Questions - Edexcel M2 - A-Level Maths

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Edexcel M2 2010 June Q4

Moderate -0.3

A car of mass 750 kg is moving up a straight road inclined at an angle $\theta$ to the horizontal, where $\sin \theta = \frac{1}{15}$. The resistance to motion of the car from non-gravitational forces has constant magnitude $R$ newtons. The power developed by the car's engine is 15 kW and the car is moving at a constant speed of 20 m s$^{-1}$.

Show that $R = 260$. (4)

The power developed by the car's engine is now increased to 18 kW. The magnitude of the resistance to motion from non-gravitational forces remains at 260 N. At the instant when the car is moving up the road at 20 m s$^{-1}$ the car's acceleration is $a$ m s$^{-2}$.

Find the value of $a$. (4)

Edexcel M2 2010 June Q5

Moderate -0.3

[In this question $\mathbf{i}$ and $\mathbf{j}$ are perpendicular unit vectors in a horizontal plane.] A ball of mass 0.5 kg is moving with velocity $(10\mathbf{i} + 24\mathbf{j})$ m s$^{-1}$ when it is struck by a bat. Immediately after the impact the ball is moving with velocity $20\mathbf{i}$ m s$^{-1}$. Find

the magnitude of the impulse of the bat on the ball, (4)
the size of the angle between the vector $\mathbf{i}$ and the impulse exerted by the bat on the ball, (2)
the kinetic energy lost by the ball in the impact. (3)

Edexcel M2 2010 June Q6

Standard +0.3

\includegraphics{figure_2} Figure 2 shows a uniform rod $AB$ of mass $m$ and length $4a$. The end $A$ of the rod is freely hinged to a point on a vertical wall. A particle of mass $m$ is attached to the rod at $B$. One end of a light inextensible string is attached to the rod at $C$, where $AC = 3a$. The other end of the string is attached to the wall at $D$, where $AD = 2a$ and $D$ is vertically above $A$. The rod rests horizontally in equilibrium in a vertical plane perpendicular to the wall and the tension in the string is $T$.

Show that $T = mg\sqrt{13}$. (5)

The particle of mass $m$ at $B$ is removed from the rod and replaced by a particle of mass $M$ which is attached to the rod at $B$. The string breaks if the tension exceeds $2mg\sqrt{13}$. Given that the string does not break,

show that $M \leq \frac{5}{2}m$. (3)

Edexcel M2 2010 June Q7

Standard +0.3

\includegraphics{figure_3} A ball is projected with speed 40 m s$^{-1}$ from a point $P$ on a cliff above horizontal ground. The point $O$ on the ground is vertically below $P$ and $OP$ is 36 m. The ball is projected at an angle $\theta°$ to the horizontal. The point $Q$ is the highest point of the path of the ball and is 12 m above the level of $P$. The ball moves freely under gravity and hits the ground at the point $R$, as shown in Figure 3. Find

the value of $\theta$, (3)
the distance $OR$, (6)
the speed of the ball as it hits the ground at $R$. (3)

Edexcel M2 2010 June Q8

Standard +0.3

A small ball $A$ of mass $3m$ is moving with speed $u$ in a straight line on a smooth horizontal table. The ball collides directly with another small ball $B$ of mass $m$ moving with speed $u$ towards $A$ along the same straight line. The coefficient of restitution between $A$ and $B$ is $\frac{1}{2}$. The balls have the same radius and can be modelled as particles.

Find
1. the speed of $A$ immediately after the collision,
2. the speed of $B$ immediately after the collision.
(7)

After the collision $B$ hits a smooth vertical wall which is perpendicular to the direction of motion of $B$. The coefficient of restitution between $B$ and the wall is $\frac{2}{3}$.

Find the speed of $B$ immediately after hitting the wall. (2)

The first collision between $A$ and $B$ occurred at a distance $4a$ from the wall. The balls collide again $T$ seconds after the first collision.

Show that $T = \frac{112a}{15u}$. (6)

the speed of the ball immediately after the impact, [4]
the angle, in degrees, between the velocity of the ball immediately after the impact and the vector $\mathbf{i}$, [2]
the kinetic energy gained by the ball as a result of the impact. [2]

Edexcel M2 2011 June Q4

7 marks Standard +0.3

\includegraphics{figure_1} Figure 1 shows a uniform lamina $ABCDE$ such that $ABDE$ is a rectangle, $BC = CD$, $AB = 4a$ and $AE = 2a$. The point $F$ is the midpoint of $BD$ and $FC = a$.

Find, in terms of $a$, the distance of the centre of mass of the lamina from $AE$. [4]

The lamina is freely suspended from $A$ and hangs in equilibrium.

Find the angle between $AB$ and the downward vertical. [3]

Edexcel M2 2011 June Q5

10 marks Standard +0.3

\includegraphics{figure_2} A particle $P$ of mass 0.5 kg is projected from a point $A$ up a line of greatest slope $AB$ of a fixed plane. The plane is inclined at 30° to the horizontal and $AB = 2$ m with $B$ above $A$, as shown in Figure 2. The particle $P$ passes through $B$ with speed 5 m s$^{-1}$. The plane is smooth from $A$ to $B$.

Find the speed of projection. [4]

The particle $P$ comes to instantaneous rest at the point $C$ on the plane, where $C$ is above $B$ and $BC = 1.5$ m. From $B$ to $C$ the plane is rough and the coefficient of friction between $P$ and the plane is $\mu$. By using the work-energy principle,

find the value of $\mu$. [6]

Edexcel M2 2011 June Q6

11 marks Moderate -0.3

A particle $P$ moves on the $x$-axis. The acceleration of $P$ at time $t$ seconds is $(t - 4)$ m s$^{-2}$ in the positive $x$-direction. The velocity of $P$ at time $t$ seconds is $v$ m s$^{-1}$. When $t = 0$, $v = 6$. Find

$v$ in terms of $t$, [4]
the values of $t$ when $P$ is instantaneously at rest, [3]
the distance between the two points at which $P$ is instantaneously at rest. [4]

Edexcel M2 2011 June Q7

13 marks Standard +0.3

\includegraphics{figure_3} A uniform rod $AB$, of mass $3m$ and length $4a$, is held in a horizontal position with the end $A$ against a rough vertical wall. One end of a light inextensible string $BD$ is attached to the rod at $B$ and the other end of the string is attached to the wall at the point $D$ vertically above $A$, where $AD = 3a$. A particle of mass $3m$ is attached to the rod at $C$, where $AC = x$. The rod is in equilibrium in a vertical plane perpendicular to the wall as shown in Figure 3. The tension in the string is $\frac{25}{4}mg$. Show that

$x = 3a$, [5]
the horizontal component of the force exerted by the wall on the rod has magnitude $5mg$. [3]

The coefficient of friction between the wall and the rod is $\mu$. Given that the rod is about to slip,

find the value of $\mu$. [5]

Edexcel M2 2011 June Q8

13 marks Standard +0.3

A particle is projected from a point $O$ with speed $u$ at an angle of elevation $\alpha$ above the horizontal and moves freely under gravity. When the particle has moved a horizontal distance $x$, its height above $O$ is $y$.

Show that $$y = x \tan \alpha - \frac{gx^2}{2u^2 \cos^2 \alpha}$$ [4]

A girl throws a ball from a point $A$ at the top of a cliff. The point $A$ is 8 m above a horizontal beach. The ball is projected with speed 7 m s$^{-1}$ at an angle of elevation of 45°. By modelling the ball as a particle moving freely under gravity,

find the horizontal distance of the ball from $A$ when the ball is 1 m above the beach. [5]

A boy is standing on the beach at the point $B$ vertically below $A$. He starts to run in a straight line with speed $v$ m s$^{-1}$, leaving $B$ 0.4 seconds after the ball is thrown. He catches the ball when it is 1 m above the beach.

Find the value of $v$. [4]

Edexcel M2 2013 June Q1

7 marks Moderate -0.3

A caravan of mass 600 kg is towed by a car of mass 900 kg along a straight horizontal road. The towbar joining the car to the caravan is modelled as a light rod parallel to the road. The total resistance to motion of the car is modelled as having magnitude 300 N. The total resistance to motion of the caravan is modelled as having magnitude 150 N. At a given instant the car and the caravan are moving with speed 20 m s$^{-1}$ and acceleration 0.2 m s$^{-2}$.

Find the power being developed by the car's engine at this instant. [5]
Find the tension in the towbar at this instant. [2]

Edexcel M2 2013 June Q2

6 marks Standard +0.3

A ball of mass 0.2 kg is projected vertically upwards from a point $O$ with speed 20 m s$^{-1}$. The non-gravitational resistance acting on the ball is modelled as a force of constant magnitude 1.24 N and the ball is modelled as a particle. Find, using the work-energy principle, the speed of the ball when it first reaches the point which is 8 m vertically above $O$. [6]

Edexcel M2 2013 June Q3

9 marks Moderate -0.3

A particle $P$ moves along a straight line in such a way that at time $t$ seconds its velocity $v$ m s$^{-1}$ is given by $$v = \frac{1}{2}t^2 - 3t + 4$$ Find

the times when $P$ is at rest, [4]
the total distance travelled by $P$ between $t = 0$ and $t = 4$. [5]

Edexcel M2 2013 June Q4

11 marks Standard +0.8

A rough circular cylinder of radius $4a$ is fixed to a rough horizontal plane with its axis horizontal. A uniform rod $AB$, of weight $W$ and length $6a\sqrt{3}$, rests with its lower end $A$ on the plane and a point $C$ of the rod against the cylinder. The vertical plane through the rod is perpendicular to the axis of the cylinder. The rod is inclined at 60° to the horizontal, as shown in Figure 1. \includegraphics{figure_1}

Show that $AC = 4a\sqrt{3}$ [2]

The coefficient of friction between the rod and the cylinder is $\frac{\sqrt{3}}{3}$ and the coefficient of friction between the rod and the plane is $\mu$. Given that friction is limiting at both $A$ and $C$,

find the value of $\mu$. [9]

Edexcel M2 2013 June Q5

13 marks Standard +0.3

Two particles $P$ and $Q$, of masses $2m$ and $m$ respectively, are on a smooth horizontal table. Particle $Q$ is at rest and particle $P$ collides directly with it when moving with speed $u$. After the collision the total kinetic energy of the two particles is $\frac{3}{4}mu^2$. Find

the speed of $Q$ immediately after the collision, [10]
the coefficient of restitution between the particles. [3]

Edexcel M2 2013 June Q6

13 marks Standard +0.8

\includegraphics{figure_2} A uniform triangular lamina $ABC$ of mass $M$ is such that $AB = AC$, $BC = 2a$ and the distance of $A$ from $BC$ is $h$. A line, parallel to $BC$ and at a distance $\frac{2h}{3}$ from $A$, cuts $AB$ at $D$ and cuts $AC$ at $E$, as shown in Figure 2. It is given that the mass of the trapezium $BCED$ is $\frac{5M}{9}$.

Show that the centre of mass of the trapezium $BCED$ is $\frac{7h}{45}$ from $BC$. [5]

\includegraphics{figure_3} The portion $ADE$ of the lamina is folded through 180° about $DE$ to form the folded lamina shown in Figure 3.

Find the distance of the centre of mass of the folded lamina from $BC$. [4]

The folded lamina is freely suspended from $D$ and hangs in equilibrium. The angle between $DE$ and the downward vertical is $\alpha$.

Find tan $\alpha$ in terms of $a$ and $h$. [4]

Edexcel M2 2013 June Q7

16 marks Standard +0.3

\includegraphics{figure_4} A small ball is projected from a fixed point $O$ so as to hit a target $T$ which is at a horizontal distance $9a$ from $O$ and at a height $6a$ above the level of $O$. The ball is projected with speed $\sqrt{(27ag)}$ at an angle $\theta$ to the horizontal, as shown in Figure 4. The ball is modelled as a particle moving freely under gravity.

Show that tan$^2 \theta - 6$ tan $\theta + 5 = 0$ [7]

The two possible angles of projection are $\theta_1$ and $\theta_2$, where $\theta_1 > \theta_2$.

Find tan $\theta_1$ and tan $\theta_2$. [3]

The particle is projected at the larger angle $\theta_1$.

Show that the time of flight from $O$ to $T$ is $\sqrt{\left(\frac{78a}{g}\right)}$. [3]
Find the speed of the particle immediately before it hits $T$. [3]