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Edexcel M4 2013 June Q3

10 marks Challenging +1.8

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

10 marks Challenging +1.2

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

8 marks Challenging +1.2

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

14 marks Challenging +1.8

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

12 marks Challenging +1.2

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

11 marks Challenging +1.2

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

8 marks Challenging +1.2

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

15 marks Challenging +1.2

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

14 marks Challenging +1.8

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

16 marks Standard +0.8

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 $x$.

Edexcel M4 2014 June Q1

8 marks Standard +0.3

A particle $A$ has constant velocity $( 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and a particle $B$ has constant velocity $( \mathbf { i } - \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. At time $t = 0$ seconds, the position vectors of the particles $A$ and $B$ with respect to a fixed origin $O$ are $( - 6 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } ) \mathrm { m }$ and $( - 2 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { m }$ respectively.
1. Show that, in the subsequent motion, the minimum distance between $A$ and $B$ is $4 \sqrt { } 2 \mathrm {~m}$.
2. Find the position vector of $A$ at the instant when the distance between $A$ and $B$ is a minimum.
3. A car of mass 1000 kg is moving along a straight horizontal road. The engine of the car is working at a constant rate of 25 kW . When the speed of the car is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the resistance to motion has magnitude 10 v newtons.
4. Show that, at the instant when $v = 20$, the acceleration of the car is $1.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
5. Find the distance travelled by the car as it accelerates from a speed of $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ to a speed of $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
6. A small ball is moving on a smooth horizontal plane when it collides obliquely with a smooth plane vertical wall. The coefficient of restitution between the ball and the wall is $\frac { 1 } { 3 }$. The speed of the ball immediately after the collision is half the speed of the ball immediately before the collision.
Find the angle through which the path of the ball is deflected by the collision.

Edexcel M4 2014 June Q4

8 marks Challenging +1.2

4. At noon two ships $A$ and $B$ are 20 km apart with $A$ on a bearing of $230 ^ { \circ }$ from $B$. Ship $B$ is moving at $6 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ on a bearing of $015 ^ { \circ }$. The maximum speed of $A$ is $12 \mathrm {~km} \mathrm {~h} ^ { - 1 }$. Ship $A$ sets a course to intercept $B$ as soon as possible.

Find the course set by $A$, giving your answer as a bearing to the nearest degree.
Find the time at which $A$ intercepts $B$.

Edexcel M4 2014 June Q5

12 marks Challenging +1.2

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{904c44f8-bd97-4a1d-8eb1-73cb52ddc8c5-07_478_1185_267_376} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Two smooth uniform spheres $A$ and $B$ have equal radii. The mass of $A$ is $m$ and the mass of $B$ is $3 m$. The spheres are moving on a smooth horizontal plane when they collide obliquely. Immediately before the collision, $A$ is moving with speed $3 u$ at angle $\alpha$ to the line of centres and $B$ is moving with speed $u$ at angle $\beta$ to the line of centres, as shown in Figure 1. The coefficient of restitution between the two spheres is $\frac { 1 } { 5 }$. It is given that $\cos \alpha = \frac { 1 } { 3 }$ and $\cos \beta = \frac { 2 } { 3 }$ and that $\alpha$ and $\beta$ are both acute angles.

Find the magnitude of the impulse on $A$ due to the collision in terms of $m$ and $u$.
Express the kinetic energy lost by $A$ in the collision as a fraction of its initial kinetic energy.

Edexcel M4 2014 June Q6

13 marks Challenging +1.2

6. A particle of mass $m \mathrm {~kg}$ is attached to one end of a light elastic string of natural length a metres and modulus of elasticity 5ma newtons. The other end of the string is attached to a fixed point $O$ on a smooth horizontal plane. The particle is held at rest on the plane with the string stretched to a length $2 a$ metres and then released at time $t = 0$. During the subsequent motion, when the particle is moving with speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the particle experiences a resistance of magnitude $4 m v$ newtons. At time $t$ seconds after the particle is released, the length of the string is ( $a + x$ ) metres, where $0 \leqslant x \leqslant a$.

Show that, from $t = 0$ until the string becomes slack, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 0$$
Hence express $x$ in terms of $a$ and $t$.
Find the speed of the particle at the instant when the string first becomes slack, giving your answer in the form $k a$, where $k$ is a constant to be found correct to 2 significant figures.

Edexcel M4 2014 June Q7

15 marks Challenging +1.2

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{904c44f8-bd97-4a1d-8eb1-73cb52ddc8c5-11_595_552_260_712} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A bead $B$ of mass $m$ is threaded on a smooth circular wire of radius $r$, which is fixed in a vertical plane. The centre of the circle is $O$, and the highest point of the circle is $A$. A light elastic string of natural length $r$ and modulus of elasticity $k m g$ has one end attached to the bead and the other end attached to $A$. The angle between the string and the downward vertical is $\theta$, and the extension in the string is $x$, as shown in Figure 2. Given that the string is taut,

show that the potential energy of the system is $$2 m g r \left\{ ( k - 1 ) \cos ^ { 2 } \theta - k \cos \theta \right\} + \text { constant }$$ Given also that $k = 3$,
find the positions of equilibrium and determine their stability. \includegraphics[max width=\textwidth, alt={}, center]{904c44f8-bd97-4a1d-8eb1-73cb52ddc8c5-12_109_127_2480_1818}

Edexcel M4 2015 June Q1

7 marks Standard +0.3

Particles $P$ and $Q$ move in a plane with constant velocities. At time $t = 0$ the position vectors of $P$ and $Q$, relative to a fixed point $O$ in the plane, are $( 16 \mathbf { i } - 12 \mathbf { j } ) \mathrm { m }$ and $( - 5 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m }$ respectively. The velocity of $P$ is $( \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and the velocity of $Q$ is $( 2 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$

Find the shortest distance between $P$ and $Q$ in the subsequent motion.

Edexcel M4 2015 June Q2

6 marks Challenging +1.2

When a woman walks due North at a constant speed of $4 \mathrm {~km} \mathrm {~h} ^ { - 1 }$, the wind appears to be blowing from due East. When she runs due South at a constant speed of $8 \mathrm {~km} \mathrm {~h} ^ { - 1 }$, the speed of the wind appears to be $20 \mathrm {~km} \mathrm {~h} ^ { - 1 }$.

Assuming that the velocity of the wind relative to the earth is constant, find

the speed of the wind,
the direction from which the wind is blowing.

Edexcel M4 2015 June Q3

12 marks Challenging +1.8

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{44066c44-e366-4f87-b1b2-c5a894e407fa-08_350_1123_258_408} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Two smooth uniform spheres $A$ and $B$ with equal radii have masses $m$ and $2 m$ respectively. The spheres are moving in opposite directions on a smooth horizontal surface and collide obliquely. Immediately before the collision, $A$ has speed $3 u$ with its direction of motion at an angle $\theta$ to the line of centres, and $B$ has speed $u$ with its direction of motion at an angle $\theta$ to the line of centres, as shown in Figure 1. The coefficient of restitution between the spheres is $\frac { 1 } { 8 }$ Immediately after the collision, the speed of $A$ is twice the speed of $B$.
Find the size of the angle $\theta$.

Edexcel M4 2015 June Q4

14 marks Standard +0.8

4. A car of mass 900 kg is moving along a straight horizontal road with the engine of the car working at a constant rate of 22.5 kW . At time $t$ seconds, the speed of the car is $v \mathrm {~m} \mathrm {~s} ^ { - 1 } ( 0 < v < 30 )$ and the total resistance to the motion of the car has magnitude $25 v$ newtons.

Show that when the speed of the car is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the acceleration of the car is $$\frac { 900 - v ^ { 2 } } { 36 v } \mathrm {~m} \mathrm {~s} ^ { - 2 }$$ The time taken for the car to accelerate from $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ to $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is $T$ seconds.
Show that $$T = 18 \ln \frac { 8 } { 5 }$$
Find the distance travelled by the car as it accelerates from $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ to $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

Edexcel M4 2015 June Q5

10 marks Challenging +1.8

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{44066c44-e366-4f87-b1b2-c5a894e407fa-16_193_1367_274_287} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A particle $P$ of mass 1.5 kg is attached to the midpoint of a light elastic spring $A B$, of natural length 2 m and modulus of elasticity 12 N . The end $A$ of the spring is attached to a fixed point on a smooth horizontal floor. The end $B$ is held at a point on the floor where $A B = 6 \mathrm {~m}$. At time $t = 0 , P$ is at rest on the floor at the point $O$, where $A O = 3 \mathrm {~m}$, as shown in Figure 2. The end $B$ is now moved along the floor in such a way that $A O B$ remains a straight line and at time $t$ seconds, $t \geqslant 0$, $$A B = \left( 6 + \frac { 1 } { 4 } \sin 2 t \right) \mathrm { m }$$ At time $t$ seconds, $A P = ( 3 + x ) \mathrm { m }$.

Show that, for $t \geqslant 0$, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 16 x = 2 \sin 2 t$$ The general solution of this differential equation is $$x = C \cos 4 t + D \sin 4 t + \frac { 1 } { 6 } \sin 2 t$$ where $C$ and $D$ are constants.
Find the time at which $P$ first comes to instantaneous rest. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{44066c44-e366-4f87-b1b2-c5a894e407fa-20_705_1104_116_420} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure}

Edexcel M4 2015 June Q6

13 marks Challenging +1.2

6. A smooth wire, with ends $A$ and $B$, is in the shape of a semicircle of radius $r$. The line $A B$ is horizontal and the midpoint of $A B$ is $O$. The wire is fixed in a vertical plane. A small ring $R$ of mass $2 m$ is threaded on the wire and is attached to two light inextensible strings. One string passes through a small smooth ring fixed at $A$ and is attached to a particle of mass $\sqrt { 6 } m$. The other string passes through a small smooth ring fixed at $B$ and is attached to a second particle of mass $\sqrt { 6 } \mathrm {~m}$. The particles hang freely under gravity, as shown in Figure 3. The angle between the radius $O R$ and the downward vertical is $2 \theta$, where $- \frac { \pi } { 4 } < \theta < \frac { \pi } { 4 }$

Show that the potential energy of the system is $$2 m g r ( 2 \sqrt { 3 } \cos \theta - \cos 2 \theta ) + \text { constant }$$
Find the values of $\theta$ for which the system is in equilibrium.
Determine the stability of the position of equilibrium for which $\theta > 0$

Edexcel M4 2015 June Q7

13 marks Standard +0.8

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{44066c44-e366-4f87-b1b2-c5a894e407fa-24_494_936_260_536} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 represents the plan view of part of a smooth horizontal floor, where $A B$ and $B C$ are smooth vertical walls. The angle between $A B$ and $B C$ is $120 ^ { \circ }$. A ball is projected along the floor towards $A B$ with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on a path at an angle of $60 ^ { \circ }$ to $A B$. The ball hits $A B$ and then hits $B C$. The ball is modelled as a particle. The coefficient of restitution between the ball and each wall is $\frac { 1 } { 2 }$

Show that the speed of the ball immediately after it has hit $A B$ is $\frac { \sqrt { 7 } } { 4 } u$. The speed of the ball immediately after it has hit $B C$ is $w \mathrm {~m} \mathrm {~s} ^ { - 1 }$
Find $w$ in terms of $u$.

Find the magnitude and direction of the velocity of the car relative to the girl, giving the direction as a bearing.
(6) At noon the girl is 150 m south of $X$ and the car is 800 m east of $X$.
Find the shortest distance between the car and the girl during the subsequent motion.

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