Questions - Edexcel M4 - A-Level Maths

Show that when $R$ is at a distance $x$ below the level of $P$ the potential energy of the system is $$3mg \sqrt{(x^2 + d^2)} - mgx + \text{constant}$$ [4]
Hence find $x$, in terms of $d$, when the system is in equilibrium. [3]
Determine the stability of the position of equilibrium. [3]

Edexcel M4 2013 June Q5

8 marks Standard +0.8

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

Find, giving your answer to 3 significant figures, the minimum possible value for $V$. [3]

It is now given that $V = 14$

Find the bearing of the course that $C$ takes to intercept $S$. [5]

Edexcel M4 2013 June Q6

14 marks Challenging +1.3

A particle $P$ of mass $m$ kg is attached to the end $A$ of a light elastic string $AB$, 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 $AB = a$ metres. At time $t = 0$ the end $B$ of the string is set in motion and moves at a constant speed $U$ m s$^{-1}$ in the direction $AB$. The air resistance acting on $P$ has magnitude $6mv$ newtons, where $v$ m 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,

$x + y = Ut$ [2]
$\frac{d^2x}{dt^2} + 6\frac{dx}{dt} + 9x = 6U$ [5]

You are given that the general solution of the differential equation in (b) is $$x = (A + Bt)e^{-3t} + \frac{2U}{3}$$ where $A$ and $B$ are arbitrary constants.

Find the value of $A$ and the value of $B$. [5]
Find the speed of $P$ at time $t$ seconds. [2]

Edexcel M4 2013 June Q7

12 marks Challenging +1.8

[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 $b\mathbf{i} + \mathbf{j}$ m s$^{-1}$, where $b$ is positive. The velocity of the ball immediately after the impact is $a(\mathbf{i} + \mathbf{j})$ m s$^{-1}$, where $a$ is positive.

Show that the impulse received by the ball when it strikes the wall is parallel to $(-\mathbf{i} + 2\mathbf{j})$. [1]

Find

the coefficient of restitution between the ball and the wall, [8]
the fraction of the kinetic energy of the ball that is lost due to the impact. [3]

Edexcel M4 2014 June Q1

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 $0° < \alpha < 45°$. 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 $45°$ 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. (11)

Edexcel M4 2014 June Q2

Standard +0.8

A ship $A$ is travelling at a constant speed of 30 km h$^{-1}$ on a bearing of $050°$. Another ship $B$ is travelling at a constant speed of $v$ km 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 $290°$.

Find the least possible value of $v$. (3)

Given that $v = 32$,

find the time at which $B$ intercepts $A$. (8)

Edexcel M4 2014 June Q3

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 $mkv$, 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. (8)

Edexcel M4 2014 June Q4

Challenging +1.8

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 $60°$ 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.
(12)

The angle through which the direction of motion of $S$ is deflected is $\delta°$.

Find
1. the value of $e$ for which $\delta$ takes the largest possible value,
2. the value of $\delta$ in this case.
(3)

Edexcel M4 2014 June Q5

Challenging +1.8

\includegraphics{figure_1} A uniform rod $AB$, of length $2l$ and mass $12m$, 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 $AC$ is horizontal and $AC = 2l$. A particle of mass $m$ is attached to the other end of the string and the particle hangs vertically below $C$. The angle $BAC$ is $\theta$, where $0 < \theta < \frac{\pi}{2}$, as shown in Figure 1.

Show that the potential energy of the system is $$4mgl\left(\sin\frac{\theta}{2} - 3\sin\theta\right) + \text{constant}$$ (4)

Find the value of $\theta$ when the system is in equilibrium and determine the stability of this equilibrium position. (10)

Edexcel M4 2014 June Q6

Challenging +1.3

\includegraphics{figure_2} 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 $PQ$, which is fixed at the end $P$. The spring has a natural length $a$ and modulus of elasticity $Mn^2a$, where $n$ is a positive 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 $Mkv$, 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 $$\frac{\text{d}^2x}{\text{d}t^2} + k\frac{\text{d}x}{\text{d}t} + n^2x = 0$$ (4)

It is given that $k = \frac{5n}{2}$

Find $x$ in terms of $U$, $n$ and $t$. (7)

Find, in terms of $U$ and $n$, the greatest value of $x$. (5)

Edexcel M4 2014 June Q1

8 marks Standard +0.8

A particle $A$ has constant velocity $(3\mathbf{i} + \mathbf{j})$ m s$^{-1}$ and a particle $B$ has constant velocity $(\mathbf{i} - \mathbf{k})$ m s$^{-1}$. At time $t = 0$ seconds, the position vectors of the particles $A$ and $B$ with respect to a fixed origin $O$ are $(-6\mathbf{i} + 4\mathbf{j} - 3\mathbf{k})$ m and $(-2\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})$ m respectively.

Show that, in the subsequent motion, the minimum distance between $A$ and $B$ is $4\sqrt{2}$ m. [6]
Find the position vector of $A$ at the instant when the distance between $A$ and $B$ is a minimum. [2]

Edexcel M4 2014 June Q2

11 marks Standard +0.8

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 $v$ m s$^{-1}$, the resistance to motion has magnitude $10v$ newtons.

Show that, at the instant when $v = 20$, the acceleration of the car is 1.05 m s$^{-2}$. [3]
Find the distance travelled by the car as it accelerates from a speed of 10 m s$^{-1}$ to a speed of 20 m s$^{-1}$. [8]

Edexcel M4 2014 June Q3

8 marks Challenging +1.2

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. [8]

Edexcel M4 2014 June Q4

8 marks Challenging +1.2

At noon two ships $A$ and $B$ are 20 km apart with $A$ on a bearing of 230° from $B$. Ship $B$ is moving at 6 km h$^{-1}$ on a bearing of 015°. The maximum speed of $A$ is 12 km 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. [4]
Find the time at which $A$ intercepts $B$. [4]

Edexcel M4 2014 June Q5

12 marks Challenging +1.8

\includegraphics{figure_1} Two smooth uniform spheres $A$ and $B$ have equal radii. The mass of $A$ is $m$ and the mass of $B$ is $3m$. The spheres are moving on a smooth horizontal plane when they collide obliquely. Immediately before the collision, $A$ is moving with speed $3u$ 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 $u$. [8]
Express the kinetic energy lost by $A$ in the collision as a fraction of its initial kinetic energy. [4]

Edexcel M4 2014 June Q6

13 marks Challenging +1.8

A particle of mass $m$ 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 $2a$ metres and then released at time $t = 0$. During the subsequent motion, when the particle is moving with speed $v$ m s$^{-1}$, the particle experiences a resistance of magnitude $4mv$ 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, $$\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} + 4\frac{\mathrm{d}x}{\mathrm{d}t} + 5x = 0$$ [3]
Hence express $x$ in terms of $a$ and $t$. [6]
Find the speed of the particle at the instant when the string first becomes slack, giving your answer in the form $ka$, where $k$ is a constant to be found correct to 2 significant figures. [4]