Questions - Edexcel M4 - A-Level Maths

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.

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