. $P$ and $Q$ are $10\,\text{m}$ apart, with $P$ at a lower level than $Q$.

May/June 2020

The winch works through a force applied by a rope. It is used to haul a load of mass $50\,\text{kg}$ along the line of greatest slope of a plane that is inclined at $60^\circ$ to the horizontal. The winch moves the load a distance of $5\,\text{m}$ up the plane at constant speed. A constant resistance to motion of $100\,\text{N}$ acts.

May/June 2021

Particles $A$ and $B$ have masses $m\,\text{kg}$ and $0.1\,\text{kg}$ respectively, with $m > 0.1$. They are fastened to the two ends of a light inextensible string, which runs over a fixed smooth pulley so that the particles hang vertically beneath it (see diagram). Initially, each particle is $0.9\,\text{m}$ above level ground, and the system is released from rest; during the time that both particles are moving, the string tension is $1.5\,\text{N}$. Particle $B$ does not reach the pulley.

May/June 2021

A playground slide slopes downward at a fixed angle of $30^\circ$ over 2.5 m. After this, it continues as a horizontal part lying in the same vertical plane as the inclined part. A child of mass 35 kg, modelled as a particle P, begins from rest at the top of the slide and moves directly down the sloping part. She then travels along the horizontal part until she stops (see diagram). As the child moves from the sloping part to the horizontal part, her speed does not change instantaneously. A resistive force acts on the horizontal part of the slide, and the work done against the resistive force on the horizontal part of the slide is 250 J per metre. The sloping part of the slide is smooth.

May/June 2021

A particle with mass $0.6\,\text{kg}$ is projected at a speed of $4\,\text{m s}^{-1}$ along the line of greatest slope of a smooth plane that is inclined at $10^\circ$ to the horizontal.

May/June 2021

A particle of mass $12\,\text{kg}$ rests on a rough plane inclined at $25^\circ$ to the horizontal. A force of magnitude $P\,\text{N}$ is applied at $8^\circ$ above the line of greatest slope of the plane. This force keeps the particle in equilibrium. The coefficient of friction between the particle and the plane is $0.3$.

May/June 2021

A car with mass $1250\,\text{kg}$ is towing a caravan whose mass is $800\,\text{kg}$ on a straight road. The resistive forces acting on the car and the caravan are $440\,\text{N}$ and $280\,\text{N}$ respectively. The car and caravan are joined by a light rigid tow-bar.

May/June 2021

A cyclist moves along a straight level road. She is producing power at a steady rate of $150\,\text{W}$. At the instant when her speed is $4\,\text{m s}^{-1}$, her acceleration is $0.25\,\text{m s}^{-2}$. The resistance opposing motion is $20\,\text{N}$.

May/June 2021

A car with mass $1400\,\text{kg}$ is pulling a trailer with mass $500\,\text{kg}$ down a straight hill that is inclined at an angle of $5^\circ$ to the horizontal. The car and trailer are joined by a light rigid tow-bar. At the top of the hill the speed of the car and trailer is $20\,\text{m s}^{-1}$ and at the bottom of the hill their speed is $30\,\text{m s}^{-1}$.

May/June 2021

A car with mass $900\,\text{kg}$ is travelling uphill on a slope of angle $\sin^{-1}(0.12)$ above the horizontal. Its starting speed is $11\,\text{m s}^{-1}$. After $12\,\text{s}$, it has covered $150\,\text{m}$ up the slope and is moving at $16\,\text{m s}^{-1}$. The car’s engine is outputting power at a steady rate of $24\,\text{kW}$.

May/June 2022

A cyclist travels on a straight, level road. The combined mass of the cyclist and her bicycle is $70\text{ kg}$. At the moment when her speed is $4\text{ m s}^{-1}$, her acceleration is $0.3\text{ m s}^{-2}$. A steady resistive force of magnitude $30\text{ N}$ acts.

May/June 2022

Particle $P$, with mass $0.4\,\text{kg}$, is fired straight up from level ground at a speed of $10\,\text{m s}^{-1}$.

May/June 2023

A car with mass $1200\,\text{kg}$ is moving along a straight horizontal road. The car’s engine has constant power equal to $16\,\text{kW}$. A constant resistive force of magnitude $500\,\text{N}$ acts on the car.

May/June 2023

A particle with mass $1.6\text{ kg}$ is released from rest at a point $9\text{ m}$ above the horizontal ground. Immediately before it strikes the ground, the particle’s speed is $12\text{ m s}^{-1}$.

May/June 2023

An athlete with mass $84\,\text{kg}$ is moving along a straight road.

May/June 2023

A lorry with mass $15\,000\,\text{kg}$ travels along a straight horizontal road from $A$ to $B$. Its speed is $20\,\text{m s}^{-1}$ at $A$ and $25\,\text{m s}^{-1}$ at $B$. The engine power of the lorry is constant, and the resistance to motion has constant magnitude $6000\,\text{N}$. The acceleration of the lorry at $B$ is $0.5$ times the acceleration of the lorry at $A$.

May/June 2023

The diagram presents the vertical cross-section $XYZ$ of a rough slide. $YZ$ is a straight segment of length $2\,\text{m}$ that makes an angle of $\alpha$ with the horizontal, where $\sin\alpha = 0.28$. At $Y$, $YZ$ touches the curved part $XY$ tangentially, and $X$ is $1.8\,\text{m}$ above level of $Y$. A child of mass $25\,\text{kg}$ slides down the slide, beginning from rest at $X$. The work done by the child against the resistance force in moving from $X$ to $Y$ is $50\,\text{J}$.

May/June 2023

A train with mass $180\,000\,\text{kg}$ climbs a straight hill of length $1.5\,\text{km}$, which is tilted at an angle of $1.5^\circ$ to the horizontal. During the climb, the total work needed to overcome the resistance to motion is $12\,000\,\text{kJ}$, and the train's speed drops from $45\,\text{m s}^{-1}$ to $40\,\text{m s}^{-1}$.

May/June 2024

A straight slope with length $60\,\text{m}$ is inclined at an angle of $12^\circ$ to the horizontal. A bobsled begins at the top of the slope with a speed of $5\,\text{m s}^{-1}$. It travels directly down the slope.

May/June 2024