Questions - Edexcel M2 - A-Level Maths

find, in terms of $\pi$, the mass of water discharged by the pipe every second,
find in kJ , correct to 3 significant figures, the total mechanical energy gained by the water per second.

Edexcel M2 Q3

3. A particle moves in a straight horizontal line such that its velocity, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, at time $t$ seconds is given by $v = 2 t ^ { 2 } - 9 t + 4$. Initially, the particle has displacement 9 m from a fixed point $O$ on the line.

Find the initial velocity of the particle.
Show that the particle is at rest when $t = 4$ and find the other value of $t$ when it is at rest.
Find the displacement of the particle from $O$ when $t = 6$.

Edexcel M2 Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f8ece90a-5042-4db1-9855-ffe74333a899-3_407_341_201_635} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} Figure 1 shows a uniform ladder of mass $m$ and length $2 a$ resting against a rough vertical wall with its lower end on rough horizontal ground. The coefficient of friction between the ladder and the wall is $\frac { 1 } { 2 }$ and the coefficient of friction between the ladder and the ground is $\frac { 1 } { 3 }$. Given that the ladder is in limiting equilibrium when it is inclined at an angle $\theta$ to the horizontal, show that $\tan \theta = \frac { 5 } { 4 }$.
(9 marks)

Edexcel M2 Q5

5. A firework company is testing its new brand of firework, the Sputnik Special. One of the company's employees lights a Sputnik Special on a large area of horizontal ground and it takes off at a small angle to the vertical. After a flight lasting 8 seconds it lands at a distance of 24 metres from the point where it was launched. The employee models the firework as a particle and ignores air resistance and any loss of mass which the Sputnik Special experiences. Using this model, find for this flight of the Sputnik Special,

the horizontal and vertical components of the initial velocity,
the initial speed, correct to 3 significant figures,
the maximum height attained.
Comment on the suitability of the modelling assumptions made by the employee.

Edexcel M2 Q6

6. Three uniform spheres $A , B$ and $C$ of equal radius have masses $3 m , 2 m$ and $2 m$ respectively. Initially, the spheres are at rest on a smooth horizontal table with their centres in a straight line and with $B$ between $A$ and $C$. Sphere $A$ is projected directly towards $B$ with speed $u$. Given that the coefficient of restitution between $A$ and $B$ is $\frac { 2 } { 3 }$,

show that the speeds of $A$ and $B$ after the collision are $\frac { 1 } { 3 } u$ and $u$ respectively.
(6 marks)
The coefficient of restitution between $B$ and $C$ is $e$. Given that $A$ and $B$ collide again,
show that $e > \frac { 1 } { 3 }$.
(8 marks)

Edexcel M2 Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f8ece90a-5042-4db1-9855-ffe74333a899-4_542_625_959_589} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} Figure 2 shows a uniform lamina $A B C D$ formed by removing an isosceles triangle $B C D$ from an equilateral triangle $A B D$ of side $2 d$. The point $C$ is the centroid of triangle $A B D$.

Find the area of triangle $B C D$ in terms of $d$.
Show that the distance of the centre of mass of the lamina from $B D$ is $\frac { 4 } { 9 } \sqrt { 3 } d$.
(8 marks)
The lamina is freely suspended from the point $B$ and hangs at rest.
Find in degrees, correct to 1 decimal place, the acute angle that the side $A B$ makes with the vertical.

Edexcel M2 Q1

A particle $P$ moves such that at time $t$ seconds its position vector, $\mathbf { r }$ metres, relative to a fixed origin $O$ is given by

$$\mathbf { r } = \left( \frac { 3 } { 2 } t ^ { 2 } - 3 t \right) \mathbf { i } + \left( \frac { 1 } { 3 } t ^ { 3 } - k t \right) \mathbf { j } ,$$ where $k$ is a constant and $\mathbf { i }$ and $\mathbf { j }$ are perpendicular horizontal unit vectors.

Find an expression for the velocity of $P$ at time $t$.
Given that $P$ comes to rest instantaneously, find the value of $k$.

Edexcel M2 Q2

2. Two smooth spheres $P$ and $Q$ of equal radius and of mass $2 m$ and $5 m$ respectively, are moving towards each other along a horizontal straight line when they collide. After the collision, $P$ and $Q$ travel in opposite directions with speeds of $3 \mathrm {~ms} ^ { - 1 }$ and $4 \mathrm {~ms} ^ { - 1 }$ respectively. Given that the coefficient of restitution between the two particles is $\frac { 1 } { 2 }$, find the speeds of $P$ and $Q$ before the collision.
(6 marks)

Edexcel M2 Q3

3. A car of mass 1200 kg experiences a resistance to motion, $R$ newtons, which is proportional to its speed, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When the power output of the car engine is 90 kW and the car is travelling along a horizontal road, its maximum speed is $50 \mathrm {~ms} ^ { - 1 }$.

Show that $R = 36 v$. The car ascends a hill inclined at an angle $\theta$ to the horizontal where $\sin \theta = \frac { 1 } { 14 }$.
Find, correct to 3 significant figures, the maximum speed of the car up the hill assuming that the power output of the engine is unchanged.
(6 marks)

Edexcel M2 Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f5449ec3-ead0-464f-9d03-f225cd21bca6-3_390_725_191_575} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} Figure 1 shows a uniform rod $A B$ of mass 2 kg and length $2 a$. The end $A$ is attached by a smooth hinge to a fixed point on a vertical wall so that the rod can rotate freely in a vertical plane. A mass of 6 kg is placed at $B$ and the rod is held in a horizontal position by a light string joining the midpoint of the rod to a point $C$ on the wall, vertically above $A$. The string is inclined at an angle of $60 ^ { \circ }$ to the wall.

Show that the tension in the string is $28 g$.
Find the magnitude and direction of the force exerted by the hinge on the rod, giving your answers correct to 3 significant figures.

Edexcel M2 Q5

5. A particle $P$ moves in a straight line with an acceleration of $( 6 t - 10 ) \mathrm { m } \mathrm { s } ^ { - 2 }$ at time $t$ seconds. Initially $P$ is at $O$, a fixed point on the line, and has velocity $3 \mathrm {~ms} ^ { - 1 }$.

Find the values of $t$ for which the velocity of $P$ is zero.
Show that, during the first two seconds, $P$ travels a distance of $6 \frac { 26 } { 27 } \mathrm {~m}$.

Edexcel M2 Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f5449ec3-ead0-464f-9d03-f225cd21bca6-4_412_770_198_507} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} A football player strikes a ball giving it an initial speed of $14 \mathrm {~ms} ^ { - 1 }$ at an angle $\alpha$ to the horizontal as shown in Figure 2. At the instant he strikes the ball it is 0.6 m vertically above the point $P$ on the ground. The trajectory of the ball is in a vertical plane containing $P$ and $M$, the middle of the goal-line. The distance between $P$ and $M$ is 12 m and the ground is horizontal. Given that the ball passes over the point $M$ without bouncing,

find, to the nearest degree, the minimum value of $\alpha$. Given that the crossbar of the goal is 2.4 m above $M$ and that $\tan \alpha = \frac { 4 } { 3 }$,
show that the ball passes 4.2 m vertically above the crossbar.

Edexcel M2 Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f5449ec3-ead0-464f-9d03-f225cd21bca6-5_536_848_191_397} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} Figure 3 shows a hotel 'key' consisting of a rectangle $O A B D$, where $O A = 8 \mathrm {~cm}$ and $O D = 4 \mathrm {~cm}$, joined to a semicircle whose diameter $B C$ is 4 cm long. The thickness of the key is negligible and the same material is used throughout. The key is modelled as a uniform lamina.
Using this model,

find, correct to 3 significant figures, the distance of the centre of mass from
1. OD ,
2. $O A$. A small circular hole of negligible diameter is made at the mid-point of $B C$ so that the key can be hung on a smooth peg. When the key is freely suspended from the peg,
find, correct to 3 significant figures, the acute angle made by $O A$ with the vertical.

Edexcel M2 Q1

A ball of mass 0.6 kg bounces against a wall and is given an impulse of $( 12 \mathbf { i } - 9 \mathbf { j } ) \mathrm { Ns }$ where $\mathbf { i }$ and $\mathbf { j }$ are perpendicular horizontal unit vectors. The velocity of the particle after the impact is $( 5 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$.

Find the velocity of the particle before the impact.
(4 marks)

Edexcel M2 Q2

2. A particle $P$ moves along the $x$-axis such that its displacement, $x$ metres, from the origin $O$ at time $t$ seconds is given by $$x = 2 + t - \frac { 1 } { 10 } \mathrm { e } ^ { t }$$

Find the distance of $P$ from $O$ when $t = 0$.
Find, correct to 1 decimal place, the value of $t$ when the velocity of $P$ is zero.
(4 marks)

Edexcel M2 Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ef2dd10c-5a3c-4868-af00-aaf7f2937d7e-2_421_474_1080_664} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} Figure 1 shows a ladder of mass 20 kg and length 6 m leaning against a rough vertical wall with its lower end on smooth horizontal ground. The ladder is prevented from slipping along the ground by a light rope which is attached to the ladder 2 m from its bottom end and fastened to the wall so that the rope is horizontal and perpendicular to the wall. The ladder is at an angle $\theta$ to the horizontal where $\tan \theta = \frac { 5 } { 2 }$ and the coefficient of friction between the ladder and the wall is $\frac { 1 } { 3 }$.

Draw a diagram showing all the forces acting on the ladder.
Show that the magnitude of the tension in the rope is $5 g$. A man wishes to use the ladder but fears the rope will snap as he climbs the ladder.
Suggest, giving a reason for your answer, a more suitable position for the rope.
(2 marks)

Edexcel M2 Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ef2dd10c-5a3c-4868-af00-aaf7f2937d7e-3_222_350_242_788} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} Figure 2 shows an earring consisting of a uniform wire $A B C D$ of length $6 a$ bent to form right angles at $B$ and $C$ such that $A B$ and $C D$ are of length $2 a$ and $a$ respectively.

Find, in terms of $a$, the distance of the centre of mass from
1. $\quad A B$,
2. $B C$. The earring is to be worn such that it hangs in equilibrium suspended from the point $A$.
Find, to the nearest degree, the angle made by $A B$ with the downward vertical.
(4 marks)

Edexcel M2 Q5

5. A lorry of mass 40 tonnes moves up a straight road inclined at an angle $\alpha$ to the horizontal where $\sin \alpha = \frac { 1 } { 20 }$. The lorry moves at a constant speed of $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. In a model of the motion of the lorry, the non-gravitational resistance to motion is assumed to be constant and of magnitude 4400 N .

Show that the engine of the lorry is working at a rate of 480 kW . The road becomes horizontal. The lorry's engine continues to work at the same rate and the resistance to motion is assumed to remain unchanged.
Find the initial acceleration of the lorry.
Find, correct to 3 significant figures, the maximum speed of the lorry.
Using your answer to part (c), comment on the suitability of the modelling assumption.

Edexcel M2 Q6

6. Particle $S$ of mass $2 M$ is moving with speed $U \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on a smooth horizontal plane when it collides directly with a particle $T$ of mass $5 M$ which is lying at rest on the plane. The coefficient of restitution between $S$ and $T$ is $\frac { 3 } { 4 }$. Given that the speed of $T$ after the collision is $4 \mathrm {~ms} ^ { - 1 }$,

find $U$. As a result of the collision, $T$ is projected horizontally from the top of a building of height 19.6 m and falls freely under gravity. $T$ strikes the ground at the point $X$ as shown in Figure 3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ef2dd10c-5a3c-4868-af00-aaf7f2937d7e-4_663_928_740_523} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
Find the time taken for $T$ to reach $X$.
Show that the angle between the horizontal and the direction of motion of $T$, just before it strikes the ground at $X$, is $78.5 ^ { \circ }$ correct to 3 significant figures.
(4 marks)

Edexcel M2 Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ef2dd10c-5a3c-4868-af00-aaf7f2937d7e-5_495_604_214_580} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure} Figure 4 shows a particle $P$ projected from the point $A$ up the line of greatest slope of a rough plane which is inclined at an angle $\alpha$ to the horizontal where $\sin \alpha = \frac { 4 } { 5 } . P$ is projected with speed $5.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the coefficient of friction between $P$ and the plane is $\frac { 4 } { 7 }$. Given that $P$ first comes to rest at point $B$,

use the Work-Energy principle to show that the distance $A B$ is 1.4 m . The particle then slides back down the plane.
Find, correct to 2 significant figures, the speed of $P$ when it returns to $A$.

Edexcel M2 Q1

An ice hockey puck of mass 0.5 kg is moving with velocity $( 5 \mathbf { i } - 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$, where $\mathbf { i }$ and $\mathbf { j }$ are perpendicular horizontal unit vectors, when it is struck by a stick. After the impact, the puck travels with velocity $( 13 \mathbf { i } + 7 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$.

Find the magnitude of the impulse exerted by the stick on the puck.
(5 marks)

Edexcel M2 Q2

2. A car of mass 1 tonne is climbing a hill inclined at an angle $\theta$ to the horizontal where $\sin \theta = \frac { 1 } { 7 }$. When the car passes a point $X$ on the hill, it is travelling at $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When the car passes the point $Y , 200 \mathrm {~m}$ further up the hill, it has speed $10 \mathrm {~ms} ^ { - 1 }$. In a preliminary model of the situation, the car engine is assumed only to be doing work against gravity. Using this model,

find the change in the total mechanical energy of the car as it moves from $X$ to $Y$.
(6 marks)
In a more sophisticated model, the car engine is also assumed to work against other forces.
Write down two other forces which this model might include.
(2 marks)

Edexcel M2 Q3

3. A particle moves along a straight horizontal track such that its displacement, $s$ metres, from a fixed point $O$ on the line after $t$ seconds is given by $$s = 2 t ^ { 3 } - 13 t ^ { 2 } + 20 t$$

Find the values of $t$ for which the particle is at $O$.
Find the values of $t$ at which the particle comes instantaneously to rest.

Edexcel M2 Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0ea2267e-6c46-4a4f-9a38-c242de57901d-3_378_730_196_609} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} Figure 1 shows a uniform rod $A B$ of length 2 m and mass 6 kg inclined at an angle of $30 ^ { \circ }$ to the horizontal with $A$ on smooth horizontal ground and $B$ supported by a rough peg. The rod is in limiting equilibrium and the coefficient of friction between $B$ and the peg is $\mu$.

Find, in terms of $g$, the magnitude of the reactions at $A$ and $B$.
Show that $\mu = \frac { 1 } { \sqrt { 3 } }$.

Edexcel M2 Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0ea2267e-6c46-4a4f-9a38-c242de57901d-3_405_718_1169_555} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} During a cricket match, a batsman hits the ball giving it an initial velocity of $22 \mathrm {~ms} ^ { - 1 }$ at an angle $\alpha$ to the horizontal where $\sin \alpha = \frac { 7 } { 8 }$. When the batsman strikes the ball it is 1.6 metres above the ground, as shown in Figure 2, and it subsequently moves freely under gravity.

Find, correct to 3 significant figures, the maximum height above the ground reached by the ball. The ball is caught by a fielder when it is 0.2 metres above the ground.
Find the length of time for which the ball is in the air. Assuming that the fielder who caught the ball ran at a constant speed of $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$,
find, correct to 3 significant figures, the maximum distance that the fielder could have been from the ball when it was struck.

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