Questions M2 - A-Level Maths

Assume that the two sections of the slide, $P Q$ and $Q R$, are both smooth.
1. Find the kinetic energy of Simon when he reaches the point $R$.
2. Hence find the speed of Simon when he reaches the point $R$.
In fact, the section $Q R$ is rough. Assume that the section $P Q$ is smooth.
Find the coefficient of friction between Simon and the section $Q R$ if Simon comes to rest at the point $R$.
[0pt] [4 marks]
\includegraphics[max width=\textwidth, alt={}]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-06_923_1707_1784_153}

AQA M2 2015 June Q4

4 A particle, $P$, of mass 5 kg is attached to two light inextensible strings, $A P$ and $B P$. The other ends of the strings are attached to the fixed points $A$ and $B$. The point $A$ is vertically above the point $B$. The particle moves at a constant speed, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, in a horizontal circle of radius 0.6 metres with centre $B$. The string $A P$ is inclined at $20 ^ { \circ }$ to the vertical, as shown in the diagram. Both strings are taut when the particle is moving.
\includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-08_835_568_568_719}

Find the tension in the string $A P$.
The speed of the particle is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Show that the tension, $T _ { B P }$, in the string $B P$ is given by $$T _ { B P } = \frac { 25 } { 3 } v ^ { 2 } - 5 g \tan 20 ^ { \circ }$$
Find $v$ when the tensions in the two strings are equal.

AQA M2 2015 June Q5

6 marks

5 An item of clothing is placed inside a washing machine. The drum of the washing machine has radius 30 cm and rotates, about a fixed horizontal axis, at a constant angular speed of 900 revolutions per minute. Model the item of clothing as a particle of mass 0.8 kg and assume that the clothing travels in a vertical circle with constant angular speed. Find the minimum magnitude of the normal reaction force exerted by the drum on the clothing and find the maximum magnitude of the normal reaction force exerted by the drum on the clothing.
[0pt] [6 marks]

\includegraphics[max width=\textwidth, alt={}]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-10_1883_1709_824_153}

AQA M2 2015 June Q6

9 marks

6 A van, of mass 1400 kg , is accelerating at a constant rate of $0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ as it travels up a slope inclined at an angle $\theta$ to the horizontal. The van experiences total resistance forces of 4000 N .
When the van is travelling at a speed of $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the power output of the van's engine is 91.1 kW . Find $\theta$.
[0pt] [9 marks]

AQA M2 2015 June Q7

2 marks

7 A parachutist, of mass 72 kg , is falling vertically. He opens his parachute at time $t = 0$ when his speed is $30 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. He then experiences an air resistance force of magnitude $240 v$ newtons, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is his speed at time $t$ seconds.

When $t > 0$, show that $- \frac { 3 } { 10 } \frac { \mathrm {~d} v } { \mathrm {~d} t } = v - 2.94$.
Find $v$ in terms of $t$.
Sketch a graph to show how, for $t \geqslant 0$, the parachutist's speed varies with time.
[0pt] [2 marks]

AQA M2 2015 June Q8

8 Carol, a bungee jumper of mass 70 kg , is attached to one end of a light elastic cord of natural length 26 metres and modulus of elasticity 1456 N . The other end of the cord is attached to a fixed horizontal platform which is at a height of 69 metres above the ground. Carol steps off the platform at the point where the cord is attached and falls vertically. Hooke's law can be assumed to apply whilst the cord is taut. Model Carol as a particle and assume air resistance to be negligible.
When Carol has fallen $x \mathrm {~m}$, her speed is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

By considering energy, show that $$5 v ^ { 2 } = 306 x - 4 x ^ { 2 } - 2704 \text { for } x \geqslant 26$$
Why is the expression found in part (a) not true when $x$ takes values less than 26?
Find the maximum value of $x$.
1. Find the distance fallen by Carol when her speed is a maximum.
2. Hence find Carol's maximum speed.

AQA M2 2015 June Q9

8 marks

9 A uniform rod, $P Q$, of length $2 a$, rests with one end, $P$, on rough horizontal ground and a point $T$ resting on a rough fixed prism of semicircular cross-section of radius $a$, as shown in the diagram. The rod is in a vertical plane which is parallel to the prism's cross-section. The coefficient of friction at both $P$ and $T$ is $\mu$.
\includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-20_451_1093_477_475} The rod is on the point of slipping when it is inclined at an angle of $30 ^ { \circ }$ to the horizontal. Find the value of $\mu$.
[0pt] [8 marks]
\includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-24_2488_1728_219_141}

AQA M2 2016 June Q1

1 A stone, of mass 0.3 kg , is thrown with a speed of $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from a point at a height of 5 metres above a horizontal surface.

Calculate the initial kinetic energy of the stone.
1. Find the kinetic energy of the stone when it hits the surface.
2. Hence find the speed of the stone when it hits the surface.
3. State one modelling assumption that you have made.

AQA M2 2016 June Q2

4 marks

2 A particle moves in a horizontal plane under the action of a single force, $\mathbf { F }$ newtons.
The unit vectors $\mathbf { i }$ and $\mathbf { j }$ are directed east and north respectively.
At time $t$ seconds, the velocity of the particle, $\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }$, is given by $$\mathbf { v } = \left( 8 t - t ^ { 4 } \right) \mathbf { i } + 6 \mathrm { e } ^ { - 3 t } \mathbf { j }$$

Find an expression for the acceleration of the particle at time $t$.
The mass of the particle is 2 kg .
1. Find an expression for the force $\mathbf { F }$ acting on the particle at time $t$.
2. Find the magnitude of $\mathbf { F }$ when $t = 1$.
Find the value of $t$ when $\mathbf { F }$ acts due south.
When $t = 0$, the particle is at the point with position vector $( 3 \mathbf { i } - 5 \mathbf { j } )$ metres. Find an expression for the position vector, $\mathbf { r }$ metres, of the particle at time $t$.
[0pt] [4 marks]

AQA M2 2016 June Q3

3 The diagram shows a uniform lamina $A B C D E F G H I J K L$.
\includegraphics[max width=\textwidth, alt={}, center]{7c2c50e0-4976-4301-9898-61b2760a2aee-08_474_1378_351_370}

Explain why the centre of mass of the lamina is 35 cm from $A L$.
Find the distance of the centre of mass from $A F$.
The lamina is freely suspended from $A$. Find the angle between $A B$ and the vertical when the lamina is in equilibrium.
Explain, briefly, how you have used the fact that the lamina is uniform.

AQA M2 2016 June Q4

4 marks

4 A particle $P$, of mass 6 kg , is attached to one end of a light inextensible string. The string passes through a small smooth ring, fixed at a point $O$. A second particle $Q$, of mass 8 kg , is attached to the other end of the string. The particle $Q$ hangs at rest vertically below the ring, and the particle $P$ moves with speed $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a horizontal circle, as shown in the diagram. The angle between $O P$ and the vertical is $\theta$.
\includegraphics[max width=\textwidth, alt={}, center]{7c2c50e0-4976-4301-9898-61b2760a2aee-10_517_433_683_790}

Find the tension in the string.
$\quad$ Find $\theta$.
Find the radius of the horizontal circle.
[0pt] [4 marks]

AQA M2 2016 June Q5

4 marks

5 A particle of mass $m$ is suspended from a fixed point $O$ by a light inextensible string of length $l$. The particle hangs in equilibrium at the point $R$ vertically below $O$. The particle is set into motion with a horizontal velocity $u$ so that it moves in a complete vertical circle with centre $O$. The point $T$ on the circle is such that angle $R O T$ is $30 ^ { \circ }$, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{7c2c50e0-4976-4301-9898-61b2760a2aee-12_766_736_644_651}

Find, in terms of $g , l$ and $u$, the speed of the particle at the point $T$.
Find, in terms of $g , l , m$ and $u$, the tension in the string when the particle is at the point $T$.
Find, in terms of $g , l , m$ and $u$, the tension in the string when the particle returns to the point $R$.
The particle makes complete revolutions. Find, in terms of $g$ and $l$, the minimum value of $u$.
[0pt] [4 marks]

AQA M2 2016 June Q6

6 marks

6 A stone, of mass $m$, falls vertically downwards under gravity through still water. At time $t$, the stone has speed $v$ and it experiences a resistance force of magnitude $\lambda m v$, where $\lambda$ is a constant.

Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = g - \lambda v$$
The initial speed of the stone is $u$. Find an expression for $v$ at time $t$.
[0pt] [6 marks] $7 \quad$ A uniform ladder, of weight $W$, rests with its top against a rough vertical wall and its base on rough horizontal ground. The coefficient of friction between the wall and the ladder is $\mu$ and the coefficient of friction between the ground and the ladder is $2 \mu$. When the ladder is on the point of slipping, the ladder is inclined at an angle of $\theta$ to the horizontal.
Draw a diagram to show the forces acting on the ladder.
Find $\tan \theta$ in terms of $\mu$.

AQA M2 2016 June Q8

8 marks

8 A particle, $P$, of mass 5 kg is placed at the point $A$ on a rough plane which is inclined at $30 ^ { \circ }$ to the horizontal. The points $Q$ and $R$ are also on the surface of the inclined plane, with $Q R = 15$ metres. The point $A$ is between $Q$ and $R$ so that $A Q = 4$ metres and $A R = 11$ metres. The three points $Q , A$ and $R$ are on a line of greatest slope of the plane.
\includegraphics[max width=\textwidth, alt={}, center]{7c2c50e0-4976-4301-9898-61b2760a2aee-20_391_882_676_587} The particle is attached to two light elastic strings, $P Q$ and $P R$.
One of the strings, $P Q$, has natural length 4 metres and modulus of elasticity 160 N , the other string, $P R$, has natural length 6 metres and modulus of elasticity 120 N . The particle is released from rest at the point $A$.
The coefficient of friction between the particle and the plane is 0.4 .
Find the distance of the particle from $Q$ when it is next at rest.
[0pt] [8 marks]

\includegraphics[max width=\textwidth, alt={}]{7c2c50e0-4976-4301-9898-61b2760a2aee-23_2488_1709_219_153}

\section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}

Edexcel M2 Q1

A car of mass 1200 kg decelerates from $30 \mathrm {~ms} ^ { - 1 }$ to $20 \mathrm {~ms} ^ { - 1 }$ in 6 seconds at a constant rate.
1. Find the magnitude, in N , of the decelerating force.
2. Find the loss, in J , in the car's kinetic energy.
3. A particle moves in a straight line from $A$ to $B$ in 5 seconds. At time $t$ seconds after leaving $A$, the velocity of the particle is $\left( 32 t - 3 t ^ { 2 } \right) \mathrm { ms } ^ { - 1 }$.
4. Calculate the straight-line distance $A B$.
5. Find the acceleration of the particle when $t = 3$.
6. Eddie, whose mass is 71 kg , rides a bicycle of mass 25 kg up a hill inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = \frac { 1 } { 12 }$. When Eddie is working at a rate of 600 W , he is moving at a constant speed of $6 \mathrm {~ms} ^ { - 1 }$.
  Find the magnitude of the non-gravitational resistance to his motion.
7. A boat leaves the point $O$ and moves such that, $t$ seconds later, its position vector relative to $O$ is $\left( t ^ { 2 } - 2 \right) \mathbf { i } + 2 t \mathbf { j }$, where the vectors $\mathbf { i }$ and $\mathbf { j }$ both have magnitude 1 metre and are directed parallel and perpendicular to the shoreline through $O$.
8. Find the speed with which the boat leaves $O$.
9. Show that the boat has constant acceleration and state the magnitude of this acceleration.
10. Find the value of $t$ when the boat is 40 m from $O$.
11. Comment on the limitations of the given model of the boat's motion.
\includegraphics[max width=\textwidth, alt={}]{996976f3-2a97-4c68-8c97-f15a3bfde9a2-1_446_595_1965_349}
The diagram shows a body which may be modelled as a uniform lamina. The body is suspended from the point marked $A$ and rests in equilibrium.
Calculate, to the nearest degree, the angle which the edge $A B$ then makes with the vertical.
(8 marks) Frank suggests that the angle between $A B$ and the vertical would be smaller if the lamina were made from lighter material.
State, with a brief explanation, whether Frank is correct.
(2 marks) \section*{MECHANICS 2 (A) TEST PAPER 1 Page 2}

Edexcel M2 Q6

A uniform rod $A B$, of mass 0.8 kg and length $10 a$, is supported at the end $A$ by a light inextensible vertical string and rests in limiting equilibrium on a rough fixed peg at $C$, where $A C = 7 a$.
\includegraphics[max width=\textwidth, alt={}, center]{996976f3-2a97-4c68-8c97-f15a3bfde9a2-2_319_638_228_1293}
Two particles $A$ and $B$, of mass $m$ and $k m$ respectively, are moving in the same direction on a smooth horizontal surface. $A$ has speed $4 u$ and $B$ has speed $u$. The coefficient of restitution between $A$ and $B$ is $e \quad A$ collides directly with $B$, and in the collision the direction of $A$ 's motion is reversed. Immediately after the impact, $B$ has speed $2 u$.
1. Show that the speed of $A$ immediately after the impact is $u ( 3 e - 2 )$.
2. Deduce the range of possible values of $e$.
3. Show that $4 < k \leq 5$.
4. A ball is projected from ground level with speed $34 \mathrm {~ms} ^ { - 1 }$ at an angle $\alpha$ above the horizontal, where $\tan \alpha = \frac { 8 } { 15 }$.
5. Find the greatest height reached by the ball above ground level.
While it is descending, the ball hits a horizontal ledge 6 metres above ground level.
Find the horizontal distance travelled by the ball before it hits the ledge.
Find the speed of the ball at the instant when it hits the ledge.

Edexcel M2 Q1

\begin{enumerate} \item A constant force acts on a particle of mass 200 grams, moving it 50 cm in a straight line on a rough horizontal surface at a constant speed. The coefficient of friction between the particle and the surface is $\frac { 1 } { 4 }$.
Calculate, in J , the work done by the force. \item A stone, of mass 0.9 kg , is projected vertically upwards with speed $10 \mathrm {~ms} ^ { - 1 }$ in a medium which exerts a constant resistance to motion. It comes to rest after rising a distance of 3.75 m . Find the magnitude of the non-gravitational resisting force acting on the stone. \item A particle $P$, of mass 0.4 kg , moves in a straight line such that, at time $t$ seconds after passing through a fixed point $O$, its distance from $O$ is $x$ metres, where $x = 3 t ^ { 2 } + 8 t$.

Show that $P$ never returns to $O$.
Find the value of $t$ when $P$ has velocity $20 \mathrm {~ms} ^ { - 1 }$.
Show that the force acting on $P$ is constant, and find its magnitude. \item Two smooth spheres $A$ and $B$, of masses $2 m$ and $3 m$ respectively, are moving on a smooth horizontal table with velocities $( 3 \mathbf { i } - \mathbf { j } ) \mathrm { ms } ^ { - 1 }$ and $( 4 \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }$, where $\mathbf { i }$ and $\mathbf { j }$ are perpendicular unit vectors. They collide, after which $A$ has velocity $( 5 \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }$.

Edexcel M2 Q6

A uniform wire $A B C D$ is bent into the shape shown, where the sections $A B , B C$ and $C D$ are straight and of length $3 a , 10 a$ and $5 a$ respectively and $A D$ is parallel to $B C$.
1. Show that the cosine of angle $B C D$ is $\frac { 4 } { 5 }$.
  \includegraphics[max width=\textwidth, alt={}, center]{70e0f6f0-9016-45a1-9ae2-b908f6b3911e-2_261_479_283_1462}
2. Find the distances of the centre of mass of the bent wire from (i) $A B$, (ii) $B C$.
The wire is hung over a smooth peg at $B$ and rests in equilibrium.
Find, to the nearest $0.1 ^ { \circ }$, the angle between $B C$ and the vertical in this position.

Edexcel M2 Q7

7. Two particles $P$ and $Q$, of masses 0.3 kg and 0.2 kg respectively, are moving towards each other along a straight line. $P$ has speed $4 \mathrm {~ms} ^ { - 1 }$. They collide directly. After the collision the direction of motion of both particles has been reversed, and $Q$ has speed $2 \mathrm {~ms} ^ { - 1 }$. The coefficient of restitution between $P$ and $Q$ is $\frac { 1 } { 3 }$. Find

the speed of $Q$ before the collision,
the speed of $P$ after the collision,
the kinetic energy, in J , lost in the impact.

Edexcel M2 Q8

8. In a fairground game, a contestant bowls a ball at a coconut 6 metres away on the same horizontal level. The ball is thrown with an initial speed of $8 \mathrm {~ms} ^ { - 1 }$ in a direction making an angle of $30 ^ { \circ }$ with the horizontal.
\includegraphics[max width=\textwidth, alt={}, center]{70e0f6f0-9016-45a1-9ae2-b908f6b3911e-2_293_641_1604_1254}

Find the time taken by the ball to travel 6 m horizontally.
Showing your method clearly, decide whether or not the ball will hit the coconut.
Find the greatest height reached by the ball above the level from which it was thrown.
Find the maximum horizontal distance from which it is possible to hit the coconut if the ball is thrown with the same initial speed of $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
State two assumptions that you have made about the ball and the forces which act on it as it travels towards the coconut.

Edexcel M2 Q1

A ball, of mass $m \mathrm {~kg}$, is moving with velocity $( 5 \mathbf { i } - 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$ when it receives an impulse of $( - 2 \mathbf { i } - 4 \mathbf { j } )$ Ns. Immediately after the impulse is applied, the ball has velocity $( 3 \mathbf { i } + k \mathbf { j } ) \mathrm { ms } ^ { - 1 }$. Find the values of the constants $k$ and $m$.
A particle $P$, initially at rest at the point $O$, moves in a straight line such that at time $t$ seconds after leaving $O$ its acceleration is $( 12 t - 15 ) \mathrm { ms } ^ { - 2 }$. Find
1. the velocity of $P$ at time $t$ seconds after it leaves $O$,
2. the value of $t$ when the speed of $P$ is $36 \mathrm {~ms} ^ { - 1 }$.
3. A non-uniform ladder $A B$, of length $3 a$, has its centre of mass at $G$, where $A G = 2 a$. The ladder rests in limiting equilibrium with the end $B$ against a smooth vertical wall and the end $A$ resting on rough horizontal ground. The angle between $A B$ and the horizontal in this position is $\alpha$, where $\tan \alpha = \frac { 14 } { 9 }$. Calculate the coefficient of friction between the ladder and the ground.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6198c550-927b-4554-9ddf-ef166fc9f2dd-1_355_330_1019_1615} \captionsetup{labelformat=empty} \caption{(7 marks)}
\end{figure}

Edexcel M2 Q4

A particle $P$ starts from the point $O$ and moves such that its position vector $\mathbf { r } \mathbf { m }$ relative to $O$ after $t$ seconds is given by $\mathbf { r } = a t ^ { 2 } \mathbf { i } + b t \mathbf { j }$.
60 seconds after $P$ leaves $O$ it is at the point $Q$ with position vector $( 90 \mathbf { i } + 30 \mathbf { j } ) \mathrm { m }$.
1. Find the values of the constants $a$ and $b$.
2. Find the speed of $P$ when it is at $Q$.
3. Sketch the path followed by $P$ for $0 \leq t \leq 60$.
4. A lorry of mass 4200 kg can develop a maximum power of 84 kW . On any road the lorry experiences a non-gravitational resisting force which is directly proportional to its speed. When the lorry is travelling at $20 \mathrm {~ms} ^ { - 1 }$ the resisting force has magnitude 2400 N . Find the maximum speed of the lorry when it is
5. travelling on a horizontal road,
6. climbing a hill inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = \frac { 1 } { 7 }$.
\section*{MECHANICS 2 (A)TEST PAPER 3 Page 2}

Edexcel M2 Q6

Two railway trucks, $P$ and $Q$, of equal mass, are moving towards each other with speeds $4 u$ and $5 u$ respectively along a straight stretch of rail which may be modelled as being smooth. They collide and move apart. The coefficient of restitution between $P$ and $Q$ is $e$.
1. Find, in terms of $u$ and $e$, the speed of $Q$ after the collision.
2. Show that $e > \frac { 1 } { 9 }$.
  $Q$ now hits a fixed buffer and rebounds along the track. $P$ continues to move with the speed that it had immediately after it collided with $Q$.
3. Prove that it is impossible for a further collision between $P$ and $Q$ to occur.
4. A uniform lamina is in the form of a trapezium $A B C D$, as shown. $A B$ and $D C$ are perpendicular to $B C . A B = 17 \mathrm {~cm} , B C = 21 \mathrm {~cm}$ and $C D = 8 \mathrm {~cm}$.
5. Find the distances of the centre of mass of the lamina from
  \includegraphics[max width=\textwidth, alt={}, center]{6198c550-927b-4554-9ddf-ef166fc9f2dd-2_273_426_948_1537}
  1. $A B$,
  2. $B C$.
The lamina is freely suspended from $C$ and rests in equilibrium.
Find the angle between $C D$ and the vertical.

Edexcel M2 Q8

8. A stone, of mass 1.5 kg , is projected horizontally with speed $4 \mathrm {~ms} ^ { - 1 }$ from a height of 7 m above horizontal ground.

Show that the stone travels about 4.78 m horizontally before it hits the ground.
Find the height of the stone above the ground when it has travelled half of this horizontal distance.
Calculate the potential energy lost by the stone as it moves from its point of projection to the ground.
Showing your method clearly, use your answer to part (c) to find the speed with which the stone hits the ground.
State two modelling assumptions that you have made in answering this question.

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