Questions - Edexcel M2 - A-Level Maths

Find, in terms of $W$, the magnitude of the reaction on the rod at its point of contact with the cylinder.
Show that the resultant force acting on the rod at $A$ is inclined to the vertical at an angle $\alpha$ where $\tan \alpha = \frac { 40 } { 73 }$
5 continued
\includegraphics[max width=\textwidth, alt={}, center]{54112b4a-3727-4e5b-97e5-4291e7172438-17_81_72_2631_1873}

Edexcel M2 2018 January Q6

6. A car of mass 800 kg pulls a trailer of mass 300 kg up a straight road which is inclined to the horizontal at an angle $\alpha$, where $\sin \alpha = \frac { 1 } { 14 }$. The trailer is attached to the car by a light inextensible towbar which is parallel to the direction of motion of the car. The car's engine works at a constant rate of $P \mathrm {~kW}$. The non-gravitational resistances to motion are constant and of magnitude 600 N on the car and 200 N on the trailer. At a given instant the car is moving at $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and is accelerating at $0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$

Find the value of $P$. When the car is moving up the road at $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the towbar breaks. The trailer comes to instantaneous rest after moving a distance $d$ metres up the road from the point where the towbar broke. The non-gravitational resistance to the motion of the trailer remains constant and of magnitude 200 N .
Find, using the work-energy principle, the value of $d$.

Edexcel M2 2018 January Q7

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

Show that $$y = x \tan \alpha - \frac { g x ^ { 2 } \left( 1 + \tan ^ { 2 } \alpha \right) } { 2 U ^ { 2 } }$$ \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{54112b4a-3727-4e5b-97e5-4291e7172438-22_330_857_632_548} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A small stone is projected horizontally with speed $U$ from a point $C$ at the top of a vertical cliff $A C$ so as to hit a fixed target $B$ on the horizontal ground. The point $C$ is a height $h$ above the ground, as shown in Figure 3. The time of flight of the stone from $C$ to $B$ is $T$, and the stone is modelled as a particle moving freely under gravity.
Find, in terms of $U , g$ and $T$, the speed of the stone as it hits the target at $B$. It is found that, using the same initial speed $U$, the target can also be hit by projecting the stone from $C$ at an angle $\alpha$ above the horizontal. The stone is again modelled as a particle moving freely under gravity and the distance $A B = d$.
Using the result in part (a), or otherwise, show that $$d = \frac { 1 } { 2 } g T ^ { 2 } \tan \alpha$$

Edexcel M2 2019 January Q1

Three particles of masses $3 m , m$ and $2 m$ are positioned at the points with coordinates $( a , 8 ) , ( - 4,0 )$ and $( 5 , - 2 )$ respectively.

Given that the centre of mass of the three particles is at the point with coordinates $( k , 2 k )$, where $k$ is a constant, find the value of $a$.
(5)

Edexcel M2 2019 January Q2

A particle of mass 0.75 kg is moving with velocity ( $4 \mathbf { i } + \mathbf { j }$ ) $\mathrm { m } \mathrm { s } ^ { - 1 }$ when it receives an impulse ( $- 6 \mathbf { i } + 4 \mathbf { j }$ ) N s. impulse $( - 6 \mathbf { i } + 4 \mathbf { j } )$ N s.

\section*{Find
Find} $$\begin{aligned} & \text { (a) the velocity of the particle immediately after receiving the impulse, }
& \text { (b) the size of the angle through which the path of the particle is deflected as a result of }
& \text { the impulse. } \end{aligned}$$ (3)

Edexcel M2 2019 January Q3

A car of mass 900 kg is moving on a straight road that is inclined at an angle $\theta$ to the horizontal, where $\sin \theta = \frac { 1 } { 49 }$. When the car is moving up the road, with the engine of the car working at a constant rate of 10.8 kW , the car has a constant speed of $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude $R$ newtons.

When the car is moving down the road, with the engine of the car working at a constant rate of 10.8 kW , the car has a constant speed of $2 v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The resistance to the motion of the car is still modelled as a constant force of magnitude $R$ newtons. Find

the value of $R$,
the value of $v$.

Edexcel M2 2019 January Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b4065fe1-55fa-4a01-8ae2-006e0d529c50-10_787_814_246_566} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The uniform lamina $L$, shown shaded in Figure 1, is formed by removing a circular disc of radius $2 a$ from a uniform circular disc of radius $4 a$. The larger disc has centre $O$ and diameter $A B$. The radius $O D$ is perpendicular to $A B$. The smaller disc has centre $C$, where $C$ is on $A B$ and $B C = 3 a$

Show that the centre of mass of $L$ is $\frac { 13 } { 3 } a$ from $B$. The mass of $L$ is $M$ and a particle of mass $k M$ is attached to $L$ at $B$. When $L$, with the particle attached, is freely suspended from point $D$, it hangs in equilibrium with $A$ higher than $B$ and $A B$ at an angle $\theta$ to the horizontal, where $\tan \theta = \frac { 3 } { 4 }$
Find the value of $k$.

Edexcel M2 2019 January Q5

5. A particle moves along the $x$-axis. At time $t$ seconds, $t \geqslant 0$, the velocity of the particle is $v \mathrm {~ms} ^ { - 1 }$ in the direction of $x$ increasing, where $v = 2 t ^ { \frac { 3 } { 2 } } - 6 t + 2$ At time $t = 0$ the particle passes through the origin $O$. At the instant when the acceleration of the particle is zero, the particle is at the point $A$. Find the distance $O A$.
(8)

Edexcel M2 2019 January Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b4065fe1-55fa-4a01-8ae2-006e0d529c50-16_449_974_237_445} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A plank $A B$ rests in equilibrium against a fixed horizontal pole. The plank has length 4 m and weight 20 N and rests on the pole at $C$, where $A C = 2.5 \mathrm {~m}$. The end $A$ of the plank rests on rough horizontal ground and $A B$ makes an angle $\theta$ with the ground, as shown in
Figure 2. The coefficient of friction between the plank and the ground is $\frac { 1 } { 4 }$.
The plank is modelled as a uniform rod and the pole as a rough horizontal peg that is perpendicular to the vertical plane containing $A B$. Given that $\cos \theta = \frac { 4 } { 5 }$ and that the friction is limiting at both $A$ and $C$,

find the magnitude of the normal reaction on the plank at $C$,
find the coefficient of friction between the plank and the pole.

Edexcel M2 2019 January Q7

7. A particle $P$ of mass $3 m$ is moving in a straight line with speed $u$ on a smooth horizontal table. A second particle $Q$ of mass $2 m$ is moving with speed $2 u$ in the opposite direction to $P$ along the same straight line. Particle $P$ collides directly with $Q$. The coefficient of restitution between $P$ and $Q$ is $e$.

Show that the direction of motion of $P$ is reversed as a result of the collision with $Q$.
Find the range of values of $e$ for which the direction of motion of $Q$ is also reversed as a result of the collision. Given that $e = \frac { 1 } { 2 }$
find, in terms of $m$ and $u$, the kinetic energy lost in the collision between $P$ and $Q$.

Edexcel M2 2019 January Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b4065fe1-55fa-4a01-8ae2-006e0d529c50-24_286_1317_251_317} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A rough ramp $A B$ is fixed to horizontal ground at $A$. The ramp is inclined at $20 ^ { \circ }$ to the ground. The line $A B$ is a line of greatest slope of the ramp and $A B = 6 \mathrm {~m}$. The point $B$ is at the top of the ramp, as shown in Figure 3. A particle $P$ of mass 3 kg is projected with speed $15 \mathrm {~ms} ^ { - 1 }$ from $A$ towards $B$. At the instant $P$ reaches the point $B$ the speed of $P$ is $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The force due to friction is modelled as a constant force of magnitude $F$ newtons.

Use the work-energy principle to find the value of $F$. After leaving the ramp at $B$, the particle $P$ moves freely under gravity until it hits the horizontal ground at the point $C$. The speed of $P$ as it hits the ground at $C$ is $w \mathrm {~ms} ^ { - 1 }$. Find
1. the value of $w$,
2. the direction of motion of $P$ as it hits the ground at $C$,
the greatest height of $P$ above the ground as $P$ moves from $A$ to $C$.

Edexcel M2 2020 January Q1

A cyclist and his bicycle have a total mass of 75 kg . The cyclist is moving down a straight road that is inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = \frac { 1 } { 15 }$

The cyclist is working at a constant rate of 56 W . The magnitude of the resistance to motion is modelled as a constant force of magnitude 40 N . At the instant when the speed of the cyclist is $\mathrm { Vm } \mathrm { s } ^ { - 1 }$, his acceleration is $\frac { 1 } { 3 } \mathrm {~ms} ^ { - 2 }$ Find the value of $V$.
(5)

Edexcel M2 2020 January Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c16c17b6-2c24-4939-b3b5-63cd63646b76-04_239_796_246_577} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A rough straight ramp is fixed to horizontal ground. The ramp is inclined at an angle $\theta$ to the horizontal, where $\sin \theta = \frac { 1 } { 7 }$. The points $A$ and $B$ are on a line of greatest slope of the ramp with $A B = 2.5 \mathrm {~m}$ and $B$ above $A$, as shown in Figure 1. A package of mass 2 kg is projected up the ramp from $A$ with speed $4 \mathrm {~ms} ^ { - 1 }$ and first comes to instantaneous rest at $B$. The coefficient of friction between the package and the ramp is $\mu$. The package is modelled as a particle. Use the work-energy principle to find the value of $\mu$.
(6)

Edexcel M2 2020 January Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c16c17b6-2c24-4939-b3b5-63cd63646b76-06_291_481_255_733} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A particle $P$ of mass 0.75 kg is moving along a straight line on a horizontal surface. At the instant when the speed of $P$ is $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, it receives an impulse of magnitude $\sqrt { 24 } \mathrm { Ns }$. The impulse acts in the plane of the horizontal surface. At the instant when $P$ receives the impulse, the line of action of the impulse makes an angle of $60 ^ { \circ }$ with the direction of motion of $P$, as shown in Figure 2. Find

the speed of $P$ immediately after receiving the impulse,
the size of the angle between the direction of motion of $P$ immediately before receiving the impulse and the direction of motion of $P$ immediately after receiving the impulse.
\includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-06_2252_51_311_1980}
\includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-07_36_65_2722_109}

Edexcel M2 2020 January Q4

4. [The centre of mass of a uniform semicircular lamina of radius $r$ is $\frac { 4 r } { 3 \pi }$ from the centre.] \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c16c17b6-2c24-4939-b3b5-63cd63646b76-08_437_563_347_701} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} The uniform rectangular lamina $A B C D E F$ has sides $A C = F D = 6 a$ and $A F = C D = 3 a$. The point $B$ lies on $A C$ with $A B = 2 a$ and the point $E$ lies on $F D$ with $F E = 2 a$. The template, $T$, shown shaded in Figure 3, is formed by removing the semicircular lamina with diameter $B C$ from the rectangular lamina and then fixing this semicircular lamina to the opposite side, $F D$, of the rectangular lamina. The diameter of the semicircular lamina coincides with $E D$ and the semicircular arc $E D$ is outside the rectangle $A B C D E F$. All points of $T$ lie in the same plane.

Show that the centre of mass of $T$ is a distance $\left( \frac { 9 + 2 \pi } { 6 } \right)$ a from $A C$. The mass of $T$ is $M$. A particle of mass $k M$ is attached to $T$ at $C$. The loaded template is freely suspended from $A$ and hangs in equilibrium with $A F$ at angle $\phi$ to the downward vertical through $A$. Given that $\tan \phi = \frac { 3 } { 2 }$
find the value of $k$.
\section*{\textbackslash section*\{Question 4 continued\}} \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-11_149_142_2604_1816}

Edexcel M2 2020 January Q5

5. A t time $t$ seconds ( $t \geqslant 0$ ), a particle $P$ has velocity $\mathbf { v m ~ s } ^ { - 1 }$, where $$\mathbf { v } = \left( 3 t ^ { 2 } - 4 \right) \mathbf { i } + ( 2 t - 4 ) \mathbf { j }$$ When $t = 0 , P$ is at the fixed point $O$.

Find the acceleration of $P$ at the instant when $t = 0$
Find the exact speed of $P$ at the instant when $P$ is moving in the direction of the vector $( 11 \mathbf { i } + \mathbf { j } )$ for the second time.
Show that $P$ never returns to $O$.
\includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-14_2658_1938_107_123} \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-15_149_140_2604_1818}

Edexcel M2 2020 January Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c16c17b6-2c24-4939-b3b5-63cd63646b76-16_358_967_248_484} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} A uniform rod, $A B$, of weight $W$ and length $8 a$, rests in equilibrium with the end $A$ on rough horizontal ground. The rod rests on a smooth cylinder. The cylinder is fixed to the ground with its axis horizontal. The point of contact between the rod and the cylinder is $C$, where $A C = 7 a$, as shown in Figure 4. The rod is resting in a vertical plane that is perpendicular to the axis of the cylinder. The rod makes an angle $\alpha$ with the horizontal .

Show that the normal reaction of the ground on the rod at $A$ has $$\text { magnitude } W \left( 1 - \frac { 4 } { 7 } \cos ^ { 2 } \alpha \right)$$ Given that the coefficient of friction between the rod and the ground is $\mu$ and that $\cos \alpha = \frac { 3 } { \sqrt { 10 } }$
find the range of possible values of $\mu$.
\section*{\textbackslash section*\{Question 6 continued\}} \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-19_147_142_2606_1816}

Edexcel M2 2020 January Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c16c17b6-2c24-4939-b3b5-63cd63646b76-20_360_1026_246_466} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} At time $t = 0$ a particle $P$ is projected from a fixed point $A$ on horizontal ground. The particle is projected with speed $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle $\alpha$ to the ground. The particle moves freely under gravity. At time $t = 3$ seconds, $P$ is passing through the point $B$ with speed $15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and is moving downwards at an angle $\beta$ to the horizontal, as shown in Figure 5.

By considering energy, find the height of $B$ above the ground.
Find the size of angle $\alpha$.
Find the size of angle $\beta$.
Find the least speed of $P$ as $P$ travels from $A$ to $B$. As $P$ travels from $A$ to $B$, the speed, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, of $P$ is such that $v \leqslant 15$ for an interval of $T$ seconds.
Find the value of $T$.
\section*{\textbackslash section*\{Question 7 continued\}}

Edexcel M2 2020 January Q8

A particle $A$ has mass $4 m$ and a particle $B$ has mass $3 m$. The particles are moving along the same straight line on a smooth horizontal plane. They are moving in opposite directions towards each other and collide directly.

Immediately before the collision the speed of $A$ is $2 u$ and the speed of $B$ is $3 u$.
The direction of motion of each particle is reversed by the collision.
The total kinetic energy lost in the collision is $\frac { 473 } { 24 } m u ^ { 2 }$
Find

the coefficient of restitution between $A$ and $B$,
the magnitude of the impulse received by $A$ in the collision.
\section*{\textbackslash section*\{Question 8 continued\}} \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-28_2642_1833_118_118}

Edexcel M2 2021 January Q1

A particle $P$ of mass 1.5 kg is moving with velocity $( 4 \mathbf { i } + 6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ when it receives an impulse of magnitude 15Ns. Immediately after $P$ receives the impulse, the velocity of $P$ is $\boldsymbol { v } \mathrm { m } \mathrm { s } ^ { - 1 }$.

Find the two possible values of $v$.

Edexcel M2 2021 January Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3eb71ecb-fa88-4cca-a2b6-bcf11f1d689b-04_760_669_118_641} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The uniform lamina $A B C$ has sides $A B = A C = 13 a$ and $B C = 10 a$. The lamina is freely suspended from $A$. A horizontal force of magnitude $F$ is applied to the lamina at $B$, as shown in Figure 1. The line of action of the force lies in the vertical plane containing the lamina. The lamina is in equilibrium with $A B$ vertical. The weight of the lamina is $W$. Find $F$ in terms of $W$.
VILM SIHI NI JAIUM ION OC
VANV SIHI NI I III M LON OO
VI4V SIHI NI JAIUM ION OC

Edexcel M2 2021 January Q3

3. A car of mass 600 kg travels along a straight horizontal road with the engine of the car working at a constant rate of $P$ watts. The resistance to the motion of the car is modelled as a constant force of magnitude $R$ newtons. At the instant when the speed of the car is $15 \mathrm {~ms} ^ { - 1 }$, the magnitude of the acceleration of the car is $0.2 \mathrm {~ms} ^ { - 2 }$. Later the same car travels up a straight road inclined at angle $\theta$ to the horizontal, where $\sin \theta = \frac { 1 } { 20 }$. The resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude $R$ newtons. When the engine of the car is working at a constant rate of $P$ watts, the car has a constant speed of $10 \mathrm {~ms} ^ { - 1 }$. Find the value of $P$.

Edexcel M2 2021 January Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3eb71ecb-fa88-4cca-a2b6-bcf11f1d689b-10_517_371_260_790} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The number "4", shown in Figure 2, is a rigid framework made from three uniform rods, $A B , B C$ and $C D$, where $$A B = 6 a , B C = 5 a \text { and } C D = 4 a$$ The point $E$ is on $A B$ and $C D$, where $B E = 4 a , C E = 3 a$ and angle $C E B = 90 ^ { \circ }$ The three rods are all made from the same material and they all lie in the same plane. The framework is suspended from $B$ and hangs in equilibrium with $B A$ at an angle $\theta$ to the downward vertical. Find $\theta$ to the nearest degree.
VILU SIHI NI JAIUM ION OC
VIUV SIHI NI JAHM ION OC
VIIV SIHI NI EIIIM ION OC

VIIV SIHI NI III HM ION OC

VIUV SIHI NI JIHM I ON OO

VI4V SIHI NI JIIIM ION OO

Edexcel M2 2021 January Q5

5. At time $t$ seconds, $t \geqslant 0$, a particle $P$ has velocity $\mathbf { v } \mathrm { ms } ^ { - 1 }$, where $$\mathbf { v } = \left( 5 t ^ { 2 } - 12 t + 15 \right) \mathbf { i } + \left( t ^ { 2 } + 8 t - 10 \right) \mathbf { j }$$ When $t = 0 , P$ is at the origin $O$.
At time $T$ seconds, $P$ is moving in the direction of $( \mathbf { i } + \mathbf { j } )$.

Find the value of $T$. When $t = 3 , P$ is at the point $A$.
Find the magnitude of the acceleration of $P$ as it passes through $A$.
Find the position vector of $A$.

Edexcel M2 2021 January Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3eb71ecb-fa88-4cca-a2b6-bcf11f1d689b-16_639_561_246_689} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A ladder $A B$ has length 6 m and mass 30 kg . The ladder rests in equilibrium at $60 ^ { \circ }$ to the horizontal with the end $A$ on rough horizontal ground and the end $B$ against a smooth vertical wall, as shown in Figure 3. A man of mass 70 kg stands on the ladder at the point $C$, where $A C = 2 \mathrm {~m}$, and the ladder remains in equilibrium. The ladder is modelled as a uniform rod in a vertical plane perpendicular to the wall. The man is modelled as a particle.

Find the magnitude of the force exerted on the ladder by the ground. The man climbs further up the ladder. When he is at the point $D$ on the ladder, the ladder is about to slip. Given that the coefficient of friction between the ladder and the ground is 0.4
find the distance $A D$.
State how you have used the modelling assumption that the ladder is a rod.

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