Questions - Edexcel M4 - A-Level Maths

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.

Edexcel M4 2012 June Q1

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 $\frac { 3 } { 4 }$.
1. Find, in terms of $V$, the speed of $S$
  1. immediately before the collision,
  2. immediately after the collision.
2. Find the angle through which the direction of motion of $S$ is deflected as a result of the collision.

Edexcel M4 2012 June Q2

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

the time at which $B$ will be due east of $A$,
the distance between the ships at that time.

Edexcel M4 2012 June Q3

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.
1. Show that $\frac { \mathrm { d } } { \mathrm { d } x } \left( v ^ { 2 } \right) + \frac { 4 k } { 3 m } v ^ { 2 } = \frac { 2 g } { 3 }$
2. Find $v ^ { 2 }$ in terms of $x$.
3. Deduce that the tension in the string, $T$, satisfies
$$\frac { 4 m g } { 3 } \leqslant T < \frac { 3 m g } { 2 }$$

Edexcel M4 2012 June Q4

4. 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

the course the rescue boat sets,
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 $P$.

Edexcel M4 2012 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{07536810-a589-4820-a330-78c35022eb68-10_977_1224_205_360} \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.

Edexcel M4 2012 June Q6

6. 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.

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 2013 June Q1

\hspace{0pt} [In this question $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors due east and due north respectively.]

Boat $A$ is moving with velocity ( $3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$ and boat $B$ is moving with velocity $( 6 \mathbf { i } - 5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$. Find

the magnitude of the velocity of $A$ relative to $B$,
the direction of the velocity of $A$ relative to $B$, giving your answer as a bearing.

Edexcel M4 2013 June Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2a3ae838-b58e-4957-8d98-f7d8a65df99a-03_604_741_123_605} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A smooth fixed plane is inclined at an angle $\alpha$ to the horizontal. A smooth ball $B$ falls vertically and hits the plane. Immediately before the impact the speed of $B$ is $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$, as shown in Figure 1. Immediately after the impact the direction of motion of $B$ is horizontal. The coefficient of restitution between $B$ and the plane is $\frac { 1 } { 3 }$. Find the size of angle $\alpha$.

Edexcel M4 2013 June Q3

A smooth uniform sphere $A$, of mass $5 m$ and radius $r$, is at rest on a smooth horizontal plane. A second smooth uniform sphere $B$, of mass $3 m$ and radius $r$, is moving in a straight line on the plane with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and strikes $A$. Immediately before the impact the direction of motion of $B$ makes an angle of $60 ^ { \circ }$ with the line of centres of the spheres. The direction of motion of $B$ is turned through an angle of $30 ^ { \circ }$ by the impact.

Find

the speed of $B$ immediately after the impact,
the coefficient of restitution between the spheres.

Edexcel M4 2013 June Q4

At 10 a.m. two walkers $A$ and $B$ are 4 km apart with $A$ due north of $B$. Walker $A$ is moving due east at a constant speed of $6 \mathrm {~km} \mathrm {~h} ^ { - 1 }$. Walker $B$ is moving with constant speed $5 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ and walks in the straight line which allows him to pass as close as possible to $A$.

Find

the direction of motion of $B$, giving your answer as a bearing,
the least distance between $A$ and $B$,
the time when the distance between $A$ and $B$ is least.

Edexcel M4 2013 June Q5

5. A van of mass 1200 kg travels along a straight horizontal road against a resistance to motion which is proportional to the speed of the van. The engine of the van is working at a constant rate of 40 kW . The van starts from rest at time $t = 0$. At time $t$ seconds, the speed of the van is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When the speed of the van is $40 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the acceleration of the van is $0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.

Show that $$75 v \frac { \mathrm {~d} v } { \mathrm {~d} t } = 2500 - v ^ { 2 }$$
Find $v$ in terms of $t$.

Edexcel M4 2013 June Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2a3ae838-b58e-4957-8d98-f7d8a65df99a-11_573_679_248_685} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A uniform rod $A B$ has mass $4 m$ and length $4 l$. The rod can turn freely in a vertical plane about a fixed smooth horizontal axis through $A$. A particle of mass $k m$, where $k < 7$, is attached to the rod at $B$. One end of a light elastic string, of natural length $l$ and modulus of elasticity 4 mg , is attached to the point $D$ of the rod, where $A D = 3 l$. The other end of the string is attached to a fixed point $E$ which is vertically above $A$, where $A E = 3 l$, as shown in Figure 2. The angle between the rod and the upward vertical is $2 \theta$, where $\arcsin \left( \frac { 1 } { 6 } \right) < \theta \leqslant \frac { \pi } { 2 }$.

Show that, while the string is stretched, the potential energy of the system is $$8 m g l \left\{ ( 7 - k ) \sin ^ { 2 } \theta - 3 \sin \theta \right\} + \text { constant }$$ There is a position of equilibrium with $\theta \leqslant \frac { \pi } { 6 }$.
Show that $k \leqslant 4$ Given that $k = 4$,
show that this position of equilibrium is stable.

Edexcel M4 2013 June Q7

7. A particle $P$ of mass 0.5 kg is attached to the end $A$ of a light elastic spring $A B$, of natural length 0.6 m and modulus of elasticity 2.7 N . At time $t = 0$ the end $B$ of the spring is held at rest and $P$ hangs at rest at the point $C$ which is vertically below $B$. The end $B$ is then moved along the line of the spring so that, at time $t$ seconds, the downwards displacement of $B$ from its initial position is $4 \sin 2 t$ metres. At time $t$ seconds, the extension of the spring is $x$ metres and the displacement of $P$ below $C$ is $y$ metres.

Show that $$y + \frac { 49 } { 45 } = x + 4 \sin 2 t$$
Hence show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 9 y = 36 \sin 2 t$$ Given that $y = \frac { 36 } { 5 } \sin 2 t$ is a particular integral of this differential equation,
find $y$ in terms of $t$,
find the speed of $P$ when $t = \frac { 1 } { 3 } \pi$.

Edexcel M4 2013 June Q1

A particle $P$ of mass 0.5 kg falls vertically from rest. After $t$ seconds it has speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. A resisting force of magnitude 1.5 v newtons acts on $P$ as it falls.
1. Show that $3 v = 9.8 \left( 1 - \mathrm { e } ^ { - 3 t } \right)$.
2. Find the distance that $P$ falls in the first two seconds of its motion.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a4cdf2b0-8dd0-4c12-9259-95b78875c6cb-03_410_919_219_532} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A river is 50 m wide and flows between two straight parallel banks. The river flows with a uniform speed of $\frac { 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ parallel to the banks. The points $A$ and $B$ are on opposite banks of the river and $A B$ is perpendicular to both banks of the river, as shown in Figure 1. Keith and Ian decide to swim across the river. The speed relative to the water of both swimmers is $\frac { 10 } { 9 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Keith sets out from $A$ and crosses the river in the least possible time, reaching the opposite bank at the point $C$. Find
the time taken by Keith to reach $C$,
the distance $B C$. Ian sets out from $A$ and swims in a straight line so as to land on the opposite bank at $B$.
Find the time taken by Ian to reach $B$.

Edexcel M4 2013 June Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a4cdf2b0-8dd0-4c12-9259-95b78875c6cb-05_643_1155_118_360} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Two smooth uniform spheres $A$ and $B$, of equal radius $r$, have masses $3 m$ and $2 m$ respectively. The spheres are moving on a smooth horizontal plane when they collide. Immediately before the collision they are moving with speeds $u$ and $2 u$ respectively. The centres of the spheres are moving towards each other along parallel paths at a distance $1.6 r$ apart, as shown in Figure 2. The coefficient of restitution between the two spheres is $\frac { 1 } { 6 }$.
Find, in terms of $m$ and $u$, the magnitude of the impulse received by $B$ in the collision.

Edexcel M4 2013 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a4cdf2b0-8dd0-4c12-9259-95b78875c6cb-07_768_666_123_609} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A small smooth peg $P$ is fixed at a distance $d$ from a fixed smooth vertical wire. A particle of mass $3 m$ is attached to one end of a light inextensible string which passes over $P$. The particle hangs vertically below $P$. The other end of the string is attached to a small ring $R$ of mass $m$, which is threaded on the wire, as shown in Figure 3.

Show that when $R$ is at a distance $x$ below the level of $P$ the potential energy of the system is $$3 m g \sqrt { } \left( x ^ { 2 } + d ^ { 2 } \right) - m g x + \text { constant }$$
Hence find $x$, in terms of $d$, when the system is in equilibrium.
Determine the stability of the position of equilibrium.

Edexcel M4 2013 June Q5

5. A coastguard ship $C$ is due south of a ship $S$. Ship $S$ is moving at a constant speed of $12 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ on a bearing of $140 ^ { \circ }$. Ship $C$ moves in a straight line with constant speed $V \mathrm {~km} \mathrm {~h} ^ { - 1 }$ in order to intercept $S$.

Find, giving your answer to 3 significant figures, the minimum possible value for $V$. It is now given that $V = 14$
Find the bearing of the course that $C$ takes to intercept $S$.

Edexcel M4 2013 June Q6

6. A particle $P$ of mass $m \mathrm {~kg}$ is attached to the end $A$ of a light elastic string $A B$, of natural length $a$ metres and modulus of elasticity 9ma newtons. Initially the particle and the string lie at rest on a smooth horizontal plane with $A B = a$ metres. At time $t = 0$ the end $B$ of the string is set in motion and moves at a constant speed $U \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in the direction $A B$. The air resistance acting on $P$ has magnitude 6mv newtons, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the speed of $P$. At time $t$ seconds, the extension of the string is $x$ metres and the displacement of $P$ from its initial position is $y$ metres. Show that, while the string is taut,

$x + y = U t$
$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 6 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 9 x = 6 \mathrm { U }$ You are given that the general solution of the differential equation in (b) is $$x = ( A + B t ) U e ^ { - 3 t } + \frac { 2 U } { 3 }$$ where $A$ and $B$ are arbitrary constants.
Find the value of $A$ and the value of $B$.
Find the speed of $P$ at time $t$ seconds.

Edexcel M4 2013 June Q7

7. [In this question $\mathbf { i }$ and $\mathbf { j }$ are perpendicular unit vectors in a horizontal plane] A small smooth ball of mass $m$ kg is moving on a smooth horizontal plane and strikes a fixed smooth vertical wall. The plane and the wall intersect in a straight line which is parallel to the vector $2 \mathbf { i } + \mathbf { j }$. The velocity of the ball immediately before the impact is $b \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where $b$ is positive. The velocity of the ball immediately after the impact is $a ( \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$, where $a$ is positive.

Show that the impulse received by the ball when it strikes the wall is parallel to $( - \mathbf { i } + 2 \mathbf { j } )$. Find
the coefficient of restitution between the ball and the wall,
the fraction of the kinetic energy of the ball that is lost due to the impact.

Edexcel M4 2014 June Q1

A small smooth ball of mass $m$ is falling vertically when it strikes a fixed smooth plane which is inclined to the horizontal at an angle $\alpha$, where $0 ^ { \circ } < \alpha < 45 ^ { \circ }$. Immediately before striking the plane the ball has speed $u$. Immediately after striking the plane the ball moves in a direction which makes an angle of $45 ^ { \circ }$ with the plane. The coefficient of restitution between the ball and the plane is $e$. Find, in terms of $m , u$ and $e$, the magnitude of the impulse of the plane on the ball.
A ship $A$ is travelling at a constant speed of $30 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ on a bearing of $050 ^ { \circ }$. Another ship $B$ is travelling at a constant speed of $v \mathrm {~km} \mathrm {~h} ^ { - 1 }$ and sets a course to intercept $A$. At 1400 hours $B$ is 20 km from $A$ and the bearing of $A$ from $B$ is $290 ^ { \circ }$.
1. Find the least possible value of $v$.
Given that $v = 32$,
find the time at which $B$ intercepts $A$.

Edexcel M4 2014 June Q3

A small ball of mass $m$ is projected vertically upwards from a point $O$ with speed $U$. The ball is subject to air resistance of magnitude $m k v$, where $v$ is the speed of the ball and $k$ is a positive constant.

Find, in terms of $U , g$ and $k$, the maximum height above $O$ reached by the ball.

Edexcel M4 2014 June Q4

4. A smooth uniform sphere $S$ is moving on a smooth horizontal plane when it collides obliquely with an identical sphere $T$ which is at rest on the plane. Immediately before the collision $S$ is moving with speed $U$ in a direction which makes an angle of $60 ^ { \circ }$ with the line joining the centres of the spheres. The coefficient of restitution between the spheres is $e$.

Find, in terms of $e$ and $U$ where necessary,
1. the speed and direction of motion of $S$ immediately after the collision,
2. the speed and direction of motion of $T$ immediately after the collision. The angle through which the direction of motion of $S$ is deflected is $\delta ^ { \circ }$.
Find
1. the value of $e$ for which $\delta$ takes the largest possible value,
2. the value of $\delta$ in this case.

Edexcel M4 2014 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{656fb620-e80b-4598-a8cd-0f5b8a11e487-08_581_784_221_589} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A uniform rod $A B$, of length $2 l$ and mass $12 m$, has its end $A$ smoothly hinged to a fixed point. One end of a light inextensible string is attached to the other end $B$ of the rod. The string passes over a small smooth pulley which is fixed at the point $C$, where $A C$ is horizontal and $A C = 2 l$. A particle of mass $m$ is attached to the other end of the string and the particle hangs vertically below $C$. The angle $B A C$ is $\theta$, where $0 < \theta < \frac { \pi } { 2 }$, as shown in Figure 1.

Show that the potential energy of the system is $$4 m g l \left( \sin \frac { \theta } { 2 } - 3 \sin \theta \right) + \mathrm { constant }$$
Find the value of $\theta$ when the system is in equilibrium and determine the stability of this equilibrium position.

Edexcel M4 2014 June Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{656fb620-e80b-4598-a8cd-0f5b8a11e487-10_403_933_276_516} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A railway truck of mass $M$ approaches the end of a straight horizontal track and strikes a buffer. The buffer is parallel to the track, as shown in Figure 2. The buffer is modelled as a light horizontal spring $P Q$, which is fixed at the end $P$. The spring has a natural length $a$ and modulus of elasticity $M n ^ { 2 } a$, where $n$ is a postive constant. At time $t = 0$, the spring has length $a$ and the truck strikes the end $Q$ with speed $U$. A resistive force whose magnitude is $M k v$, where $v$ is the speed of the truck at time $t$, and $k$ is a positive constant, also opposes the motion of the truck. At time $t$, the truck is in contact with the buffer and the compression of the buffer is $x$.

Show that, while the truck is compressing the buffer $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + k \frac { \mathrm {~d} x } { \mathrm {~d} t } + n ^ { 2 } x = 0$$ It is given that $k = \frac { 5 n } { 2 }$
Find $x$ in terms of $U , n$ and $t$.
Find, in terms of $U$ and $n$, the greatest value of $x$.

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