Questions - Edexcel M2 - A-Level Maths

Show that the centre of mass of the lamina $B C D E$ is $6 \frac { 2 } { 3 } \mathrm {~cm}$ from $B C$.
(6 marks)
The lamina $B C D E$ is freely suspended from $D$ and hangs in equilibrium.
Find, in degrees to one decimal place, the angle which $D E$ makes with the vertical.
(3 marks)

Edexcel M2 Q4

4. The resistance to the motion of a cyclist is modelled as $k v ^ { 2 } \mathrm {~N}$, where $k$ is a constant and $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the speed of the cyclist. The total mass of the cyclist and his bicycle is 100 kg . The cyclist freewheels down a slope inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = \frac { 1 } { 20 }$, at a constant speed of $3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Show that $k = 4$. The cyclist ascends a slope inclined at an angle $\beta$ to the horizontal, where $\sin \beta = \frac { 1 } { 40 }$, at a constant speed of $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find the rate at which the cyclist is working.
(6 marks)

Edexcel M2 Q5

5. A smooth sphere $S$ of mass $m$ is moving with speed $u$ on a smooth horizontal plane. The sphere $S$ collides with another smooth sphere $T$, of equal radius to $S$ but of mass $k m$, moving in the same straight line and in the same direction with speed $\lambda u , 0 < \lambda < \frac { 1 } { 2 }$. The coefficient of restitution between $S$ and $T$ is $e$. Given that $S$ is brought to rest by the impact,

show that $e = \frac { 1 + k \lambda } { k ( 1 - \lambda ) }$.
Deduce that $k > 1$.

Edexcel M2 Q6

6. At time $t$ seconds the acceleration, a $\mathrm { m } \mathrm { s } ^ { - 2 }$, of a particle $P$ relative to a fixed origin $O$, is given by $\mathbf { a } = 2 \mathbf { i } + 6 t \mathbf { j }$. Initially the velocity of $P$ is $( 2 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$.

Find the velocity of $P$ at time $t$ seconds. At time $t = 2$ seconds the particle $P$ is given an impulse ( $3 \mathbf { i } - 1.5 \mathbf { j }$ ) Ns. Given that the particle $P$ has mass 0.5 kg ,
find the speed of $P$ immediately after the impulse has been applied.

Edexcel M2 Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9126ebb1-eaa7-4a40-953f-5dc819c9f479-6_675_1243_392_415} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} A shot is projected upwards from the top of a cliff with a velocity of $28 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $30 ^ { \circ }$ above the horizontal. It strikes the ground 52.5 m vertically below the level of the point of projection, as shown in Fig. 3. The motion of the shot is modelled as that of a particle moving freely under gravity. Find, to 3 significant figures,

the horizontal distance from the point of projection at which the shot strikes the ground,
the speed of the shot as it strikes the ground.

Edexcel M2 Q8

8. A particle $P$ is projected up a line of greatest slope of a rough plane which is inclined at an angle $\alpha$ to the horizontal, where $\tan \alpha = \frac { 3 } { 4 }$. The coefficient of friction between $P$ and the plane is $\frac { 1 } { 2 }$. The particle is projected from the point $O$ with a speed of $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and comes to instantaneous rest at the point $A$. By Using the Work-Energy principle, or otherwise,

find, to 3 significant figures, the length $O A$.
Show that $P$ will slide back down the plane.
Find, to 3 significant figures, the speed of $P$ when it returns to $O$.

Edexcel M2 Specimen Q1

The vectors $\mathbf { i }$ and $\mathbf { j }$ are perpendicular unit vectors in a horizontal plane. A ball of mass 0.5 kg is moving with velocity $- 20 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when it is struck by a bat. The bat gives the ball an impulse of $( 15 \mathbf { i } + 10 \mathbf { j } )$ Ns.

Find, to 3 significant figures, the speed of the ball immediately after it has been struck.
(5)

Edexcel M2 Specimen Q2

2. A bullet of mass 6 grams passes horizontally through a fixed, vertical board. After the bullet has travelled 2 cm through the board its speed is reduced from $400 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ to $250 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The board exerts a constant resistive force on the bullet. Find, to 3 significant figures, the magnitude of this resistive force.
(5)

Edexcel M2 Specimen Q3

3. At time $t$ seconds, a particle $P$ has position vector $\mathbf { r }$ metres relative to a fixed origin $O$, where $$\mathbf { r } = \left( t ^ { 3 } - 3 t \right) \mathbf { i } + 4 t ^ { 2 } \mathbf { j } , t \geq 0$$ Find

the velocity of $P$ at time $t$ seconds,
the time when $P$ is moving parallel to the vector $\mathbf { i } + \mathbf { j }$.
(5)

Edexcel M2 Specimen Q4

4. \section*{Figure 1}

\includegraphics[max width=\textwidth, alt={}]{0d3d35b1-e3c5-47ac-b05e-78cdf1eb3083-3_714_565_262_749}

A uniform ladder, of mass $m$ and length $2 a$, has one end on rough horizontal ground. The other end rests against a smooth vertical wall. A man of mass $3 m$ stands at the top of the ladder and the ladder is in equilibrium. The coefficient of friction between the ladder and the ground is $\frac { 1 } { 4 }$, and the ladder makes an angle $\alpha$ with the vertical, as shown in Fig. 1. The ladder is in a vertical plane perpendicular to the wall. Show that $\tan \alpha \leq \frac { 2 } { 7 }$.

Edexcel M2 Specimen Q5

5. A straight road is inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = \frac { 1 } { 20 }$. A lorry of mass 4800 kg moves up the road at a constant speed of $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The non-gravitational resistance to the motion of the lorry is constant and has magnitude 2000 N .

Find, in kW to 3 significant figures, the rate of working of the lorry's engine.
(5) The road becomes horizontal. The lorry's engine continues to work at the same rate and the resistance to motion remains the same. Find
the acceleration of the lorry immediately after the road becomes horizontal,
(3)
the maximum speed, in $\mathrm { m } \mathrm { s } ^ { - 1 }$ to 3 significant figures, at which the lorry will go along the horizontal road.
(3)

Edexcel M2 Specimen Q6

6. A cricket ball is hit from a height of 0.8 m above horizontal ground with a speed of $26 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle $\alpha$ above the horizontal, where $\tan \alpha = \frac { 5 } { 12 }$. The motion of the ball is modelled as that of a particle moving freely under gravity.

Find, to 2 significant figures, the greatest height above the ground reached by the ball. When the ball has travelled a horizontal distance of 36 m , it hits a window.
Find, to 2 significant figures, the height above the ground at which the ball hits the window.
State one physical factor which could be taken into account in any refinement of the model which would make it more realistic. Figure 2

Edexcel M2 Specimen Q7

7.
\includegraphics[max width=\textwidth, alt={}, center]{0d3d35b1-e3c5-47ac-b05e-78cdf1eb3083-4_360_472_1105_815} A uniform plane lamina $A B C D E$ is formed by joining a uniform square $A B D E$ with a uniform triangular lamina $B C D$, of the same material, along the side $B D$, as shown in Fig. 2. The lengths $A B , B C$ and $C D$ are $18 \mathrm {~cm} , 15 \mathrm {~cm}$ and 15 cm respectively.

Find the distance of the centre of mass of the lamina from $A E$. The lamina is freely suspended from $B$ and hangs in equilibrium.
Find, in degrees to one decimal place, the angle which $B D$ makes with the vertical.

Edexcel M2 Specimen Q8

8. A particle $A$ of mass $m$ is moving with speed $3 u$ on a smooth horizontal table when it collides directly with a particle $B$ of mass $2 m$ which is moving in the opposite direction with speed $u$. The direction of motion of $A$ is reversed by the collision. The coefficient of restitution between $A$ and $B$ is $e$.

Show that the speed of $B$ immediately after the collision is $\frac { 1 } { 3 } ( 1 + 4 e ) u$.
(6)
Show that $e > \frac { 1 } { 8 }$.
(3) Subsequently $B$ hits a wall fixed at right angles to the line of motion of $A$ and $B$. The coefficient of restitution between $B$ and the wall is $\frac { 1 } { 2 }$. After $B$ rebounds from the wall, there is a further collision between $A$ and $B$.
Show that $e < \frac { 1 } { 4 }$.
(4) END

Edexcel M2 Q2

2. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{173a2029-a0b8-437f-9339-5a1b6f30a8e3-004_378_652_294_630}

\end{figure} A thin uniform wire, of total length 20 cm , is bent to form a frame. The frame is in the shape of a trapezium $A B C D$, where $A B = A D = 4 \mathrm {~cm} , C D = 5 \mathrm {~cm}$, and $A B$ is perpendicular to $B C$ and $A D$, as shown in Figure 1.

Find the distance of the centre of mass of the frame from $A B$. The frame has mass $M$. A particle of mass $k M$ is attached to the frame at $C$. When the frame is freely suspended from the mid-point of $B C$, the frame hangs in equilibrium with $B C$ horizontal.
Find the value of $k$.

Edexcel M2 Q3

3．A particle $P$ moves in a horizontal plane．At time $t$ seconds，the position vector of $P$ is $\mathbf { r }$ metres relative to a fixed origin $O$ ，and $\mathbf { r }$ is given by $$\mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } ,$$ where $c$ is a positive constant．When $t = 1.5$ ，the speed of $P$ is $15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ ．Find
（a）the value of $c$ ，（b）the acceleration of $P$ when $t = 1.5$ ． $\mathbf { r }$ metres relative to a fixed origin $O$ ，and $\mathbf { r }$ is given by $$\begin{aligned} \mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } ,
\text { where } c \text { is a positive constant．When } t = 1.5 \text { ，the speed of } P \text { is } 15 \mathrm {~m} \mathrm {~s} ^ { - 1 } \text { ．Find } \end{aligned}$$ （a）the value of $c$ ， 3．A particle $P$ moves in a horizontal plane．At time $t$ seconds，the position vector of $P$ is D墐
（b）the acceleration of $P$ when $t = 1.5$ ．

Edexcel M2 Q6

6. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{173a2029-a0b8-437f-9339-5a1b6f30a8e3-010_442_689_292_632}

\end{figure} A uniform pole $A B$, of mass 30 kg and length 3 m , is smoothly hinged to a vertical wall at one end $A$. The pole is held in equilibrium in a horizontal position by a light rod CD. One end $C$ of the rod is fixed to the wall vertically below $A$. The other end $D$ is freely jointed to the pole so that $\angle A C D = 30 ^ { \circ }$ and $A D = 0.5 \mathrm {~m}$, as shown in Figure 2. Find

the thrust in the rod $C D$,
the magnitude of the force exerted by the wall on the pole at $A$. The rod $C D$ is removed and replaced by a longer light rod $C M$, where $M$ is the mid-point of $A B$. The rod is freely jointed to the pole at $M$. The pole $A B$ remains in equilibrium in a horizontal position.
Show that the force exerted by the wall on the pole at $A$ now acts horizontally.

Edexcel M2 2016 January Q1

A car of mass 900 kg is travelling up a straight road inclined at an angle $\theta$ to the horizontal, where $\sin \theta = \frac { 1 } { 25 }$. The car is travelling at a constant speed of $14 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the resistance to motion from non-gravitational forces has a constant magnitude of 800 N . The car takes 10 seconds to travel from $A$ to $B$, where $A$ and $B$ are two points on the road.
1. Find the work done by the engine of the car as the car travels from $A$ to $B$.
When the car is at $B$ and travelling at a speed of $14 \mathrm {~ms} ^ { - 1 }$ the rate of working of the engine of the car is suddenly increased to $P \mathrm {~kW}$, resulting in an initial acceleration of the car of $0.7 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. The resistance to motion from non-gravitational forces still has a constant magnitude of 800 N .
Find the value of $P$.

Edexcel M2 2016 January Q2

2. A particle $P$ of mass 0.7 kg is moving in a straight line on a smooth horizontal surface. The particle $P$ collides with a particle $Q$ of mass 1.2 kg which is at rest on the surface. Immediately before the collision the speed of $P$ is $6 \mathrm {~ms} ^ { - 1 }$. Immediately after the collision both particles are moving in the same direction. The coefficient of restitution between the particles is $e$.

Show that $e < \frac { 7 } { 12 }$ Given that $e = \frac { 1 } { 4 }$
find the magnitude of the impulse exerted on $Q$ in the collision.

Edexcel M2 2016 January Q3

3．At time $t$ seconds（ $t \geqslant 0$ ）a particle $P$ has velocity $\mathbf { v } \mathrm { ms } ^ { - 1 }$ ，where When $t = 0$ the particle $P$ is at the origin $O$ ．At time $T$ seconds，$P$ is at the point $A$ and $\mathbf { v } = \lambda ( \mathbf { i } + \mathbf { j } )$ ，where $\lambda$ is a constant． Find
（a）the value of $T$ ，
（b）the acceleration of $P$ as it passes through the point $A$ ，
（c）the distance $O A$ ． $$\mathbf { v } = \left( 6 t ^ { 2 } + 6 t \right) \mathbf { i } + \left( 3 t ^ { 2 } + 24 \right) \mathbf { j }$$ 的 When $t = 0$ the particle $P$ is at the origin $O$ ．At time $T$ seconds，$P$ is at the point $A$ and
$\mathbf { v } = \lambda ( \mathbf { i } + \mathbf { j } )$ ，where $\lambda$ is a constant． Find
（a）the value of $T$ ，
$\_\_\_\_$ ＂

Edexcel M2 2016 January Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e6d100ff-dd4a-4591-a0a3-81761773045e-07_544_1264_251_338} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Two particles $P$ and $Q$, of mass 2 kg and 4 kg respectively, are connected by a light inextensible string. Initially $P$ is held at rest at the point $A$ on a rough fixed plane inclined at $\alpha$ to the horizontal ground, where $\sin \alpha = \frac { 3 } { 5 }$. The string passes over a small smooth pulley fixed at the top of the plane. The particle $Q$ hangs freely below the pulley and 2.5 m above the ground, as shown in Figure 1. The part of the string from $P$ to the pulley lies along a line of greatest slope of the plane. The system is released from rest with the string taut. At the instant when $Q$ hits the ground, $P$ is at the point $B$ on the plane. The coefficient of friction between $P$ and the plane is $\frac { 1 } { 4 }$.

Find the work done against friction as $P$ moves from $A$ to $B$.
Find the total potential energy lost by the system as $P$ moves from $A$ to $B$.
Find, using the work-energy principle, the speed of $P$ as it passes through $B$.

Edexcel M2 2016 January Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e6d100ff-dd4a-4591-a0a3-81761773045e-09_689_581_237_683} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The uniform lamina $A B C D E F$, shown in Figure 2, consists of two identical rectangles with sides of length $a$ and $3 a$. The mass of the lamina is $M$. A particle of mass $k M$ is attached to the lamina at $E$. The lamina, with the attached particle, is freely suspended from $A$ and hangs in equilibrium with $A F$ at an angle $\theta$ to the downward vertical. Given that $\tan \theta = \frac { 4 } { 7 }$, find the value of $k$.
(10)

Edexcel M2 2016 January Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e6d100ff-dd4a-4591-a0a3-81761773045e-11_757_1269_233_331} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A uniform rod $A B$, of mass $3 m$ and length $2 a$, is freely hinged at $A$ to a fixed point on horizontal ground. A particle of mass $m$ is attached to the rod at the end $B$. The system is held in equilibrium by a force $\mathbf { F }$ acting at the point $C$, where $A C = b$. The rod makes an acute angle $\theta$ with the ground, as shown in Figure 3. The line of action of $\mathbf { F }$ is perpendicular to the rod and in the same vertical plane as the rod.

Show that the magnitude of $\mathbf { F }$ is $\frac { 5 m g a } { b } \cos \theta$ The force exerted on the rod by the hinge at $A$ is $\mathbf { R }$, which acts upwards at an angle $\phi$ above the horizontal, where $\phi > \theta$.
Find
1. the component of $\mathbf { R }$ parallel to the rod, in terms of $m , g$ and $\theta$,
2. the component of $\mathbf { R }$ perpendicular to the rod, in terms of $a , b , m , g$ and $\theta$.
Hence, or otherwise, find the range of possible values of $b$, giving your answer in terms of $a$.

Edexcel M2 2016 January Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e6d100ff-dd4a-4591-a0a3-81761773045e-13_552_1296_255_317} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} At time $t = 0$, a particle $P$ of mass 0.7 kg is projected with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from a fixed point $O$ at an angle $\theta ^ { \circ }$ to the horizontal. The particle moves freely under gravity. At time $t = 2$ seconds, $P$ passes through the point $A$ with speed $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and is moving downwards at $45 ^ { \circ }$ to the horizontal, as shown in Figure 4. Find

the value of $\theta$,
the kinetic energy of $P$ as it reaches the highest point of its path. For an interval of $T$ seconds, the speed, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, of $P$ is such that $v \leqslant 6$
Find the value of $T$.

Questions — Edexcel M2 (551 questions)

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Edexcel M2 Specimen Q1

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Edexcel M2 2016 January Q1

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Edexcel M2 2016 January Q5

Edexcel M2 2016 January Q6

Edexcel M2 2016 January Q7