Questions - CAIE Further Paper 3

Find the value of $k$.
Find the value of $\cos \theta$.
\includegraphics[max width=\textwidth, alt={}, center]{1c53c407-25ea-43fc-a571-74ba1fffea8f-06_584_695_264_667} The diagram shows the cross-section $A B C D$ of a uniform solid object which is formed by removing a cone with cross-section $D C E$ from the top of a larger cone with cross-section $A B E$. The perpendicular distance between $A B$ and $D C$ is $h$, the diameter $A B$ is $6 r$ and the diameter $D C$ is $2 r$.
Find an expression, in terms of $h$, for the distance of the centre of mass of the solid object from $A B$.
The object is freely suspended from the point $B$ and hangs in equilibrium. The angle between $A B$ and the downward vertical through $B$ is $\theta$.
Given that $h = \frac { 13 } { 4 } r$, find the value of $\tan \theta$.

CAIE Further Paper 3 2021 November Q1

1 A particle is projected with speed $u$ at an angle $\alpha$ above the horizontal from a point $O$ on a horizontal plane. The particle moves freely under gravity.

Write down the horizontal and vertical components of the velocity of the particle at time $T$ after projection.
At time $T$ after projection, the direction of motion of the particle is perpendicular to the direction of projection.
Express $T$ in terms of $u , g$ and $\alpha$.
Deduce that $\mathrm { T } > \frac { \mathrm { u } } { \mathrm { g } }$.

CAIE Further Paper 3 2021 November Q2

2 A light spring $A B$ has natural length $a$ and modulus of elasticity 5 mg . The end $A$ of the spring is attached to a fixed point on a smooth horizontal surface. A particle $P$ of mass $m$ is attached to the end $B$ of the spring. The spring and particle $P$ are at rest on the surface. Another particle $Q$ of mass $k m$ is moving with speed $\sqrt { 4 \mathrm { ga } }$ along the horizontal surface towards $P$ in the direction $B A$. The particles $P$ and $Q$ collide directly and coalesce. In the subsequent motion the greatest amount by which the spring is compressed is $\frac { 1 } { 5 } a$. Find the value of $k$.
\includegraphics[max width=\textwidth, alt={}, center]{e34abb4b-1c6c-4f39-836d-467ed18345eb-04_307_1088_274_470} Particles $A$ and $B$, of masses $m$ and $3 m$ respectively, are connected by a light inextensible string of length $a$ that passes through a fixed smooth ring $R$. Particle $B$ hangs in equilibrium vertically below the ring. Particle $A$ moves in horizontal circles with speed $v$. Particles $A$ and $B$ are at the same horizontal level. The angle between $A R$ and $B R$ is $\theta$ (see diagram).

Show that $\cos \theta = \frac { 1 } { 3 }$.
Find an expression for $v$ in terms of $a$ and $g$.
\includegraphics[max width=\textwidth, alt={}, center]{e34abb4b-1c6c-4f39-836d-467ed18345eb-06_597_803_258_625} An object is formed by removing a solid cylinder, of height $k a$ and radius $\frac { 1 } { 2 } a$, from a uniform solid hemisphere of radius $a$. The axes of symmetry of the hemisphere and the cylinder coincide and one circular face of the cylinder coincides with the plane face of the hemisphere. $A B$ is a diameter of the circular face of the hemisphere (see diagram).

CAIE Further Paper 3 2021 November Q6

6 A particle $P$ of mass 2 kg moves along a horizontal straight line. The point $O$ is a fixed point on this line. At time $t$ s the velocity of $P$ is $v \mathrm {~ms} ^ { - 1 }$ and the displacement of $P$ from $O$ is $x \mathrm {~m}$.
A force of magnitude $\left( 8 x - \frac { 128 } { x ^ { 3 } } \right) \mathrm { N }$ acts on $P$ in the direction $O P$. When $\mathrm { t } = 0 , \mathrm { x } = 8$ and $\mathrm { v } = - 15$.

Show that $\mathrm { v } = - \frac { 2 } { \mathrm { x } } \left( \mathrm { x } ^ { 2 } - 4 \right)$.
Find an expression for $x$ in terms of $t$.

CAIE Further Paper 3 2021 November Q7

7 One end of a light inextensible string of length $a$ is attached to a fixed point $O$. The other end of the string is attached to a particle $P$ of mass $m$. The particle $P$ is held vertically below $O$ with the string taut and then projected horizontally. When the string makes an angle of $60 ^ { \circ }$ with the upward vertical, $P$ becomes detached from the string. In its subsequent motion, $P$ passes through the point $A$ which is a distance $a$ vertically above $O$.

The speed of $P$ when it becomes detached from the string is $V$. Use the equation of the trajectory of a projectile to find $V$ in terms of $a$ and $g$.
Find, in terms of $m$ and $g$, the tension in the string immediately after $P$ is initially projected horizontally.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

CAIE Further Paper 3 2021 November Q3

3 A light elastic string has natural length $a$ and modulus of elasticity 12 mg . One end of the string is attached to a fixed point $O$. The other end of the string is attached to a particle of mass $m$. The particle hangs in equilibrium vertically below $O$. The particle is pulled vertically down and released from rest with the extension of the string equal to $e$, where $\mathrm { e } > \frac { 1 } { 3 } \mathrm { a }$. In the subsequent motion the particle has speed $\sqrt { 2 \mathrm { ga } }$ when it has ascended a distance $\frac { 1 } { 3 } a$. Find $e$ in terms of $a$.
\includegraphics[max width=\textwidth, alt={}, center]{b10c65ef-dafd-4746-be5b-789130b7d030-06_488_496_269_781} A uniform lamina $A E C F$ is formed by removing two identical triangles $B C E$ and $C D F$ from a square lamina $A B C D$. The square has side $3 a$ and $E B = D F = h$ (see diagram).

Find the distance of the centre of mass of the lamina $A E C F$ from $A D$ and from $A B$, giving your answers in terms of $a$ and $h$.
The lamina $A E C F$ is placed vertically on its edge $A E$ on a horizontal plane.
Find, in terms of $a$, the set of values of $h$ for which the lamina remains in equilibrium.

CAIE Further Paper 3 2021 November Q6

6 A particle $P$, of mass $m$, is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. The particle $P$ moves in complete vertical circles about $O$ with the string taut. The points $A$ and $B$ are on the path of $P$ with $A B$ a diameter of the circle. $O A$ makes an angle $\theta$ with the downward vertical through $O$ and $O B$ makes an angle $\theta$ with the upward vertical through $O$. The speed of $P$ when it is at $A$ is $\sqrt { 5 a g }$. The ratio of the tension in the string when $P$ is at $A$ to the tension in the string when $P$ is at $B$ is $9 : 5$.

Find the value of $\cos \theta$.
Find, in terms of $a$ and $g$, the greatest speed of $P$ during its motion.
\includegraphics[max width=\textwidth, alt={}, center]{b10c65ef-dafd-4746-be5b-789130b7d030-12_613_718_251_676} The smooth vertical walls $A B$ and $C B$ are at right angles to each other. A particle $P$ is moving with speed $u$ on a smooth horizontal floor and strikes the wall $C B$ at an angle $\alpha$. It rebounds at an angle $\beta$ to the wall $C B$. The particle then strikes the wall $A B$ and rebounds at an angle $\gamma$ to that wall (see diagram). The coefficient of restitution between each wall and $P$ is $e$.
Show that $\tan \beta = e \tan \alpha$.
Express $\gamma$ in terms of $\alpha$ and explain what this result means about the final direction of motion of $P$.
As a result of the two impacts the particle loses $\frac { 8 } { 9 }$ of its initial kinetic energy.
Given that $\alpha + \beta = 90 ^ { \circ }$, find the value of $e$ and the value of $\tan \alpha$.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

CAIE Further Paper 3 2022 November Q1

1 A particle of mass 2 kg is attached to one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a fixed point on a smooth horizontal surface. The particle is moving in a circular path on the surface. The tension in the string is 20 N . Find how many revolutions the particle makes per minute.

CAIE Further Paper 3 2022 November Q2

2 A light elastic string has natural length $a$ and modulus of elasticity 4 mg . One end of the string is fixed to a point $O$ on a smooth horizontal surface. A particle $P$ of mass $m$ is attached to the other end of the string. The particle $P$ is projected along the surface in the direction $O P$. When the length of the string is $\frac { 5 } { 4 } a$, the speed of $P$ is $v$. When the length of the string is $\frac { 3 } { 2 } a$, the speed of $P$ is $\frac { 1 } { 2 } v$.

Find an expression for $v$ in terms of $a$ and $g$.
Find, in terms of $g$, the acceleration of $P$ when the stretched length of the string is $\frac { 3 } { 2 } a$.
\includegraphics[max width=\textwidth, alt={}, center]{7febbd80-4cbb-4b2e-b022-d6a20e7e13aa-04_552_1059_264_502} A smooth cylinder is fixed to a rough horizontal surface with its axis of symmetry horizontal. A uniform rod $A B$, of length $4 a$ and weight $W$, rests against the surface of the cylinder. The end $A$ of the rod is in contact with the horizontal surface. The vertical plane containing the rod $A B$ is perpendicular to the axis of the cylinder. The point of contact between the rod and the cylinder is $C$, where $A C = 3 a$. The angle between the rod and the horizontal surface is $\theta$ where $\tan \theta = \frac { 3 } { 4 }$ (see diagram). The coefficient of friction between the rod and the horizontal surface is $\frac { 6 } { 7 }$. A particle of weight $k W$ is attached to the rod at $B$. The rod is about to slip. The normal reaction between the rod and the cylinder is $N$.

CAIE Further Paper 3 2022 November Q4

4 A particle of mass 0.5 kg moves along a horizontal straight line. Its velocity is $v \mathrm {~ms} ^ { - 1 }$ at time $t \mathrm {~s}$. The forces acting on the particle are a driving force of magnitude 50 N and a resistance of magnitude $2 v ^ { 2 } \mathrm {~N}$. The initial velocity of the particle is $3 \mathrm {~ms} ^ { - 1 }$.

Find an expression for $v$ in terms of $t$.
Deduce the limiting value of $v$.

CAIE Further Paper 3 2022 November Q5

5 A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. The string is held taut with $O P$ horizontal. The particle $P$ is projected vertically downwards with speed $\sqrt { \frac { 1 } { 3 } \mathrm { ag } }$ and starts to move in a vertical circle. $P$ passes through the lowest point of the circle and reaches the point $Q$ where $O Q$ makes an angle $\theta$ with the downward vertical. At $Q$ the speed of $P$ is $\sqrt { \mathrm { kag } }$ and the tension in the string is $\frac { 11 } { 6 } \mathrm { mg }$.

Find the value of $k$ and the value of $\cos \theta$.
At $Q$ the particle $P$ becomes detached from the string.
In the subsequent motion, find the greatest height reached by $P$ above the level of the lowest point of the circle.

CAIE Further Paper 3 2022 November Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{7febbd80-4cbb-4b2e-b022-d6a20e7e13aa-10_426_1191_267_438} Two uniform smooth spheres $A$ and $B$ of equal radii have masses $m$ and $k m$ respectively. The two spheres are moving on a horizontal surface with speeds $u$ and $\frac { 5 } { 8 } u$ respectively. Immediately before the spheres collide, $A$ is travelling along the line of centres, and $B$ 's direction of motion makes an angle $\alpha$ with the line of centres (see diagram). The coefficient of restitution between the spheres is $\frac { 2 } { 3 }$ and $\tan \alpha = \frac { 3 } { 4 }$. After the collision, the direction of motion of $B$ is perpendicular to the line of centres.

Find the value of $k$.
Find the loss in the total kinetic energy as a result of the collision.

CAIE Further Paper 3 2022 November Q7

7 A particle $P$ is projected with speed $\mathrm { Vms } ^ { - 1 }$ at an angle $75 ^ { \circ }$ above the horizontal from a point $O$ on a horizontal plane. It then moves freely under gravity.

Show that the total time of flight, in seconds, is $\frac { 2 \mathrm {~V} } { \mathrm {~g} } \sin 75 ^ { \circ }$.
A smooth vertical barrier is now inserted with its lower end on the plane at a distance 15 m from $O$. The particle is projected as before but now strikes the barrier, rebounds and returns to $O$. The coefficient of restitution between the barrier and the particle is $\frac { 3 } { 5 }$.
Explain why the total time of flight is unchanged.
Find an expression for $V$ in terms of $g$.
If you use the following page to complete the answer to any question, the question number must be clearly shown.

CAIE Further Paper 3 2022 November Q1

1 A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. The string is held taut with $O P$ making an angle $\alpha$ with the downward vertical, where $\cos \alpha = \frac { 2 } { 3 }$. The particle $P$ is projected perpendicular to $O P$ in an upwards direction with speed $\sqrt { 3 a g }$. It then starts to move along a circular path in a vertical plane. Find the cosine of the angle between the string and the upward vertical when the string first becomes slack.
\includegraphics[max width=\textwidth, alt={}, center]{9b3f3add-17fd-4597-bd5d-27e3abb527be-03_671_455_255_845} A uniform lamina is in the form of a triangle $A B C$ in which angle $B$ is a right angle, $\mathrm { AB } = 9 \mathrm { a }$ and $\mathrm { BC } = 6 \mathrm { a }$. The point $D$ is on $B C$ such that $\mathrm { BD } = \mathrm { x }$ (see diagram). The region $A B D$ is removed from the lamina. The resulting shape $A D C$ is placed with the edge $D C$ on a horizontal surface and the plane $A D C$ is vertical. Find the set of values of $x$, in terms of $a$, for which the shape is in equilibrium.

CAIE Further Paper 3 2022 November Q3

3 One end of a light elastic string, of natural length $a$ and modulus of elasticity $\frac { 16 } { 3 } \mathrm { Mg }$, is attached to a fixed point $O$. A particle $P$ of mass $4 M$ is attached to the other end of the string and hangs vertically in equilibrium. Another particle of mass $2 M$ is attached to $P$ and the combined particle is then released from rest. The speed of the combined particle when it has descended a distance $\frac { 1 } { 4 } a$ is $v$. Find an expression for $v$ in terms of $g$ and $a$.

CAIE Further Paper 3 2022 November Q4

4 A particle $P$ of mass 5 kg moves along a horizontal straight line. At time $t \mathrm {~s}$, the velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and its displacement from a fixed point $O$ on the line is $x \mathrm {~m}$. The forces acting on $P$ are a force of magnitude $\frac { 500 } { v } \mathrm {~N}$ in the direction $O P$ and a resistive force of magnitude $\frac { 1 } { 2 } v ^ { 2 } \mathrm {~N}$. When $t = 0 , x = 0$ and $v = 5$.

Find an expression for $v$ in terms of $x$.
State the value that the speed approaches for large values of $x$.

CAIE Further Paper 3 2022 November Q5

5 A particle $P$ is projected with speed $u \mathrm {~ms} ^ { - 1 }$ at an angle of $\theta$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of $P$ from $O$ at a subsequent time $t \mathrm {~s}$ are denoted by $x \mathrm {~m}$ and $y \mathrm {~m}$ respectively.

Show that the equation of the trajectory is given by $$\mathrm { y } = \mathrm { x } \tan \theta - \frac { \mathrm { gx } ^ { 2 } } { 2 \mathrm { u } ^ { 2 } } \left( 1 + \tan ^ { 2 } \theta \right)$$ In the subsequent motion $P$ passes through the point with coordinates $( 30,20 )$.
Given that one possible value of $\tan \theta$ is $\frac { 4 } { 3 }$, find the other possible value of $\tan \theta$.
\includegraphics[max width=\textwidth, alt={}, center]{9b3f3add-17fd-4597-bd5d-27e3abb527be-10_451_1339_258_404} A light inextensible string is threaded through a fixed smooth ring $R$ which is at a height $h$ above a smooth horizontal surface. One end of the string is attached to a particle $A$ of mass $m$. The other end of the string is attached to a particle $B$ of mass $\frac { 6 } { 7 } m$. The particle $A$ moves in a horizontal circle on the surface. The particle $B$ hangs in equilibrium below the ring and above the surface (see diagram). When $A$ has constant angular speed $\omega$, the angle between $A R$ and $B R$ is $\theta$ and the normal reaction between $A$ and the surface is $N$. When $A$ has constant angular speed $\frac { 3 } { 2 } \omega$, the angle between $A R$ and $B R$ is $\alpha$ and the normal reaction between $A$ and the surface is $\frac { 1 } { 2 } N$.
Show that $\cos \theta = \frac { 4 } { 9 } \cos \alpha$.
Find $N$ in terms of $m$ and $g$ and find the value of $\cos \alpha$.
\includegraphics[max width=\textwidth, alt={}, center]{9b3f3add-17fd-4597-bd5d-27e3abb527be-12_413_974_255_587} Two uniform smooth spheres $A$ and $B$ of equal radii have masses $m$ and $\frac { 1 } { 2 } m$ respectively. The two spheres are moving on a horizontal surface when they collide. Immediately before the collision, sphere $A$ is travelling with speed $u$ and its direction of motion makes an angle $\alpha$ with the line of centres. Sphere $B$ is travelling with speed $2 u$ and its direction of motion makes an angle $\beta$ with the line of centres (see diagram). The coefficient of restitution between the spheres is $\frac { 5 } { 8 }$ and $\alpha + \beta = 90 ^ { \circ }$.
Find the component of the velocity of $B$ parallel to the line of centres after the collision, giving your answer in terms of $u$ and $\alpha$.
The direction of motion of $B$ after the collision is parallel to the direction of motion of $A$ before the collision.
Find the value of $\tan \alpha$.
If you use the following page to complete the answer to any question, the question number must be clearly shown.

CAIE Further Paper 3 2022 November Q2

Find an expression for $v$ in terms of $a$ and $g$.
Find, in terms of $g$, the acceleration of $P$ when the stretched length of the string is $\frac { 3 } { 2 } a$.
\includegraphics[max width=\textwidth, alt={}, center]{5e95e0c9-d47d-4f2b-89da-ab949b9661f4-04_552_1059_264_502} A smooth cylinder is fixed to a rough horizontal surface with its axis of symmetry horizontal. A uniform rod $A B$, of length $4 a$ and weight $W$, rests against the surface of the cylinder. The end $A$ of the rod is in contact with the horizontal surface. The vertical plane containing the rod $A B$ is perpendicular to the axis of the cylinder. The point of contact between the rod and the cylinder is $C$, where $A C = 3 a$. The angle between the rod and the horizontal surface is $\theta$ where $\tan \theta = \frac { 3 } { 4 }$ (see diagram). The coefficient of friction between the rod and the horizontal surface is $\frac { 6 } { 7 }$. A particle of weight $k W$ is attached to the rod at $B$. The rod is about to slip. The normal reaction between the rod and the cylinder is $N$.

CAIE Further Paper 3 2022 November Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{5e95e0c9-d47d-4f2b-89da-ab949b9661f4-10_426_1191_267_438} Two uniform smooth spheres $A$ and $B$ of equal radii have masses $m$ and $k m$ respectively. The two spheres are moving on a horizontal surface with speeds $u$ and $\frac { 5 } { 8 } u$ respectively. Immediately before the spheres collide, $A$ is travelling along the line of centres, and $B$ 's direction of motion makes an angle $\alpha$ with the line of centres (see diagram). The coefficient of restitution between the spheres is $\frac { 2 } { 3 }$ and $\tan \alpha = \frac { 3 } { 4 }$. After the collision, the direction of motion of $B$ is perpendicular to the line of centres.

Find the value of $k$.
Find the loss in the total kinetic energy as a result of the collision.

CAIE Further Paper 3 2023 November Q2

2 A ball of mass 2 kg is projected vertically downwards with speed $5 \mathrm {~ms} ^ { - 1 }$ through a liquid. At time $t \mathrm {~s}$ after projection, the velocity of the ball is $v \mathrm {~ms} ^ { - 1 }$ and its displacement from its starting point is $x \mathrm {~m}$. The forces acting on the ball are its weight and a resistive force of magnitude $0.2 v ^ { 2 } \mathrm {~N}$.

Find an expression for $v$ in terms of $t$.
Deduce what happens to $v$ for large values of $t$.
\includegraphics[max width=\textwidth, alt={}, center]{6964a0b0-8fc8-4fa8-b3fe-f51dafdaaeec-06_803_652_251_703} A uniform square lamina of side $2 a$ and weight $W$ is suspended from a light inextensible string attached to the midpoint $E$ of the side $A B$. The other end of the string is attached to a fixed point $P$ on a rough vertical wall. The vertex $B$ of the lamina is in contact with the wall. The string $E P$ is perpendicular to the side $A B$ and makes an angle $\theta$ with the wall (see diagram). The string and the lamina are in a vertical plane perpendicular to the wall. The coefficient of friction between the wall and the lamina is $\frac { 1 } { 2 }$. Given that the vertex $B$ is about to slip up the wall, find the value of $\tan \theta$.
\includegraphics[max width=\textwidth, alt={}, center]{6964a0b0-8fc8-4fa8-b3fe-f51dafdaaeec-08_581_576_269_731} A light elastic string has natural length $8 a$ and modulus of elasticity $5 m g$. A particle $P$ of mass $m$ is attached to the midpoint of the string. The ends of the string are attached to points $A$ and $B$ which are a distance $12 a$ apart on a smooth horizontal table. The particle $P$ is held on the table so that $A P = B P = L$ (see diagram). The particle $P$ is released from rest. When $P$ is at the midpoint of $A B$ it has speed $\sqrt { 80 a g }$.
Find $L$ in terms of $a$.
Find the initial acceleration of $P$ in terms of $g$.

CAIE Further Paper 3 2023 November Q5

5 A particle $P$ is projected with speed $u \mathrm {~ms} ^ { - 1 }$ at an angle $\theta$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. During its flight $P$ passes through the point which is a horizontal distance $3 a$ from $O$ and a vertical distance $\frac { 3 } { 8 } a$ above the horizontal plane. It is given that $\tan \theta = \frac { 1 } { 3 }$.

Show that $\mathrm { u } ^ { 2 } = 8 \mathrm { ag }$.
A particle $Q$ is projected with speed $V \mathrm {~ms} ^ { - 1 }$ at an angle $\alpha$ above the horizontal from $O$ at the instant when $P$ is at its highest point. Particles $P$ and $Q$ both land at the same point on the horizontal plane at the same time.
Find $V$ in terms of $a$ and $g$.

CAIE Further Paper 3 2023 November Q6

6 A particle $P$ of mass $m$ is attached to one end of a light inextensible rod of length $3 a$. An identical particle $Q$ is attached to the other end of the rod. The rod is smoothly pivoted at a point $O$ on the rod, where $\mathrm { OQ } = \mathrm { x }$. The system, of rod and particles, rotates about $O$ in a vertical plane. At an instant when the rod is vertical, with $P$ above $Q$, the particle $P$ is moving horizontally with speed $u$. When the rod has turned through an angle of $60 ^ { \circ }$ from the vertical, the speed of $P$ is $2 \sqrt { \mathrm { ag } }$, and the tensions in the two parts of the rod, $O P$ and $O Q$, have equal magnitudes.

Show that the speed of $Q$ when the rod has turned through an angle of $60 ^ { \circ }$ from the vertical is $\frac { 2 x } { 3 a - x } \sqrt { a g }$.
Find $x$ in terms of $a$.
Find $u$ in terms of $a$ and $g$.
If you use the following page to complete the answer to any question, the question number must be clearly shown.

CAIE Further Paper 3 2023 November Q1

1 One end of a light inextensible string of length $a$ is attached to a fixed point $O$. The other end of the string is attached to a particle of mass $m$. The string is taut and makes an angle $\theta$ with the downward vertical through $O$, where $\cos \theta = \frac { 2 } { 3 }$. The particle moves in a horizontal circle with speed $v$. Find $v$ in terms of $a$ and $g$.
..................................................................................................................................................

CAIE Further Paper 3 2023 November Q2

2 A particle $P$ of mass 0.5 kg moves in a straight line. At time $t \mathrm {~s}$ the velocity of $P$ is $v \mathrm {~ms} ^ { - 1 }$ and its displacement from a fixed point $O$ on the line is $x \mathrm {~m}$. The only forces acting on $P$ are a force of magnitude $\frac { 150 } { ( x + 1 ) ^ { 2 } } \mathrm {~N}$ in the direction of increasing displacement and a resistive force of magnitude $\frac { 450 } { ( x + 1 ) ^ { 3 } } \mathrm {~N}$. When $t = 0 , x = 0$ and $v = 20$.
Find $v$ in terms of $x$, giving your answer in the form $v = \frac { A x + B } { ( x + 1 ) }$, where $A$ and $B$ are constants to be determined.
\includegraphics[max width=\textwidth, alt={}, center]{0270d51a-e252-46d3-8c97-7f71ba91fa65-04_451_812_255_625} A uniform lamina is in the form of an isosceles triangle $A B C$ in which $A C = 2 \mathrm { a }$ and angle $A B C = 90 ^ { \circ }$. The point $D$ on $A B$ is such that the ratio $D B : A B = 1 : k$. The point $E$ on $C B$ is such that $D E$ is parallel to $A C$. The triangle $D B E$ is removed from the lamina (see diagram).

Find, in terms of $k$, the distance of the centre of mass of the remaining lamina $A D E C$ from the midpoint of $A C$.
When the lamina $A D E C$ is freely suspended from the vertex $A$, the edge $A C$ makes an angle $\theta$ with the downward vertical, where $\tan \theta = \frac { 5 } { 18 }$.
Find the value of $k$.
\includegraphics[max width=\textwidth, alt={}, center]{0270d51a-e252-46d3-8c97-7f71ba91fa65-06_604_798_251_635} Two smooth vertical walls meet at right angles. The smooth sphere $A$, with mass $m$, is at rest on a smooth horizontal surface and is at a distance $d$ from each wall. An identical smooth sphere $B$ is moving on the horizontal surface with speed $u$ at an angle $\theta$ with the line of centres when the spheres collide (see diagram). After the collision, the spheres take the same time to reach a wall. The coefficient of restitution between the spheres is $\frac { 1 } { 2 }$.

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