Representation of data

A particle is launched from a point on level ground.

Feb/March 2016

A stone is projected at a speed of $9\text{ m s}^{-1}$ at an angle of $60^\circ$ above the horizontal from a point on level ground. Find the distance between the two points where the path of the stone makes an angle of $45^\circ$ with the horizontal.

Feb/March 2016

A uniform lamina is formed by combining rectangle $ABCD$, where $AB = CD = 0.56\text{ m}$ and $BC = AD = 2\text{ m}$, with square $EFGA$ of side $1.2\text{ m}$. The square’s vertex $E$ lies on the side $AD$ of the rectangle (see diagram). The lamina’s centre of mass is $h\text{ m}$ from $BC$ and $v\text{ m}$ from $BAG$.

Feb/March 2016

A particle $P$ with mass $0.2\text{ kg}$ is set free from rest at point $O$ on a plane that is inclined at $30^\circ$ to the horizontal. After $t\text{ s}$ from release, $P$ has velocity $v\text{ m s}^{-1}$ and has moved a distance $x\text{ m}$ down the plane from $O$. The coefficient of friction between $P$ and the plane grows as $P$ moves down the plane, and is given by $0.1x^2$.

Feb/March 2016

A small ball is launched at $15\,\text{m s}^{-1}$ at $60^\circ$ above the horizontal. Find the distance from the point of projection when the ball is moving horizontally.

Feb/March 2017

A cylindrical container is open at the top. Its curved side and circular base are each made from the same thin material of uniform thickness. The container has radius $0.2\,\text{m}$ and height $0.9\,\text{m}$.

Feb/March 2017

The diagram depicts a uniform lamina $ABCD$ with $AB = 0.75\,\text{m}$, $AD = 0.6\,\text{m}$ and $BC = 0.9\,\text{m}$. Angle $BAD = \angle ABC = 90^\circ$.

Feb/March 2017

$O$ and $A$ are fixed points on a rough horizontal surface, where $OA = 1\,\text{m}$. A particle $P$ of mass $0.4\,\text{kg}$ is projected from $A$ horizontally in the direction $OA$ with speed $U\,\text{m s}^{-1}$ and travels along a straight line. After projection, when $P$ is $x\,\text{m}$ from $O$, its velocity is $v\,\text{m s}^{-1}$. The coefficient of friction between the surface and $P$ is $0.4$. A force of magnitude $\frac{0.8}{x}\,\text{N}$ acts on $P$ in the direction $PO$.

Feb/March 2017

An object is launched from a point on horizontal ground with speed $15\,\text{m s}^{-1}$ at an angle of $35^\circ$ above the horizontal.

Feb/March 2018

A particle $P$ is launched from point $O$ on level ground. At time $t\,\text{s}$ after the launch, its horizontal displacement from $O$ is $x\,\text{m}$ and its vertical upward displacement is $y\,\text{m}$. The path of $P$ is given by $y = 3x - 0.05x^2$.

Feb/March 2018

A tiny object with mass $0.2\,\text{kg}$ is initially at rest at point $O$ on a rough horizontal surface. The coefficient of friction between the object and the surface is $0.5$. A force of magnitude $P\,\text{N}$ acting at an angle $\theta$ below the horizontal is applied to the object. At time $t\,\text{s}$ after the force starts to act, the object's velocity is $v\,\text{m s}^{-1}$ away from $O$ (see diagram). It is given that $\tan\theta = \frac{3}{4}$ and that $P = 0.4t$ for $0 \leq t \leq 8$.

Feb/March 2018

$ABCD$ denotes a uniform square lamina with side length $0.6\text{ m}$. A circular hole of radius $r\text{ m}$ is cut in the lamina. The centre of the hole is $0.3\text{ m}$ from $AB$ and $0.25\text{ m}$ from $AD$. The lamina is freely suspended from $A$ and hangs so that the axis of symmetry makes an angle of $48^\circ$ with the horizontal (see Fig. 1).

Feb/March 2018

A small ball is launched from a point $O$ on level ground. At time $t\,\text{s}$ after launch, its horizontal displacement from $O$ is $x\,\text{m}$ and its vertical displacement upwards from $O$ is $y\,\text{m}$, where $x = 4t$ and $y = 6t - 5t^2$.

Feb/March 2019

Fig. 1 shows the cross-section of a solid cylinder from which a cylindrical hole has been drilled to form a uniform prism. The cylinder has radius $5r$ and the hole has radius $r$. The centre of the hole is $2r$ from the centre of the cylinder.

Feb/March 2019

Helen records the lengths of $150$ fish from one species in a large pond. Their lengths, measured to the nearest centimetre, are displayed in the table below.

Feb/March 2020

The driver notes the distance covered on each of 150 journeys. The distances, each rounded to the nearest km, are shown in the table below.

Feb/March 2021

In a summer camp, $250$ children sit an arithmetic test. The times taken, correct to the nearest minute, to finish the test were noted. The results are shown in the table: Time taken (in minutes): $1\!\text{-}\!30$, $31\!\text{-}\!45$, $46\!\text{-}\!65$, $66\!\text{-}\!75$, $76\!\text{-}\!100$ Frequency: $21$, $30$, $68$, $86$, $45$.

Feb/March 2022

In each year, the total number of sunshine hours, $x$, in Kintoo is measured for the month of June. The findings for the latest 60 years are shown in the table, with the classes $30 \leq x < 60$, $60 \leq x < 90$, $90 \leq x < 110$, $110 \leq x < 140$, $140 \leq x < 180$ and $180 \leq x \leq 240$ together with their frequencies.

Feb/March 2023

The times, measured in minutes, needed by $150$ students to finish a puzzle are shown in the table. The time intervals are $0 \leq t < 20$, $20 \leq t < 30$, $30 \leq t < 35$, $35 \leq t < 40$, $40 \leq t < 50$, $50 \leq t < 70$ with matching frequencies $8$, $23$, $35$, $52$, $20$, and $12$.

Feb/March 2024

For a particular plant species, the lengths of $250$ leaves are measured to the nearest centimetre. The findings are shown in the table below: Length (cm): $5$-$9$, $10$-$14$, $15$-$19$, $20$-$24$, $25$-$29$, $30$-$39$ with matching frequencies $18$, $28$, $60$, $72$, $48$, $24$.

Feb/March 2025

A frame is made from a uniform semicircular wire with radius $20\,\text{cm}$ and mass $2\,\text{kg}$, together with a uniform straight wire of length $40\,\text{cm}$ and mass $0.9\,\text{kg}$. The semicircular wire’s ends are fixed to the two ends of the straight wire (see diagram).

May/June 2010

A solid cone of uniform density has height $30\,\text{cm}$ and base radius $r\,\text{cm}$. It is positioned with its axis vertical on a rough horizontal plane. The plane is then slowly tilted, and the cone remains in equilibrium until the angle of inclination reaches $35^\circ$, when it topples. The diagram shows a cross-section of the cone.

May/June 2010

A particle is fired from a point $O$ on level ground. The initial speed is $20\,\text{m s}^{-1}$, and it is directed upwards at an angle $\theta$ to the horizontal. The particle goes through the point that is $7\,\text{m}$ above the ground and $16\,\text{m}$ horizontally from $O$, and lands on the ground at $A$.

May/June 2010

A particle $P$ with mass $0.35\,\text{kg}$ is fixed to the midpoint of a light elastic string whose natural length is $4\,\text{m}$. The two ends of the string are fastened to fixed points $A$ and $B$, which are $4.8\,\text{m}$ apart and at the same horizontal level. $P$ is in equilibrium at a point $0.7\,\text{m}$ vertically beneath the midpoint $M$ of $AB$ (see diagram).

May/June 2010

A particle $P$ with mass $0.25\,\text{kg}$ travels in a straight line on a smooth horizontal surface. $P$ begins at $O$ with speed $10\,\text{m s}^{-1}$ and travels towards a fixed point $A$ on the line. At time $t\,\text{s}$, the displacement of $P$ from $O$ is $x\,\text{m}$ and its velocity is $v\,\text{m s}^{-1}$. A resistive force of magnitude $(5 - x)\,\text{N}$ acts on $P$ in the direction towards $O$.

May/June 2010

The frame is formed from a uniform semicircular wire of radius $20\text{ cm}$ and mass $2\text{ kg}$, together with a uniform straight wire of length $40\text{ cm}$ and mass $0.9\text{ kg}$. The two ends of the semicircular wire are joined to the two ends of the straight wire (see diagram).

May/June 2010

A particle is fired from a point $O$ on level ground. Its initial speed is $20\text{ m s}^{-1}$ and it is directed upwards at an angle $\theta$ to the horizontal. The particle goes through the point that is $7\text{ m}$ above the ground and $16\text{ m}$ horizontally from $O$, and then lands on the ground at $A$.

May/June 2010

A particle $P$ with mass $0.25\text{ kg}$ travels in a straight line along a smooth horizontal surface. It begins at the point $O$ with speed $10\text{ m s}^{-1}$ and travels towards a fixed point $A$ on the line. After $t\text{ s}$, the displacement of $P$ from $O$ is $x\text{ m}$ and its velocity is $v\text{ m s}^{-1}$. A resistive force of magnitude $(5 - x)\text{ N}$ acts on $P$ in the direction towards $O$.

May/June 2010

A particle is fired horizontally with speed $12\ \text{m s}^{-1}$ from the top of a high cliff. Determine the direction of motion of the particle after $2\ \text{s}$.

May/June 2010

The cone is a uniform solid cone with height $20\ \text{cm}$ and base radius $4\ \text{cm}$. $PQ$ is a diameter of the cone’s base. The cone is in equilibrium, with $P$ resting on a horizontal surface and $PQ$ vertical, while a force is applied at $Q$. This force has magnitude $3\ \text{N}$ and acts parallel to the axis of the cone (see diagram).

May/June 2010

A particle is launched from level ground with speed $15\,\text{m s}^{-1}$ at an angle of $40^\circ$ above the horizontal. Calculate how long the particle takes to strike the ground.

May/June 2011

$AOB$ is a uniform lamina shaped as a quadrant of a circle, with centre $O$ and radius $0.6\,\text{m}$ (see diagram).

May/June 2011

A light elastic string with natural length $1.2\,\text{m}$ and modulus of elasticity $24\,\text{N}$ is fixed between points $A$ and $B$ on a smooth horizontal plane, with $AB = 1.2\,\text{m}$. A particle $P$ is attached at the midpoint of the string. $P$ is projected at $0.5\,\text{m s}^{-1}$ across the surface in a direction at right angles to $AB$ (see diagram). $P$ is momentarily at rest when it is $0.25\,\text{m}$ from $AB$.

May/June 2011

A particle $P$ leaves rest at point $O$ and moves in a straight line. The acceleration of $P$ is $(15 - 6x)\,\text{m s}^{-2}$, where $x\,\text{m}$ denotes the displacement of $P$ from $O$.

May/June 2011

A particle $P$ is launched from point $O$ on level ground. After $0.4\,\text{s}$ from the instant of projection, $P$ is $5\,\text{m}$ above the ground and $12\,\text{m}$ horizontally from $O$.

May/June 2011

A slender groove is made along a diameter on the top face of a horizontal disc with centre $O$. Particles $P$ and $Q$, with masses $0.2\,\text{kg}$ and $0.3\,\text{kg}$ respectively, are placed in the groove, and the coefficient of friction between each particle and the groove is $\mu$. The particles are joined to opposite ends of a light inextensible string of length $1\,\text{m}$. The disc turns with angular velocity $\omega\,\text{rad s}^{-1}$ about a vertical axis through $O$, and the particles travel in horizontal circles (see diagram).

May/June 2011

A uniform rod $AB$ has weight $16\,\text{N}$ and is hinged freely at $A$ to a fixed point. A force with magnitude $4\,\text{N}$, acting at right angles to the rod, is applied at $B$ (see diagram). If the rod is in equilibrium,

May/June 2011

A uniform lamina $ABCD$ is made up of a semicircle $BCD$ with centre $O$ and diameter $0.4\,\text{m}$, together with an isosceles triangle $ABD$ whose base is $BD = 0.4\,\text{m}$ and whose perpendicular height is $h\,\text{m}$. The lamina’s centre of mass is at $O$.

May/June 2011

Particle $P$, with mass $0.4\,\text{kg}$, travels in a straight line on a horizontal surface, and its velocity at time $t\,\text{s}$ is $v\,\text{m s}^{-1}$. A horizontal force of magnitude $k\sqrt{v}\,\text{N}$ acts against the motion of $P$. When $t = 0$, $v = 9$, and when $t = 2$, $v = 4$.

May/June 2011

Particle $P$ is projected from a point $O$, which is $80\,\text{m}$ vertically above horizontal ground, with speed $26\,\text{m s}^{-1}$ at an angle of $30^\circ$ below the horizontal.

May/June 2011

A stone is projected horizontally from the top of a vertical cliff $20\,\text{m}$ above the sea with speed $15\,\text{m s}^{-1}$. Calculate

May/June 2011

A smooth hemispherical shell, centred at $O$, has weight $12\,\text{N}$ and radius $0.4\,\text{m}$, and is resting on a horizontal plane. A particle of weight $W\,\text{N}$ is in equilibrium on the inside surface of the hemisphere directly below $O$. A vertically upward force of magnitude $F\,\text{N}$ is applied at the topmost point of the hemisphere, and the system is in equilibrium with its axis of symmetry at an angle of $20^\circ$ to the horizontal (see diagram).

May/June 2011

$O$ and $A$ are fixed points on a horizontal surface, where $OA = 0.5\,\text{m}$. Particle $P$, of mass $0.2\,\text{kg}$, is projected horizontally from $A$ along $OA$ with speed $3\,\text{m s}^{-1}$ and travels in a straight line (see diagram). After $t\,\text{s}$, the velocity of $P$ is $v\,\text{m s}^{-1}$ and its displacement from $O$ is $x\,\text{m}$. The coefficient of friction between the surface and $P$ is $0.5$, and a force of magnitude $\frac{0.4}{x^2}\,\text{N}$ acts on $P$ in the direction $PO$.

May/June 2011

$ABCDE$ is the cross-section of a uniform prism whose centre of mass lies in the section, and it is in equilibrium with $DE$ resting on a horizontal surface. The cross-section is a square $OBCD$ of side length $a\,\text{m}$, from which a quadrant $OAE$ of a circle with radius $1\,\text{m}$ has been cut away (see diagram).

May/June 2011

The rod $AB$ has length $1.2\,\mathrm{m}$, with end $A$ freely pivoted at a fixed point. It turns about $A$ in a vertical plane.

May/June 2012

The diagram represents a circular shape assembled from a uniform semicircular lamina of weight $11\,\mathrm{N}$ together with a uniform semicircular arc of weight $9\,\mathrm{N}$. Both parts have centre $O$ and radius $0.7\,\mathrm{m}$, and they are connected at the endpoints of their shared diameter $AB$.

May/June 2012

The diagram gives the cross-section $OABCDE$ passing through the centre of mass of a uniform prism. The interior angles of the cross-section at $O$, $A$, $C$, $D$ and $E$ are each right angles. $OA = 0.4\,\mathrm{m}$, $AB = 0.5\,\mathrm{m}$ and $BC = CD = 1\,\mathrm{m}$. The prism has weight $120\,\mathrm{N}$. A force of magnitude $F\,\mathrm{N}$ acting along $DE$ keeps the prism in equilibrium while $OA$ lies on a rough horizontal surface.

May/June 2012

A light ball $B$ is launched from point $O$ with speed $15\,\mathrm{m\,s^{-1}}$ at $41^\circ$ above the horizontal. Point $O$ is $1.6\,\mathrm{m}$ above level ground. After $t\,\mathrm{s}$ from launch, the horizontal displacement of $B$ from $O$ is $x\,\mathrm{m}$ and its vertical upward displacement from $O$ is $y\,\mathrm{m}$. A vertical fence stands $1.5\,\mathrm{m}$ from $O$ and is perpendicular to the plane of motion of $B$. The ball just clears the fence and then lands on the ground at $A$.

May/June 2012

A particle $P$ of mass $0.6\,\text{kg}$ is projected horizontally from point $O$ on a smooth horizontal surface with velocity $2\,\text{m s}^{-1}$. A horizontal force of magnitude $0.3x\,\text{N}$ acts on $P$ in the direction $OP$, where $x$ m gives the distance of $P$ from $O$.

May/June 2012

A particle $P$ with mass $0.25\,\text{kg}$ travels along a straight line on a smooth horizontal surface. At time $t$ s, its velocity is $v\,\text{m s}^{-1}$. A variable force of magnitude $3t\,\text{N}$ acts in the opposite direction to the motion of $P$.

May/June 2012

A ball is launched with velocity $25\,\text{m s}^{-1}$ at an angle of $70^{\circ}$ above the horizontal from a point $O$ on level ground. It then rebounds once on the ground at a point $P$ before coming to rest at a point $Q$. The distance $PQ$ is $17.1\,\text{m}$.

May/June 2012

The diagram depicts a uniform lamina $ABCDEF$ made by taking a semicircle with centre $O$ and radius $1\,\text{m}$ and cutting away a semicircular part with centre $O$ and radius $r\,\text{m}$. The centre of mass of the lamina is on the arc $ABC$. When the lamina is freely suspended at $F$, it hangs in equilibrium.

May/June 2012

A particle $P$ is fired with speed $25\,\text{m s}^{-1}$ at an angle of $30^\circ$ above the horizontal from point $O$ on horizontal ground. Calculate the distance $OP$ after $2\,\text{s}$.

May/June 2012

The diagram represents a uniform object $ABC$ with weight $3\,\text{N}$, shaped as an arc of a circle with centre $O$ and radius $0.7\,\text{m}$. The angle $AOC$ is $2$ radians. The object is in equilibrium with $A$ resting on a horizontal surface and $C$ positioned vertically above $A$. This equilibrium is kept by a horizontal force of magnitude $F\,\text{N}$ acting at $C$ in the plane of the object.

May/June 2012

A particle $P$ with mass $0.2\,\text{kg}$ is fired horizontally from a fixed point $O$ and travels in a straight line on a smooth horizontal surface. A force of magnitude $0.4x\,\text{N}$ acts on $P$ towards $O$, where $x\,\text{m}$ denotes the displacement of $P$ from $O$.

May/June 2012

A light elastic string, with natural length $3\,\text{m}$ and modulus of elasticity $45\,\text{N}$, has a particle $P$ of mass $0.6\,\text{kg}$ fixed at its mid-point. Its two ends are fastened to points $A$ and $B$ on the line of greatest slope of a smooth plane inclined at $30^\circ$ to the horizontal. The separation $AB$ is $4\,\text{m}$, and $A$ is above $B$.

May/June 2012

A uniform lamina $ABCDE$ is made up of a rectangle $BCDE$ together with an isosceles triangle $ABE$, the two sharing the side $BE$. For the triangle, $AB = AE$, $BE = a\,\text{m}$ and the perpendicular height is $h\,\text{m}$. For the rectangle, $BC = DE = 0.5\,\text{m}$ and $CD = BE = a\,\text{m}$ (see diagram).

May/June 2012

The projectile’s path is given by $y = 0.6x - 0.017x^2$, with the axes taken as horizontal and vertically upward through the point of projection.

May/June 2012

A small ball is launched from point $O$ on level ground with speed $20\,\text{m s}^{-1}$ at an angle of $45^\circ$ above the horizontal. After time $t$ from projection, its horizontal displacement from $O$ is $x$ m and its upward vertical displacement from $O$ is $y$ m.

May/June 2013

A uniform object $ABC$ is made from two rods $AB$ and $BC$ fixed rigidly together at right angles at $B$. Rod $AB$ is $0.3\,\text{m}$ long and rod $BC$ is $0.2\,\text{m}$ long. The object is in contact with a rough horizontal surface at $A$, with rod $AB$ standing vertically. It is kept in equilibrium by a horizontal force of magnitude $4\,\text{N}$ applied at $B$ and directed along $CB$ (see diagram).

May/June 2013

The cross-section $OABC$ passes through the centre of mass of a uniform prism whose weight is $20\,\text{N}$. This cross-section has the form of a sector of a circle with centre $O$, radius $OA = r\,\text{m}$ and angle $AOC = \frac{2}{3}\pi$ radians. The prism rests on a plane inclined at an angle $\theta$ radians to the horizontal, where $\theta < \tfrac{1}{3}\pi$. The line $OC$ is along a line of greatest slope, with $O$ higher than $C$. The prism is freely hinged to the plane at $O$. A force of magnitude $15\,\text{N}$ acts at $A$, directed towards the plane and at right angles to it (see diagram).

May/June 2013

A uniform semicircular lamina with radius $0.25\,\text{m}$ has diameter $AB$. It is hung freely from a fixed point at $A$ and comes to equilibrium.

May/June 2013

A ball $B$ is launched from point $O$ on horizontal ground at an angle of $40^{\circ}$ above the horizontal. $B$ lands on the ground $1.8\,\text{s}$ after projection. Calculate

May/June 2013

A uniform solid cone, with height $0.6\,\text{m}$ and mass $0.5\,\text{kg}$, is arranged so that its axis of symmetry is vertical and its vertex $V$ is at the top. The cone has a semi-vertical angle of $60^{\circ}$ and its surface is smooth. It is fixed on a horizontal surface. A particle $P$ of mass $0.2\,\text{kg}$ is attached to $V$ by a light inextensible string of length $0.4\,\text{m}$ (see diagram).

May/June 2013

A particle $P$ is projected at $15\,\text{m s}^{-1}$ at $60^\circ$ above the horizontal. Determine the direction of motion of $P$ at the instant $0.9\,\text{s}$ after projection.

May/June 2013

A ball is launched horizontally at speed $5\,\text{m s}^{-1}$ from the top of a $30\,\text{m}$ tower. The tower is built on horizontal ground.

May/June 2013

A uniform solid cone with height $1.2\,\text{m}$ and semi-vertical angle $\theta^\circ$ is split into two sections by a cut that is parallel to the circular base and lies $0.4\,\text{m}$ above it. The top conical section, $C$, has weight $16\,\text{N}$, while the bottom section, $L$, has weight $38\,\text{N}$. The solid is then in equilibrium, with the larger plane face of $L$ resting on a horizontal surface and the smaller plane face of $L$ covered by the base of $C$ (see diagram).

May/June 2013

A uniform metal frame $OABC$ is formed from a semicircular arc $ABC$ with radius $1.8\text{ m}$ together with a straight rod $AOC$ where $AO = OC = 1.8\text{ m}$ (see diagram). A uniform semicircular lamina of radius $1.8\text{ m}$ weighs $27.5\text{ N}$. A non-uniform object is produced by fixing the frame $OABC$ around the edge of the lamina. The object is hung freely from a fixed point at $A$ and comes to equilibrium. The diameter $AOC$ of the object is inclined at an angle of $22^\circ$ to the vertical.

May/June 2014

A non-uniform rod $AB$ has weight $6\,\text{N}$ and is in limiting equilibrium, with end $A$ touching a rough vertical wall. $AB = 1.2\,\text{m}$, the centre of mass of the rod is $0.8\,\text{m}$ from $A$, and the angle between $AB$ and the downward vertical is $\theta^\circ$. At $B$, a force of magnitude $10\,\text{N}$ is applied at an angle of $30^\circ$ to the upwards vertical (see diagram). The rod and the line of action of the $10\,\text{N}$ force both lie in a vertical plane perpendicular to the wall.

May/June 2014

A uniform metal frame $OABC$ is constructed from a semicircular arc $ABC$ with radius $1.8\,\text{m}$ and a straight rod $AOC$, where $AO = OC = 1.8\,\text{m}$ (see diagram). A uniform semicircular lamina of radius $1.8\,\text{m}$ has weight $27.5\,\text{N}$. By attaching the frame $OABC$ around the boundary of the lamina, a non-uniform object is produced. The object is then freely hung from a fixed point at $A$ and is in equilibrium. The diameter $AOC$ of the object is at an angle of $22^\circ$ to the vertical.

May/June 2014

A uniform lamina $ABC$, shaped as an isosceles triangle, has weight $24\,\text{N}$. The perpendicular height from $A$ to $BC$ is $12\,\text{cm}$. The lamina is positioned in a vertical plane in equilibrium, with vertex $A$ touching a horizontal surface. Angle $BAC = 100^\circ$ and $AB$ is inclined at an angle of $10^\circ$ to the horizontal. Equilibrium is upheld by a force of magnitude $F\,\text{N}$ acting along $BC$ (see diagram).

May/June 2014

A small block $B$ of mass $0.2\,\text{kg}$ is set at a fixed point $O$ on a smooth horizontal surface. A horizontal force of magnitude $0.42\,\text{N}$ acts on $B$. At time $t\,\text{s}$ after the force is first applied, the velocity of $B$ away from $O$ is $v\,\text{m s}^{-1}$.

May/June 2014

A small ball is projected horizontally at $5\,\text{m s}^{-1}$ from point $O$ on the roof of a building. After $t\,\text{s}$ have passed since it was projected, its horizontal displacement from $O$ is $x\,\text{m}$ and its vertical displacement downwards from $O$ is $y\,\text{m}$.

May/June 2014

The diagram depicts a container made up of a bowl with weight $14\,\text{N}$ and a handle with weight $8\,\text{N}$. The bowl is a uniform hemispherical shell with centre $O$ and radius $0.3\,\text{m}$. The handle takes the form of a uniform semicircular arc of radius $0.3\,\text{m}$ and is freely hinged to the bowl at $A$ and $B$, where $AB$ is a diameter of the bowl.

May/June 2014

One end of a light elastic string with natural length $0.7\,\text{m}$ is fixed at point $A$ on a smooth horizontal surface. The other end is fastened to a particle $P$ of mass $0.3\,\text{kg}$, which is initially held at point $B$ on the horizontal surface, where $AB = 1.2\,\text{m}$. It is given that $P$ is released from rest at $B$ and that when $AP = 0.9\,\text{m}$, the particle has speed $4\,\text{m s}^{-1}$.

May/June 2015

From a point $O$ on horizontal ground, a stone is projected. Its trajectory is described by $y = 1.2x - 0.15x^2$, where $x\,\text{m}$ and $y\,\text{m}$ represent respectively the stone's horizontal displacement from $O$ and its upward vertical displacement from $O$.

May/June 2015

One end of a light inextensible string is fastened to a fixed point $A$, while the other end is joined to a particle $P$. Particle $P$ travels at constant angular speed $5\,\text{rad s}^{-1}$ in a horizontal circle with centre $O$ directly beneath $A$. The string is at angle $\theta$ to the vertical (see diagram). The tension in the string is three times the weight of $P$.

May/June 2015

A small ball $B$ is launched from a point $O$ above horizontal ground, with initial speed $15\,\text{m s}^{-1}$ at an angle of projection of $30^\circ$ above the horizontal (see diagram). The ball hits the ground $3\,\text{s}$ after projection.

May/June 2015

The diagram presents the cross-section $OABCDE$ passing through the centre of mass of a uniform prism resting on a rough inclined plane. The section $ADEO$ forms a rectangle with $AD = OE = 0.6\,\text{m}$ and $DE = AO = 0.8\,\text{m}$. The section $BCD$ is an isosceles triangle in which angle $BCD$ is a right angle, and $A$ is the mid-point of $BD$. The plane is at $45^\circ$ to the horizontal, $BC$ is along a line of greatest slope of the plane and $DE$ is horizontal. The prism weighs $21\,\text{N}$, and equilibrium is maintained by a horizontal force of magnitude $P\,\text{N}$ acting along $ED$.

May/June 2015

A triangular frame $ABC$ is formed by two uniform rigid rods, each of length $0.8\,\text{m}$ and weight $3\,\text{N}$, together with a longer uniform rod of weight $4\,\text{N}$. In the frame, $AB = BC$, and $\angle BAC = \angle BCA = 30\degree$.

May/June 2015

One end of a light inextensible string with length $0.5\,\text{m}$ is fixed to point $A$. The string’s other end is joined to a particle $P$ whose weight is $6\,\text{N}$. A second light inextensible string of length $0.5\,\text{m}$ links $P$ to a fixed point $B$, and $B$ is $0.8\,\text{m}$ vertically beneath $A$. Particle $P$ moves at constant speed in a horizontal circle centred at the midpoint of $AB$. Both strings are taut.

May/June 2015

A small ball $B$ is launched from $O$ with speed $U\,\text{m s}^{-1}$ at an angle of $\theta\degree$ above the horizontal. Two seconds later, $B$ hits a smooth wall inclined at $60\degree$ to the horizontal. At that instant, the speed of $B$ is $18\,\text{m s}^{-1}$ and its direction of motion is perpendicular to the wall (see Fig. 1). $B$ rebounds from the wall with speed $V\,\text{m s}^{-1}$, again moving perpendicular to the wall. After $0.8\,\text{s}$, $B$ meets the wall again at a lower point $A$ (see Fig. 2).

May/June 2015

The diameter $AB$ of a uniform semicircular lamina is $0.8\text{ m}$. The lamina is placed in a vertical plane, with point $B$ touching a rough horizontal surface and $A$ positioned vertically above $B$. A force of magnitude $6\text{ N}$ acting in the plane of the lamina is applied at $A$ at an angle of $20^\circ$ below the horizontal, and this keeps the lamina in equilibrium.

May/June 2015

Starting from point $O$ on horizontal ground, a particle $P$ is projected with speed $V\text{ m s}^{-1}$ at an angle of $60^\circ$ above the horizontal. $1.5\text{ s}$ after projection, $P$ is travelling at an angle of $45^\circ$ above the horizontal.

May/June 2015

A small ball $B$ is launched from a point $1.5\text{ m}$ above horizontal ground with initial speed $29\text{ m s}^{-1}$ at an angle of $30^\circ$ above the horizontal.

May/June 2015

A uniform triangular prism with weight $20\text{ N}$ is at rest on a horizontal table. $ABC$ is the cross-section through the prism’s centre of mass, with $BC = 0.5\text{ m}$, $AB = 0.4\text{ m}$, $AC = 0.3\text{ m}$ and angle $BAC = 90^\circ$. The vertical plane $ABC$ is perpendicular to the table edge. Point $D$ on $AC$ lies at the table edge, and $AD = 0.25\text{ m}$. One end of a light elastic string of natural length $0.6\text{ m}$ and modulus of elasticity $48\text{ N}$ is fixed to $C$ and a particle of mass $2.5\text{ kg}$ is attached to the other end of the string. The particle is let go from rest at $C$ and moves down vertically.

May/June 2015

A cyclist and her bicycle together have a combined mass of $60\text{ kg}$. She moves along a horizontal straight line, while a constant force of $150\text{ N}$ acts in the direction of travel. The motion is resisted by a force with magnitude $12v\text{ N}$, where $v\text{ m s}^{-1}$ is the cyclist’s speed at time $t\text{ s}$ after she goes past a fixed point $A$.

May/June 2015

A small ball is fired from a point on horizontal ground with speed $16\text{ m s}^{-1}$ at an angle of $45^{\circ}$ above the horizontal.

May/June 2016

A uniform wire is bent into a semicircular arc, and the diameter $AB$ has length $0.8\text{ m}$. The wire is joined to a vertical wall by a smooth hinge at $A$. It is in equilibrium with $AB$ making an angle of $70^{\circ}$ to the upward vertical, supported by a light string fixed to $B$. The other end of the string is fixed at point $C$ on the wall, $0.8\text{ m}$ vertically above $A$. The tension in the string is $15\text{ N}$ (see diagram).

May/June 2016

A uniform solid cone has base radius $0.4\text{ m}$ and height $4.4\text{ m}$. A uniform solid cylinder has radius $0.4\text{ m}$ and weight equal to the weight of the cone. The cone and cylinder are joined together so that the cone’s base and one circular end of the cylinder touch, with their circumferences matching exactly. The combined body is in equilibrium with its circular base resting on a plane that is inclined at $20^{\circ}$ to the horizontal (see diagram).

May/June 2016

A small ball $B$ is launched from a point $O$ on horizontal ground with speed $12\,\text{m s}^{-1}$ at an angle of $30^\circ$ above the horizontal. After $0.8\,\text{s}$, $B$ is $0.5\,\text{m}$ vertically above the top of a vertical post.

May/June 2016

The point $O$ lies $8\,\text{m}$ above a horizontal plane. A particle $P$ is projected from $O$. Once projected, the horizontal displacement of $P$ from $O$ is $x\,\text{m}$ and its vertically upward displacement from $O$ is $y\,\text{m}$. The path of $P$ is given by $y = 2x - x^2$.

May/June 2016

A uniform body is obtained by boring a cylindrical hole right through a rectangular block. The axis of the cylindrical hole is perpendicular to the cross-section $ABCD$ that passes through the centre of mass of the body. $AB = CD = 0.7\,\text{m}$, $BC = AD = 0.4\,\text{m}$, and the centre of the hole is $0.1\,\text{m}$ from $AB$ and $0.2\,\text{m}$ from $AD$ (see diagram). The hole has cross-sectional area $0.03\,\text{m}^2$.

May/June 2016

Particle P, of mass $0.4\,\text{kg}$, is initially at rest at point A on a rough horizontal surface. A horizontal force of magnitude $0.6t\,\text{N}$ acts on P in the direction away from A, where $t\,\text{s}$ denotes the time after P is placed at A. The coefficient of friction between A and the surface is $0.3$, and at time $t\,\text{s}$ the displacement of P from A is $x\,\text{m}$.

May/June 2016

$OA$ is a rod that turns in a horizontal circle about a vertical axis passing through $O$. A particle $P$ with mass $0.2\,\text{kg}$ is fixed to the midpoint of a light inextensible string. One end of the string is fastened to the rod at $A$, and the other end is fastened to a point $B$ on the axis. It is given that $OA = OB$, angle $OAP =$ angle $OBP = 30^\circ$, and $P$ is $0.4\,\text{m}$ from the axis. The rod and the particle rotate together about the axis, with $P$ lying in the plane $OAB$ (see diagram).

May/June 2016

A small ball is launched at a speed of $16\,\text{m s}^{-1}$ at an angle of $45^\circ$ above the horizontal from a point on level ground.

May/June 2016

A uniform wire is shaped into a semicircular arc, and its diameter AB has a length of $0.8\,\text{m}$. It is fixed to a vertical wall by a smooth hinge at $A$. The wire is in equilibrium, with $AB$ inclined at $70^\circ$ to the upward vertical, and it is supported by a light string attached at $B$. The other end of the string is fastened to the point $C$ on the wall, $0.8\,\text{m}$ vertically above $A$. The tension in the string is $15\,\text{N}$ (see diagram).

May/June 2016

A particle $P$ with mass $0.4\,\text{kg}$ is let go from rest at point $O$ on a smooth plane inclined at $30^\circ$ to the horizontal. If the displacement of $P$ from $O$ is $x\,\text{m}$ down the plane, then its velocity is $v\,\text{m s}^{-1}$. A force of magnitude $0.8e^{-x}\,\text{N}$ acts on $P$ up the plane along the line of greatest slope through $O$.

May/June 2016

One end of a light inextensible string is fixed at point $A$. Its other end is fastened to a particle $P$ of mass $m\,\text{kg}$, which hangs vertically beneath $A$. $P$ is also connected to one end of a light elastic string with natural length $0.25\,\text{m}$. The other end of that string is attached to point $B$, which is $0.6\,\text{m}$ from $P$ and lies on the same horizontal level as $P$. A horizontal force of magnitude $7\,\text{N}$ acting on $P$ keeps the system in equilibrium (see Fig. 1).

May/June 2017

An object is formed by taking a uniform solid hemisphere of radius $0.56\,\text{m}$ with centre $O$ and then removing from it a hemisphere of radius $0.28\,\text{m}$ with centre $O$. The diagram shows a cross-section of the object through $O$.

May/June 2017