Questions M4 - A-Level Maths

OCR MEI M4 2014 June Q3

3 A uniform rigid rod AB of mass $m$ and length $2 a$ is freely hinged to a horizontal floor at A . The end B is attached to a light elastic string of modulus $\lambda$ and natural length $5 a$. The other end of the string is attached to a small, light, smooth ring C which can slide along a horizontal rail. The rail is a distance $7 a$ above the floor and C is always vertically above B . The angle that AB makes with the floor is $\theta$. The system is shown in Fig. 3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c3ac9277-d34d-4d0e-9f9b-d0bce8c741af-3_664_773_584_648} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure}

Find the potential energy, $V$, of the system and hence show that $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = a \cos \theta \left( m g - \frac { 4 \lambda } { 5 } ( 1 - \sin \theta ) \right) .$$
Show that there is a position of equilibrium when $\theta = \frac { 1 } { 2 } \pi$ and determine whether or not it is stable. There are two further positions of equilibrium when $0 < \theta < \pi$.
Find the magnitude of the tension in the string and the vertical force of the hinge on the rod in these positions.
Show that $\lambda > \frac { 5 m g } { 4 }$.
Show that these positions of equilibrium are stable.

OCR MEI M4 2014 June Q4

4

A pulley consists of a central cylinder of wood and an outer ring of steel. The density of the wood is $700 \mathrm {~kg} \mathrm {~m} ^ { - 3 }$ and the density of the steel is $7800 \mathrm {~kg} \mathrm {~m} ^ { - 3 }$. The pulley has a radius of 20 cm and is 10 cm thick (see Fig. 4.1). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c3ac9277-d34d-4d0e-9f9b-d0bce8c741af-4_359_661_404_742} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
\end{figure} Find the radius that the central cylinder must have in order that the moment of inertia of the pulley about the axis of symmetry shown in Fig. 4.1 is $1.5 \mathrm {~kg} \mathrm {~m} ^ { 2 }$.
Two blocks P and Q of masses 10 kg and 20 kg are connected by a light inextensible string. The string passes over a heavy rough pulley of radius 25 cm . The pulley can rotate freely and the string does not slip. Block P is held at rest in smooth contact with a plane inclined at $30 ^ { \circ }$ to the horizontal, and block Q is at rest below the pulley (see Fig. 4.2). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c3ac9277-d34d-4d0e-9f9b-d0bce8c741af-5_341_917_438_541} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
\end{figure} At $t \mathrm {~s}$ after the system is released from rest, the pulley has angular velocity $\omega \mathrm { rad } \mathrm { s } ^ { - 1 }$ and block P has constant acceleration of $2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ up the slope.
1. Show that the net loss of energy of the two blocks in the first $t$ seconds of motion is $87 t ^ { 2 } \mathrm {~J}$ and use the principle of conservation of energy to show that the moment of inertia of the pulley about its axis of rotation is $\frac { 87 } { 32 } \mathrm {~kg} \mathrm {~m} ^ { 2 }$. When $t = 3$ a resistive couple is applied to the pulley. This resistive couple has magnitude $( 2 \omega + k ) \mathrm { Nm }$, where $k$ is a constant. The couple on the pulley due to tensions in the sections of string is $\left( \frac { 147 } { 4 } - \frac { 15 } { 8 } \frac { \mathrm {~d} \omega } { \mathrm {~d} t } \right) \mathrm { Nm }$ in the direction of positive $\omega$.
2. Write down a first order differential equation for $\omega$ when $t \geqslant 3$ and show by integration that $$\omega = \frac { 1 } { 8 } \left( ( 45 + 4 k ) \mathrm { e } ^ { \frac { 64 } { 147 } ( 3 - t ) } + 147 - 4 k \right) .$$
3. By considering the equation given in part (ii), find the value or set of values of $k$ for which the pulley
  (A) continues to rotate with constant angular velocity,
  (B) rotates with decreasing angular velocity without coming to rest,
  (C) rotates with decreasing angular velocity and comes to rest if there is sufficient distance between P and the pulley. \section*{END OF QUESTION PAPER}

OCR MEI M4 2015 June Q1

1 A rocket is launched vertically upwards from rest. The initial mass of the rocket, including fuel and payload, is $m _ { 0 }$ and the propulsion system ejects mass at a constant mass rate $k$ with constant speed $u$ relative to the rocket. The only other force acting on the rocket is its weight. The acceleration due to gravity is constant throughout the motion. At time $t$ after launch the speed of the rocket is $v$.

Show that while mass is being ejected from the rocket $v = u \ln \left( \frac { m _ { 0 } } { m _ { 0 } - k t } \right) - g t$. The rocket initially has 2400 kg of fuel which is ejected at a constant rate of $100 \mathrm {~kg} \mathrm {~s} ^ { - 1 }$ with constant speed $3000 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ relative to the rocket.
Given that the rocket must reach a speed of $7910 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ before releasing its payload, find the maximum possible value of $m _ { 0 }$.

OCR MEI M4 2015 June Q2

2 Fig. 2 shows a system in a vertical plane. A uniform rod AB of length $2 a$ and mass $m$ is freely hinged at A . The angle that AB makes with the horizontal is $\theta$, where $- \frac { 2 } { 3 } \pi < \theta < \frac { 2 } { 3 } \pi$. Attached at B is a light spring BC of natural length $a$ and stiffness $\frac { m g } { a }$. The other end of the spring is attached to a small light smooth ring C which can slide freely along a vertical rail. The rail is at a distance of $a$ from A and the spring is always horizontal. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8ea28e6f-528c-4e3c-9562-6c964043747e-2_737_703_1356_680} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure}

Find the potential energy, $V$, of the system and hence show that $\frac { \mathrm { d } V } { \mathrm {~d} \theta } = m g a \cos \theta ( 1 - 4 \sin \theta )$.
Hence find the positions of equilibrium of the system and investigate their stability.

OCR MEI M4 2015 June Q3

3 A particle of mass 4 kg moves along the $x$-axis. At time $t$ seconds the particle is $x \mathrm {~m}$ from the origin O and has velocity $v \mathrm {~ms} ^ { - 1 }$. A driving force of magnitude $20 t \mathrm { t } ^ { - t } \mathrm {~N}$ is applied in the positive $x$ direction. Initially $v = 2$ and the particle is at O .

Find, in either order, the impulse of the force over the first 3 seconds and the velocity of the particle after 3 seconds. From time $t = 3$ a resistive force of magnitude $\frac { 1 } { 2 } t \mathrm {~N}$ and the driving force are applied until the particle comes to rest.
Show that, after the resistive force is applied, the only time at which the resultant force on the particle is zero is when $t = \ln 40$. Hence find the maximum velocity of the particle during the motion.
Given that the time $T$ seconds at which the particle comes to rest is given by the equation $T = \sqrt { 121 - 80 \mathrm { e } ^ { - T } ( 1 + T ) }$, without solving the equation deduce that $T \approx 11$.
Use a numerical method to find $T$ correct to 4 decimal places.

OCR MEI M4 2015 June Q4

4 A solid cylinder of radius $a \mathrm {~m}$ and length $3 a \mathrm {~m}$ has density $\rho \mathrm { kg } \mathrm { m } ^ { - 3 }$ given by $\rho = k \left( 2 + \frac { x } { a } \right)$ where $x \mathrm {~m}$ is the distance from one end and $k$ is a positive constant. The mass of the cylinder is $M \mathrm {~kg}$ where $M = \frac { 21 } { 2 } \pi a ^ { 3 } k$. Let A and B denote the circular faces of the cylinder where $x = 0$ and $x = 3 a$, respectively.

Show by integration that the moment of inertia, $I _ { \mathrm { A } } \mathrm { kg } \mathrm { m } ^ { 2 }$, of the cylinder about a diameter of the face A is given by $I _ { \mathrm { A } } = \frac { 109 } { 28 } M a ^ { 2 }$.
Show that the centre of mass of the cylinder is $\frac { 12 } { 7 } a \mathrm {~m}$ from A .
Using the parallel axes theorem, or otherwise, show that the moment of inertia, $I _ { \mathrm { B } } \mathrm { kg } \mathrm { m } ^ { 2 }$, of the cylinder about a diameter of the face B is given by $I _ { \mathrm { B } } = \frac { 73 } { 28 } M a ^ { 2 }$. You are now given that $M = 4$ and $a = 0.7$. The cylinder is at rest and can rotate freely about a horizontal axis which is a diameter of the face B as shown in Fig. 4. It is struck at the bottom of the curved surface by a small object of mass 0.2 kg which is travelling horizontally at speed $20 \mathrm {~ms} ^ { - 1 }$ in the vertical plane which is both perpendicular to the axis of rotation and contains the axis of symmetry of the cylinder. The object sticks to the cylinder at the point of impact. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8ea28e6f-528c-4e3c-9562-6c964043747e-4_606_435_1087_817} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
Find the initial angular speed of the combined object after the collision. \section*{END OF QUESTION PAPER}

OCR MEI M4 2016 June Q1

1 A car of mass $m$ moves horizontally in a straight line. At time $t$ the car is a distance $x$ from a point O and is moving away from O with speed $v$. There is a force of magnitude $k v ^ { 2 }$, where $k$ is a constant, resisting the motion of the car. The car's engine has a constant power $P$. The terminal speed of the car is $U$.

Show that $$m v ^ { 2 } \frac { \mathrm {~d} v } { \mathrm {~d} x } = P \left( 1 - \frac { v ^ { 3 } } { U ^ { 3 } } \right)$$
Show that the distance moved while the car accelerates from a speed of $\frac { 1 } { 4 } U$ to a speed of $\frac { 1 } { 2 } U$ is $$\frac { m U ^ { 3 } } { 3 P } \ln A$$ stating the exact value of the constant $A$. Once the car attains a speed of $\frac { 1 } { 2 } U$, no further power is supplied by the car's engine.
Find, in terms of $m , P$ and $U$, the time taken for the speed of the car to reduce from $\frac { 1 } { 2 } U$ to $\frac { 1 } { 4 } U$.

OCR MEI M4 2016 June Q2

2 A thin rigid rod PQ has length $2 a$. Its mass per unit length, $\rho$, is given by $\rho = k \left( 1 + \frac { x } { 2 a } \right)$ where $x$ is the distance from P and $k$ is a positive constant. The mass of the rod is $M$ and the moment of inertia of the rod about an axis through P perpendicular to PQ is $I$.

Show that $I = \frac { 14 } { 9 } M a ^ { 2 }$. The rod is initially at rest with Q vertically below P . It is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through P . The rod is struck a horizontal blow perpendicular to the fixed axis at the point where $x = \frac { 3 } { 2 } a$. The magnitude of the impulse of this blow is $J$.
Find, in terms of $a , J$ and $M$, the initial angular speed of the rod.
Find, in terms of $a , g$ and $M$, the set of values of $J$ for which the rod makes complete revolutions.

OCR MEI M4 2016 June Q3

3 Fig. 3 shows a uniform rigid rod AB of length $2 a$ and mass $2 m$. The rod is freely hinged at A so that it can rotate in a vertical plane. One end of a light inextensible string of length $l$ is attached to B . The string passes over a small smooth fixed pulley at C , where C is vertically above A and $\mathrm { AC } = 6 a$. A particle of mass $\lambda m$, where $\lambda$ is a positive constant, is attached to the other end of the string and hangs freely, vertically below C . The rod makes an angle $\theta$ with the upward vertical, where $0 \leqslant \theta \leqslant \pi$. You may assume that the particle does not interfere with the rod AB or the section of the string BC . \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3fdb2cff-0f74-4c88-b25a-759bfab1675a-3_878_615_667_717} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure}

Find the potential energy, $V$, of the system relative to a situation in which the rod AB is horizontal, and hence show that $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = 2 m g a \sin \theta \left( \frac { 3 \lambda } { \sqrt { 10 - 6 \cos \theta } } - 1 \right) .$$
Show that $\theta = 0$ and $\theta = \pi$ are the only values of $\theta$ for which the system is in equilibrium whatever the value of $\lambda$.
Show that, if there is a third value of $\theta$ for which the system is in equilibrium, then $\frac { 2 } { 3 } < \lambda < \frac { 4 } { 3 }$.
Given that there are three positions of equilibrium, establish whether each of these positions is stable or unstable. It is given that, for small values of $\theta$, $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } \approx 2 m g a \left[ \left( \frac { 3 } { 2 } \lambda - 1 \right) \theta - \left( \frac { 13 } { 16 } \lambda - \frac { 1 } { 6 } \right) \theta ^ { 3 } \right] .$$
Investigate the stability of the equilibrium position given by $\theta = 0$ in the case when $\lambda = \frac { 2 } { 3 }$.

OCR MEI M4 2016 June Q4

4 A raindrop falls from rest through a stationary cloud. The raindrop has mass $m$ and speed $v$ when it has fallen a distance $x$. You may assume that resistances to motion are negligible.

Derive the equation of motion $$m v \frac { \mathrm {~d} v } { \mathrm {~d} x } + v ^ { 2 } \frac { \mathrm {~d} m } { \mathrm {~d} x } = m g .$$ Initially the mass of the raindrop is $m _ { 0 }$. Two different models for the mass of the raindrop are suggested.
In the first model $m = m _ { 0 } \mathrm { e } ^ { k _ { 1 } x }$, where $k _ { 1 }$ is a positive constant.
Show that $$v ^ { 2 } = \frac { g } { k _ { 1 } } \left( 1 - \mathrm { e } ^ { - 2 k _ { 1 } x } \right) ,$$ and hence state, in terms of $g$ and $k _ { 1 }$, the terminal velocity of the raindrop according to this first model. In the second model $m = m _ { 0 } \left( 1 + k _ { 2 } x \right)$, where $k _ { 2 }$ is a positive constant.
By considering the expression obtained from differentiating $v ^ { 2 } \left( 1 + k _ { 2 } x \right) ^ { 2 }$ with respect to $x$, show that, for the second model, the equation of motion in part (i) may be written as $$\frac { d } { d x } \left[ v ^ { 2 } \left( 1 + k _ { 2 } x \right) ^ { 2 } \right] = 2 g \left( 1 + k _ { 2 } x \right) ^ { 2 } .$$ Hence find an expression for $v ^ { 2 }$ in terms of $g , k _ { 2 }$ and $x$.
Suppose that the models give the same value for the speed of the raindrop at the instant when it has doubled its initial mass. Find the exact value of $\frac { k _ { 1 } } { k _ { 2 } }$, giving your answer in the form $\frac { a } { b }$ where $a$ and $b$
are integers. are integers. \section*{END OF QUESTION PAPER}

Edexcel M4 2005 January Q5

Show that, when $\angle B F O = \theta$, the potential energy of the system is $$\frac { 1 } { 10 } m g a ( 8 \cos \theta - 5 ) ^ { 2 } - 2 m g a \cos ^ { 2 } \theta + \text { constant } .$$
Hence find the values of $\theta$ for which the system is in equilibrium.
Determine the nature of the equilibrium at each of these positions.

OCR M4 2008 June Q6

Show that the moment of inertia of the lamina about the axis through $X$ is $\frac { 4 } { 3 } m a ^ { 2 }$.
At an instant when $\cos \theta = \frac { 3 } { 5 }$, show that $\omega ^ { 2 } = \frac { 6 g } { 5 a }$.
At an instant when $\cos \theta = \frac { 3 } { 5 }$, show that $R = 0$, and given also that $\sin \theta = \frac { 4 } { 5 }$ find $S$ in terms of $m$ and $g$.

OCR M4 2010 June Q6

Show that $\theta = 0$ is a position of stable equilibrium.
Show that the kinetic energy of the system is $4 m a ^ { 2 } \dot { \theta } ^ { 2 }$.
By differentiating the energy equation, then making suitable approximations for $\sin \theta$ and $\cos \theta$, find the approximate period of small oscillations about the equilibrium position $\theta = 0$. \section*{[Question 7 is printed overleaf.]}

OCR M4 2013 June Q5

Find the magnitude and bearing of the velocity of $U$ relative to $P$.
Find the shortest distance between $P$ and $U$ in the subsequent motion.
(ii) Plane $Q$ is flying with constant velocity $160 \mathrm {~ms} ^ { - 1 }$ in the direction which brings it as close as possible to $U$.
Find the bearing of the direction in which $Q$ is flying.
Find the shortest distance between $Q$ and $U$ in the subsequent motion.
\includegraphics[max width=\textwidth, alt={}, center]{6e3d5f5e-7ffa-4111-903d-468fb4d20192-3_771_769_262_646} A square frame $A B C D$ consists of four uniform rods $A B , B C , C D , D A$, rigidly joined at $A , B , C , D$. Each rod has mass 0.6 kg and length 1.5 m . The frame rotates freely in a vertical plane about a fixed horizontal axis passing through the mid-point $O$ of $A D$. At time $t$ seconds the angle between $A D$ and the horizontal, measured anticlockwise, is $\theta$ radians (see diagram).
1. Show that the moment of inertia of the frame about the axis through $O$ is $3.15 \mathrm {~kg} \mathrm {~m} ^ { 2 }$.
2. Show that $\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - 5.6 \sin \theta$.
3. Deduce that the frame can make small oscillations which are approximately simple harmonic, and find the period of these oscillations. The frame is at rest with $A D$ horizontal. A couple of constant moment 25 Nm about the axis is then applied to the frame.
4. Find the angular speed of the frame when it has rotated through 1.2 radians.

OCR M4 2015 June Q5

Taking $H$ as the reference level for gravitational potential energy, show that the total potential energy $V$ of the system is given by $$V = m g \left( 2 \lambda r \cos \theta - 2 r \cos ^ { 2 } \theta - \lambda a \right)$$
Find the set of possible values of $\lambda$ so that there is more than one position of equilibrium.
For the case $\lambda = \frac { 3 } { 2 }$, determine whether each equilibrium position is stable or unstable.

Edexcel M4 Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-03_457_638_233_598} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A fixed smooth plane is inclined to the horizontal at an angle of $45 ^ { \circ }$. A particle $P$ is moving horizontally and strikes the plane. Immediately before the impact, $P$ is moving in a vertical plane containing a line of greatest slope of the inclined plane. Immediately after the impact, $P$ is moving in a direction which makes an angle of $30 ^ { \circ }$ with the inclined plane, as shown in Figure 1. Find the fraction of the kinetic energy of $P$ which is lost in the impact.

Edexcel M4 Q2

2. At time $t = 0$, a particle $P$ of mass $m$ is projected vertically upwards with speed $\sqrt { \frac { g } { k } }$, where $k$ is a constant. At time $t$ the speed of $P$ is $v$. The particle $P$ moves against air resistance whose magnitude is modelled as being $m k v ^ { 2 }$ when the speed of $P$ is $v$. Find, in terms of $k$, the distance travelled by $P$ until its speed first becomes half of its initial speed.

Edexcel M4 Q3

At noon a motorboat $P$ is 2 km north-west of another motorboat $Q$. The motorboat $P$ is moving due south at $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The motorboat $Q$ is pursuing motorboat $P$ at a speed of $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and sets a course in order to get as close to motorboat $P$ as possible.
1. Find the course set by $Q$, giving your answer as a bearing to the nearest degree.
2. Find the shortest distance between $P$ and $Q$.
3. Find the distance travelled by $Q$ from its position at noon to the point of closest approach.

Edexcel M4 Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-08_479_807_246_571} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A light inextensible string of length $2 a$ has one end attached to a fixed point $A$. The other end of the string is attached to a particle $P$ of mass $m$. A second light inextensible string of length $L$, where $L > \frac { 12 a } { 5 }$, has one of its ends attached to $P$ and passes over a small smooth peg fixed at a point $B$. The line $A B$ is horizontal and $A B = 2 a$. The other end of the second string is attached to a particle of mass $\frac { 7 } { 20 } m$, which hangs vertically below $B$, as shown in Figure 2.

Show that the potential energy of the system, when the angle $P A B = 2 \theta$, is $$\frac { 1 } { 5 } m g a ( 7 \sin \theta - 10 \sin 2 \theta ) + \text { constant. }$$
Show that there is only one value of $\cos \theta$ for which the system is in equilibrium and find this value.
Determine the stability of the position of equilibrium.

Edexcel M4 Q5

Two small smooth spheres $A$ and $B$, of mass 2 kg and 1 kg respectively, are moving on a smooth horizontal plane when they collide. Immediately before the collision the velocity of $A$ is $( \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and the velocity of $B$ is $- 2 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Immediately after the collision the velocity of $A$ is $\mathbf { j } \mathrm { m } \mathrm { s } ^ { - 1 }$.
1. Show that the velocity of $B$ immediately after the collision is $2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
2. Find the impulse of $B$ on $A$ in the collision, giving your answer as a vector, and hence show that the line of centres is parallel to $\mathbf { i } + \mathbf { j }$.
3. Find the coefficient of restitution between $A$ and $B$.

Edexcel M4 Q6

6. A light elastic spring $A B$ has natural length $2 a$ and modulus of elasticity $2 m n ^ { 2 } a$, where $n$ is a constant. A particle $P$ of mass $m$ is attached to the end $A$ of the spring. At time $t = 0$, the spring, with $P$ attached, lies at rest and unstretched on a smooth horizontal plane. The other end $B$ of the spring is then pulled along the plane in the direction $A B$ with constant acceleration $f$. At time $t$ the extension of the spring is $x$.

Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + n ^ { 2 } x = f .$$
Find $x$ in terms of $n , f$ and $t$. Hence find
the maximum extension of the spring,
the speed of $P$ when the spring first reaches its maximum extension.
Turn over
1. \hspace{0pt} [In this question $\mathbf { i }$ and $\mathbf { j }$ are unit vectors due east and due north respectively]
A man cycles at a constant speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on level ground and finds that when his velocity is $u \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ the velocity of the wind appears to be $v ( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$, where $v$ is a positive constant. When the man cycles with velocity $\frac { 1 } { 5 } u ( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$, the velocity of the wind appears to be $w \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where $w$ is a positive constant. Find, in terms of $u$, the true velocity of the wind.
2. Two smooth uniform spheres $S$ and $T$ have equal radii. The mass of $S$ is 0.3 kg and the mass of $T$ is 0.6 kg . The spheres are moving on a smooth horizontal plane and collide obliquely. Immediately before the collision the velocity of $S$ is $\mathbf { u } _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the velocity of $T$ is $\mathbf { u } _ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The coefficient of restitution between the spheres is 0.5 . Immediately after the collision the velocity of $S$ is $( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and the velocity of $T$ is $( \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. Given that when the spheres collide the line joining their centres is parallel to $\mathbf { i }$,
find
1. $\mathbf { u } _ { 1 }$,
2. $\mathbf { u } _ { 2 }$. After the collision, $T$ goes on to collide with a smooth vertical wall which is parallel to $\mathbf { j }$. Given that the coefficient of restitution between $T$ and the wall is also 0.5 , find
the angle through which the direction of motion of $T$ is deflected as a result of the collision with the wall,
the loss in kinetic energy of $T$ caused by the collision with the wall.
1. At 12 noon, $\operatorname { ship } A$ is 8 km due west of $\operatorname { ship } B$. Ship $A$ is moving due north at a constant speed of $10 \mathrm {~km} \mathrm {~h} ^ { - 1 }$. Ship $B$ is moving at a constant speed of $6 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ on a bearing so that it passes as close to $A$ as possible.
2. Find the bearing on which ship $B$ moves.
3. Find the shortest distance between the two ships.
4. Find the time when the two ships are closest.
1. A particle of mass $m$ is projected vertically upwards, at time $t = 0$, with speed $U$. The particle is subject to air resistance of magnitude $\frac { m g v ^ { 2 } } { k ^ { 2 } }$, where $v$ is the speed of the particle at time $t$ and $k$ is a positive constant.
2. Show that the particle reaches its greatest height above the point of projection at time
$$\frac { k } { g } \tan ^ { - 1 } \left( \frac { U } { k } \right)$$
Find the greatest height above the point of projection attained by the particle. 5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-22_419_1212_260_365} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The end $A$ of a uniform rod $A B$, of length $2 a$ and mass $4 m$, is smoothly hinged to a fixed point. The end $B$ is attached to one end of a light inextensible string which passes over a small smooth pulley, fixed at the same level as $A$. The distance from $A$ to the pulley is $4 a$. The other end of the string carries a particle of mass $m$ which hangs freely, vertically below the pulley, with the string taut. The angle between the rod and the downward vertical is $\theta$, where $0 < \theta < \frac { \pi } { 2 }$, as shown in Figure 1.
Show that the potential energy of the system is $$2 m g a ( \sqrt { } ( 5 - 4 \sin \theta ) - 2 \cos \theta ) + \text { constant }$$
Hence, or otherwise, show that any value of $\theta$ which corresponds to a position of equilibrium of the system satisfies the equation $$4 \sin ^ { 3 } \theta - 6 \sin ^ { 2 } \theta + 1 = 0$$
Given that $\theta = \frac { \pi } { 6 }$ corresponds to a position of equilibrium, determine its stability.
1. Two points $A$ and $B$ lie on a smooth horizontal table with $A B = 4 a$. One end of a light elastic spring, of natural length $a$ and modulus of elasticity $2 m g$, is attached to $A$. The other end of the spring is attached to a particle $P$ of mass $m$. Another light elastic spring, of natural length $a$ and modulus of elasticity $m g$, has one end attached to $B$ and the other end attached to $P$. The particle $P$ is on the table at rest and in equilibrium.
2. Show that $A P = \frac { 5 a } { 3 }$.
The particle $P$ is now moved along the table from its equilibrium position through a distance $0.5 a$ towards $B$ and released from rest at time $t = 0$. At time $t , P$ is moving with speed $v$ and has displacement $x$ from its equilibrium position. There is a resistance to motion of magnitude $4 m \omega v$ where $\omega = \sqrt { } \left( \frac { g } { a } \right)$.
Show that $\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \omega \frac { \mathrm {~d} x } { \mathrm {~d} t } + 3 \omega ^ { 2 } x = 0$.
Find the velocity, $\frac { \mathrm { d } x } { \mathrm {~d} t }$, of $P$ in terms of $a , \omega$ and $t$. Turn over
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1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-27_780_1022_228_488} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Two smooth uniform spheres $A$ and $B$ have masses $2 m \mathrm {~kg}$ and $3 m \mathrm {~kg}$ respectively and equal radii. The spheres are moving on a smooth horizontal surface. Initially, sphere $A$ has velocity $( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and sphere $B$ has velocity $( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. When the spheres collide, the line joining their centres is parallel to $\mathbf { j }$, as shown in Figure 1. The coefficient of restitution between the spheres is $\frac { 3 } { 7 }$. Find, in terms of $m$, the total kinetic energy lost in the collision. 2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-29_680_853_285_543} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 represents part of the smooth rectangular floor of a sports hall. A ball is at $B$, 4 m from one wall of the hall and 5 m from an adjacent wall. These two walls are smooth and meet at the corner $C$. The ball is kicked so that it travels along the floor, bounces off the first wall at the point $X$ and hits the second wall at the point $Y$. The point $Y$ is 7.5 m from the corner $C$.
The coefficient of restitution between the ball and the first wall is $\frac { 3 } { 4 }$.
Modelling the ball as a particle, find the distance $C X$.
1. \hspace{0pt} [In this question the unit vectors $\mathbf { i }$ and $\mathbf { j }$ are due east and due north respectively.]
A coastguard patrol boat $C$ is moving with constant velocity $( 8 \mathbf { i } + u \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$. Another ship $S$ is moving with constant velocity $( 12 \mathbf { i } + 16 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$.
Find, in terms of $u$, the velocity of $C$ relative to $S$. At noon, $S$ is 10 km due west of $C$.
If $C$ is to intercept $S$,
1. find the value of $u$.
2. Using this value of $u$, find the time at which $C$ would intercept $S$. If instead, at noon, $C$ is moving with velocity $( 8 \mathbf { i } + 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$ and continues at this constant velocity,
find the distance of closest approach of $C$ to $S$.
1. A hiker walking due east at a steady speed of $5 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ notices that the wind appears to come from a direction with bearing 050. At the same time, another hiker moving on a bearing of 320 , and also walking at $5 \mathrm {~km} \mathrm {~h} ^ { - 1 }$, notices that the wind appears to come from due north.
Find
the direction from which the wind is blowing,
the wind speed.
5. A particle $Q$ of mass 6 kg is moving along the $x$-axis. At time $t$ seconds the displacement of $Q$ from the origin $O$ is $x$ metres and the speed of $Q$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The particle moves under the action of a retarding force of magnitude ( $a + b v ^ { 2 }$ ) N, where $a$ and $b$ are positive constants. At time $t = 0 , Q$ is at $O$ and moving with speed $U \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in the positive $x$-direction. The particle $Q$ comes to instantaneous rest at the point $X$.
Show that the distance $O X$ is $$\frac { 3 } { b } \ln \left( 1 + \frac { b U ^ { 2 } } { a } \right) \mathrm { m }$$ Given that $a = 12$ and $b = 3$,
find, in terms of $U$, the time taken to move from $O$ to $X$. 6. A particle $P$ of mass 4 kg moves along a horizontal straight line under the action of a force directed towards a fixed point $O$ on the line. At time $t$ seconds, $P$ is $x$ metres from $O$ and the force towards $O$ has magnitude $9 x$ newtons. The particle $P$ is also subject to air resistance, which has magnitude $12 v$ newtons when $P$ is moving with speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Show that the equation of motion of $P$ is $$4 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 12 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 9 x = 0$$ It is given that the solution of this differential equation is of the form $$x = \mathrm { e } ^ { - \lambda t } ( A t + B )$$ When $t = 0$ the particle is released from rest at the point $R$, where $O R = 4 \mathrm {~m}$. Find,
the values of the constants $\lambda , A$ and $B$,
the greatest speed of $P$ in the subsequent motion.

Edexcel M4 Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-38_451_1077_315_370} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a framework $A B C$, consisting of two uniform rods rigidly joined together at $B$ so that $\angle A B C = 90 ^ { \circ }$. The $\operatorname { rod } A B$ has length $2 a$ and mass $4 m$, and the $\operatorname { rod } B C$ has length $a$ and mass $2 m$. The framework is smoothly hinged at $A$ to a fixed point, so that the framework can rotate in a fixed vertical plane. One end of a light elastic string, of natural length $2 a$ and modulus of elasticity $3 m g$, is attached to $A$. The string passes through a small smooth ring $R$ fixed at a distance $2 a$ from $A$, on the same horizontal level as $A$ and in the same vertical plane as the framework. The other end of the string is attached to $B$. The angle $A R B$ is $\theta$, where $0 < \theta < \frac { \pi } { 2 }$.

Show that the potential energy $V$ of the system is given by $$V = 8 a m g \sin 2 \theta + 5 a m g \cos 2 \theta + \text { constant }$$
Find the value of $\theta$ for which the system is in equilibrium.
Determine the stability of this position of equilibrium. Turn over
1. A smooth uniform sphere $S$, of mass $m$, is moving on a smooth horizontal plane when it collides obliquely with another smooth uniform sphere $T$, of the same radius as $S$ but of mass $2 m$, which is at rest on the plane. Immediately before the collision the velocity of $S$ makes an angle $\alpha$, where $\tan \alpha = \frac { 3 } { 4 }$, with the line joining the centres of the spheres. Immediately after the collision the speed of $T$ is $V$. The coefficient of restitution between the spheres is $\frac { 3 } { 4 }$.
2. Find, in terms of $V$, the speed of $S$
  1. immediately before the collision,
  2. immediately after the collision.
3. Find the angle through which the direction of motion of $S$ is deflected as a result of the collision.
1. A $\operatorname { ship } A$ is moving at a constant speed of $8 \mathrm {~km} \mathrm {~h} \mathrm {~h} ^ { - 1 }$ on a bearing of $150 ^ { \circ }$. At noon a second ship $B$ is 6 km from $A$, on a bearing of $210 ^ { \circ }$. Ship $B$ is moving due east at a constant speed. At a later time, $B$ is $2 \sqrt { 3 } \mathrm {~km}$ due south of $A$.
Find
(i) the time at which $B$ will be due east of $A$,
(ii) the distance between the ships at that time.
1. Two particles, of masses $m$ and $2 m$, are connected to the ends of a long light inextensible string. The string passes over a small smooth fixed pulley and hangs vertically on either side. The particles are released from rest with the string taut. Each particle is subject to air resistance of magnitude $k v ^ { 2 }$, where $v$ is the speed of each particle after it has moved a distance $x$ from rest and $k$ is a positive constant.
2. Show that $\frac { \mathrm { d } } { \mathrm { d } x } \left( v ^ { 2 } \right) + \frac { 4 k } { 3 m } v ^ { 2 } = \frac { 2 g } { 3 }$
3. Find $v ^ { 2 }$ in terms of $x$.
4. Deduce that the tension in the string, $T$, satisfies
$$\frac { 4 m g } { 3 } \leqslant T < \frac { 3 m g } { 2 }$$
1. A rescue boat, whose maximum speed is $20 \mathrm {~km} \mathrm {~h} ^ { - 1 }$, receives a signal which indicates that a yacht is in distress near a fixed point $P$. The rescue boat is 15 km south-west of $P$. There is a constant current of $5 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ flowing uniformly from west to east. The rescue boat sets the course needed to get to $P$ as quickly as possible. Find
2. the course the rescue boat sets,
3. the time, to the nearest minute, to get to $P$.
When the rescue boat arrives at $P$, the yacht is just visible 4 km due north of $P$ and is drifting with the current. Find
the course that the rescue boat should set to get to the yacht as quickly as possible,
the time taken by the rescue boat to reach the yacht from $P$. 5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-49_977_1228_205_356} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform rod $A B$, of length $4 a$ and weight $W$, is free to rotate in a vertical plane about a fixed smooth horizontal axis which passes through the point $C$ of the rod, where $A C = 3 a$. One end of a light inextensible string of length $L$, where $L > 10 a$, is attached to the end $A$ of the rod and passes over a small smooth fixed peg at $P$ and another small smooth fixed peg at $Q$. The point $Q$ lies in the same vertical plane as $P , A$ and $B$. The point $P$ is at a distance $3 a$ vertically above $C$ and $P Q$ is horizontal with $P Q = 4 a$. A particle of weight $\frac { 1 } { 2 } W$ is attached to the other end of the string and hangs vertically below $Q$. The rod is inclined at an angle $2 \theta$ to the vertical, where $- \pi < 2 \theta < \pi$, as shown in Figure 1.
Show that the potential energy of the system is $$W a ( 3 \cos \theta - \cos 2 \theta ) + \text { constant }$$
Find the positions of equilibrium and determine their stability.
1. Two points $A$ and $B$ are in a vertical line, with $A$ above $B$ and $A B = 4 a$. One end of a light elastic spring, of natural length $a$ and modulus of elasticity $3 m g$, is attached to $A$. The other end of the spring is attached to a particle $P$ of mass $m$. Another light elastic spring, of natural length $a$ and modulus of elasticity $m g$, has one end attached to $B$ and the other end attached to $P$. The particle $P$ hangs at rest in equilibrium.
2. Show that $A P = \frac { 7 a } { 4 }$
The particle $P$ is now pulled down vertically from its equilibrium position towards $B$ and at time $t = 0$ it is released from rest. At time $t$, the particle $P$ is moving with speed $v$ and has displacement $x$ from its equilibrium position. The particle $P$ is subject to air resistance of magnitude $m k v$, where $k$ is a positive constant.
Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + k \frac { \mathrm {~d} x } { \mathrm {~d} t } + \frac { 4 g } { a } x = 0$$
Find the range of values of $k$ which would result in the motion of $P$ being a damped oscillation.

Edexcel M4 2002 January Q1

A river of width 40 m flows with uniform and constant speed between straight banks. A swimmer crosses as quickly as possible and takes 30 s to reach the other side. She is carried 25 m downstream.

Find

the speed of the river,
the speed of the swimmer relative to the water.

Edexcel M4 2002 January Q2

2. A ball of mass $m$ is thrown vertically upwards from the ground with an initial speed $u$. When the speed of the ball is $v$, the magnitude of the air resistance is $m k v$, where $k$ is a positive constant. By modelling the ball as a particle, find, in terms of $u , k$ and $g$, the time taken for the ball to reach its greatest height.

Edexcel M4 2002 January Q3

3. A smooth uniform sphere $P$ of mass $m$ is falling vertically and strikes a fixed smooth inclined plane with speed $u$. The plane is inclined at an angle $\theta , \theta < 45 ^ { \circ }$, to the horizontal. The coefficient of restitution between $P$ and the inclined plane is $e$. Immediately after $P$ strikes the plane, $P$ moves horizontally.

Show that $e = \tan ^ { 2 } \theta$.
Show that the magnitude of the impulse exerted by $P$ on the plane is $m u \sec \theta$.

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