Questions - Edexcel - A-Level Maths

\hspace{0pt} [In this question, $\mathbf { i }$ and $\mathbf { j }$ are horizontal perpendicular unit vectors.]

A particle $A$ has mass 3 kg and a particle $B$ has mass 2 kg .
The particles are moving on a smooth horizontal plane when they collide directly.
Immediately before the collision, the velocity of $A$ is $( 3 \mathbf { i } - \mathbf { j } ) \mathrm { ms } ^ { - 1 }$ and the velocity of $B$ is $( - 6 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$ Immediately after the collision the velocity of $A$ is $\left( - 2 \mathbf { i } + \frac { 2 } { 3 } \mathbf { j } \right) \mathrm { ms } ^ { - 1 }$

Find the total kinetic energy of the two particles before the collision.
Find, in terms of $\mathbf { i }$ and $\mathbf { j }$, the impulse exerted on $A$ by $B$ in the collision.
Find, in terms of $\mathbf { i }$ and $\mathbf { j }$, the velocity of $B$ immediately after the collision.

Edexcel FM1 2024 June Q2

7 marks Standard +0.3

A rough plane is inclined to the horizontal at an angle $\theta$, where $\tan \theta = \frac { 3 } { 4 }$

A particle $P$ of mass $m$ is at rest at a point on the plane. The particle is projected up the plane with speed $\sqrt { 2 a g }$
The particle moves up a line of greatest slope of the plane and comes to instantaneous rest after moving a distance $d$. The coefficient of friction between $P$ and the plane is $\frac { 1 } { 7 }$

Show that the magnitude of the frictional force acting on $P$ as it moves up the plane is $\frac { 4 m g } { 35 }$ Air resistance is assumed to be negligible.
Using the work-energy principle,
find $d$ in terms of $a$.

Edexcel FM1 2024 June Q3

12 marks Standard +0.3

A car of mass 1000 kg moves in a straight line along a horizontal road at a constant speed of $72 \mathrm {~km} \mathrm {~h} ^ { - 1 }$

The resistance to the motion of the car is modelled as a constant force of magnitude 900 N

The engine of the car is working at a constant rate of $P \mathrm {~kW}$.
Using the model,

find the value of $P$. The car now travels in a straight line up a road which is inclined to the horizontal at an angle $\alpha$, where $\sin \alpha = \frac { 2 } { 49 }$
- In a refined model, the resistance to the motion of the car from non-gravitational forces is now modelled as a force of magnitude $20 v$ newtons, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the speed of the car
At the instant when the engine of the car is working at a constant rate of 30 kW and the car is moving up the road at $10 \mathrm {~ms} ^ { - 1 }$, the acceleration of the car is $a \mathrm {~ms} ^ { - 2 }$ Using the refined model,
find the value of $a$. Later on, when the engine of the car is again working at a constant rate of 30 kW , the car is moving up the road at a constant speed $U \mathrm {~m} \mathrm {~s} ^ { - 1 }$ Using the refined model,
find the value of $U$.

Edexcel FM1 2024 June Q4

15 marks Standard +0.3

A particle $A$ of mass $2 m$ is moving in a straight line with speed $3 u$ on a smooth horizontal plane. Particle $A$ collides directly with a particle $B$ of mass $m$ which is at rest on the plane.

The coefficient of restitution between $A$ and $B$ is $e$, where $e > 0$

Show that the speed of $B$ immediately after the collision is $2 u ( 1 + e )$. After the collision, $B$ hits a smooth fixed vertical wall which is perpendicular to the direction of motion of $B$.
Show that there will be a second collision between $A$ and $B$. The coefficient of restitution between $B$ and the wall is $\frac { 1 } { 2 }$
Find, in simplified form, in terms of $m$, $u$ and $e$,
the magnitude of the impulse received by $B$ in its collision with the wall,
the loss in kinetic energy of $B$ due to its collision with the wall.

Edexcel FM1 2024 June Q5

7 marks Standard +0.8

A light elastic string has natural length $2 a$ and modulus of elasticity $2 m g$. One end of the string is attached to a fixed point $A$ on a horizontal ceiling. The other end is attached to a particle $P$ of mass $m$.

The particle $P$ hangs in equilibrium at the point $E$, where $A E = 3 a$.
The particle $P$ is then projected vertically downwards from $E$ with speed $\frac { 3 } { 2 } \sqrt { a g }$
Air resistance is assumed to be negligible.
Find the elastic energy stored in the string, when $P$ first comes to instantaneous rest. Give your answer in the form kmga, where $k$ is a constant to be found.

Edexcel FM1 2024 June Q6

10 marks Standard +0.3

\hspace{0pt} [In this question, $\mathbf { i }$ and $\mathbf { j }$ are horizontal perpendicular unit vectors.]

A particle $P$ is moving with velocity ( $4 \mathbf { i } - \mathbf { j }$ ) $\mathrm { m } \mathrm { s } ^ { - 1 }$ on a smooth horizontal plane. The particle collides with a smooth vertical wall and rebounds with velocity $( \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$ The coefficient of restitution between $P$ and the wall is $e$.

Find the value of $e$. After the collision, $P$ goes on to hit a second smooth vertical wall, which is parallel to $\mathbf { i }$.
The coefficient of restitution between $P$ and this second wall is $\frac { 1 } { 3 }$
The angle through which the direction of motion of $P$ has been deflected by its collision with this second wall is $\alpha ^ { \circ }$.
Find the value of $\alpha$, giving your answer to the nearest whole number.

Edexcel FM1 2024 June Q7

15 marks Challenging +1.8

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{58a33c19-77c6-4b08-ac09-ce6aa1e641df-20_501_703_251_680} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A smooth uniform sphere $A$ of mass $m$ is moving with speed $U$ on a smooth horizontal plane. The sphere $A$ collides obliquely with a smooth uniform sphere $B$ of mass $3 m$ which is at rest on the plane. The two spheres have the same radius. Immediately before the collision, the direction of motion of $A$ makes an angle $\alpha$, where $0 ^ { \circ } < \alpha < 90 ^ { \circ }$, with the line joining the centres of the spheres. Immediately after the collision, the direction of motion of $A$ is perpendicular to its original direction, as shown in Figure 1. The coefficient of restitution between the spheres is $e$.

Show that the speed of $B$ immediately after the collision is $$\frac { 1 } { 4 } ( 1 + e ) U \cos \alpha$$
Show that $e > \frac { 1 } { 3 }$
Show that $0 < \tan \alpha \leqslant \frac { 1 } { \sqrt { 2 } }$

Edexcel FM1 Specimen Q1

6 marks Standard +0.3

A particle $P$ of mass 0.5 kg is moving with velocity $( 4 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ when it receives an impulse $( 2 \mathbf { i } - \mathbf { j } )$ Ns.

Show that the kinetic energy gained by $P$ as a result of the impulse is 12 J .

Edexcel FM1 Specimen Q2

6 marks Standard +0.3

A parcel of mass 5 kg is projected with speed $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ up a line of greatest slope of a fixed rough inclined ramp.
The ramp is inclined at angle $\alpha$ to the horizontal, where $\sin \alpha = \frac { 1 } { 7 }$
The parcel is projected from the point $A$ on the ramp and comes to instantaneous rest at the point $B$ on the ramp, where $A B = 14 \mathrm {~m}$.

The coefficient of friction between the parcel and the ramp is $\mu$.
In a model of the parcel's motion, the parcel is treated as a particle.

Use the work-energy principle to find the value of $\mu$.
Suggest one way in which the model could be refined to make it more realistic.

Edexcel FM1 Specimen Q3

8 marks Standard +0.3

A particle of mass $m \mathrm {~kg}$ lies on a smooth horizontal surface.

Initially the particle is at rest at a point $O$ between two fixed parallel vertical walls.
The point $O$ is equidistant from the two walls and the walls are 4 m apart.
At time $t = 0$ the particle is projected from $O$ with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a direction perpendicular to the walls.
The coefficient of restitution between the particle and each wall is $\frac { 3 } { 4 }$
The magnitude of the impulse on the particle due to the first impact with a wall is $\lambda m u$ Ns.

Find the value of $\lambda$. The particle returns to $O$, having bounced off each wall once, at time $t = 7$ seconds.
Find the value of $u$.

Edexcel FM1 Specimen Q4

9 marks Standard +0.8

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{681833ac-b266-4ac8-881e-46ede398ce58-08_513_807_303_630} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 represents the plan view of part of a horizontal floor, where $A B$ and $B C$ are perpendicular vertical walls. The floor and the walls are modelled as smooth.
A ball is projected along the floor towards $A B$ with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on a path at an angle of $60 ^ { \circ }$ to $A B$. The ball hits $A B$ and then hits $B C$. The ball is modelled as a particle.
The coefficient of restitution between the ball and wall $A B$ is $\frac { 1 } { \sqrt { 3 } }$
The coefficient of restitution between the ball and wall $B C$ is $\sqrt { \frac { 2 } { 5 } }$

Show that, using this model, the final kinetic energy of the ball is $35 \%$ of the initial kinetic energy of the ball.
In reality the floor and the walls may not be smooth. What effect will the model have had on the calculation of the percentage of kinetic energy remaining?

Edexcel FM1 Specimen Q5

9 marks Standard +0.3

A car of mass 600 kg is moving along a straight horizontal road.

At the instant when the speed of the car is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the resistance to the motion of the car is modelled as a force of magnitude $( 200 + 2 v ) \mathrm { N }$. The engine of the car is working at a constant rate of 12 kW .

Find the acceleration of the car at the instant when $v = 20$ Later on the car is moving up a straight road inclined at an angle $\theta$ to the horizontal, where $\sin \theta = \frac { 1 } { 14 }$ At the instant when the speed of the car is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the resistance to the motion of the car from non-gravitational forces is modelled as a force of magnitude ( $200 + 2 v ) \mathrm { N }$. The engine is again working at a constant rate of 12 kW .
At the instant when the car has speed $w \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the car is decelerating at $0.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
Find the value of $w$.

Edexcel FM1 Specimen Q6

9 marks Standard +0.8

\hspace{0pt} [In this question $\mathbf { i }$ and $\mathbf { j }$ are perpendicular unit vectors in a horizontal plane.]

A smooth uniform sphere $A$ has mass $2 m \mathrm {~kg}$ and another smooth uniform sphere $B$, with the same radius as $A$, has mass $3 m \mathrm {~kg}$. The spheres are moving on a smooth horizontal plane when they collide obliquely.
Immediately before the collision the velocity of $A$ is $( 3 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and the velocity of $B$ is $( - 5 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. At the instant of collision, the line joining the centres of the spheres is parallel to $\mathbf { i }$.
The coefficient of restitution between the spheres is $\frac { 1 } { 4 }$

Find the velocity of $B$ immediately after the collision.
Find, to the nearest degree, the size of the angle through which the direction of motion of $B$ is deflected as a result of the collision.

Edexcel FM1 Specimen Q7

14 marks Standard +0.8

A particle $P$ of mass $m$ is attached to one end of a light elastic string of natural length $a$ and modulus of elasticity $3 m g$.

The other end of the string is attached to a fixed point $O$ on a ceiling.
The particle hangs freely in equilibrium at a distance $d$ vertically below $O$.

Show that $d = \frac { 4 } { 3 } a$. The point $A$ is vertically below $O$ such that $O A = 2 a$.
The particle is held at rest at $A$, then released and first comes to instantaneous rest at the point $B$.
Find, in terms of $g$, the acceleration of $P$ immediately after it is released from rest.
Find, in terms of $g$ and $a$, the maximum speed attained by $P$ as it moves from $A$ to $B$.
Find, in terms of $a$, the distance $O B$.

Edexcel FM1 Specimen Q8

14 marks Standard +0.8

A particle $P$ of mass $2 m$ and a particle $Q$ of mass $5 m$ 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 $P$ is $2 u$ and the speed of $Q$ is $u$.
The direction of motion of $Q$ is reversed by the collision.
The coefficient of restitution between $P$ and $Q$ is $e$.

Find the range of possible values of $e$. Given that $e = \frac { 1 } { 3 }$
show that the kinetic energy lost in the collision is $\frac { 40 m u ^ { 2 } } { 7 }$.
Without doing any further calculation, state how the amount of kinetic energy lost in the collision would change if $e > \frac { 1 } { 3 }$

Edexcel FM2 2019 June Q1

6 marks Standard +0.3

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9b995178-a4be-4d5a-95f8-6c2978ff01b3-02_330_662_349_753} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A hemispherical shell of radius $a$ is fixed with its rim uppermost and horizontal. A small bead, $B$, is moving with constant angular speed, $\omega$, in a horizontal circle on the smooth inner surface of the shell. The centre of the path of $B$ is at a distance $\frac { 1 } { 4 } a$ vertically below the level of the rim of the hemisphere, as shown in Figure 1. Find the magnitude of $\omega$, giving your answer in terms of $a$ and $g$.

Edexcel FM2 2019 June Q2

10 marks Challenging +1.2

A particle, $P$, of mass 0.4 kg is moving along the positive $x$-axis, in the positive $x$ direction under the action of a single force. At time $t$ seconds, $t > 0 , P$ is $x$ metres from the origin $O$ and the speed of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The force is acting in the direction of $x$ increasing and has magnitude $\frac { k } { v }$ newtons, where $k$ is a constant.

At $x = 3 , v = 2$ and at $x = 6 , v = 2.5$

Show that $v ^ { 3 } = \frac { 61 x + 9 } { 24 }$ The time taken for the speed of $P$ to increase from $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ to $2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is $T$ seconds.
Use algebraic integration to show that $T = \frac { 81 } { 61 }$

Edexcel FM2 2019 June Q3

11 marks Challenging +1.2

Numerical (calculator) integration is not acceptable in this question.

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9b995178-a4be-4d5a-95f8-6c2978ff01b3-08_547_550_303_753} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The shaded region $O A B$ in Figure 2 is bounded by the $x$-axis, the line with equation $x = 4$ and the curve with equation $y = \frac { 1 } { 4 } ( x - 2 ) ^ { 3 } + 2$. The point $A$ has coordinates (4, 4) and the point $B$ has coordinates $( 4,0 )$. A uniform lamina $L$ has the shape of $O A B$. The unit of length on both axes is one centimetre. The centre of mass of $L$ is at the point with coordinates $( \bar { x } , \bar { y } )$. Given that the area of $L$ is $8 \mathrm {~cm} ^ { 2 }$,

show that $\bar { y } = \frac { 8 } { 7 }$ The lamina is freely suspended from $A$ and hangs in equilibrium with $A B$ at an angle $\theta ^ { \circ }$ to the downward vertical.
Find the value of $\theta$.

Edexcel FM2 2019 June Q4

12 marks Standard +0.8

A flagpole, $A B$, is 4 m long. The flagpole is modelled as a non-uniform rod so that, at a distance $x$ metres from $A$, the mass per unit length of the flagpole, $m \mathrm {~kg} \mathrm {~m} ^ { - 1 }$, is given by $m = 18 - 3 x$.
1. Show that the mass of the flagpole is 48 kg .
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9b995178-a4be-4d5a-95f8-6c2978ff01b3-12_515_439_502_806} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The end $A$ of the flagpole is fixed to a point on a vertical wall. A cable has one end attached to the midpoint of the flagpole and the other end attached to a point on the wall that is vertically above $A$. The cable is perpendicular to the flagpole. The flagpole and the cable lie in the same vertical plane that is perpendicular to the wall. A small ball of mass 4 kg is attached to the flagpole at $B$. The cable holds the flagpole and ball in equilibrium, with the flagpole at $45 ^ { \circ }$ to the wall, as shown in Figure 3. The tension in the cable is $T$ newtons.
The cable is modelled as a light inextensible string and the ball is modelled as a particle.
Using the model, find the value of $T$.
Give a reason why the answer to part (b) is not likely to be the true value of $T$.

Edexcel FM2 2019 June Q5

11 marks Standard +0.8

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9b995178-a4be-4d5a-95f8-6c2978ff01b3-16_560_560_283_749} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} The region $R$, shown shaded in Figure 4, is bounded by part of the curve with equation $y ^ { 2 } = 2 x$, the line with equation $y = 2$ and the $y$-axis. The unit of length on both axes is one centimetre. A uniform solid, $S$, is formed by rotating $R$ through $360 ^ { \circ }$ about the $y$-axis.
Given that the volume of $S$ is $\frac { 8 } { 5 } \pi \mathrm {~cm} ^ { 3 }$,

show that the centre of mass of $S$ is $\frac { 1 } { 3 } \mathrm {~cm}$ from its plane face. A uniform solid cylinder, $C$, has base radius 2 cm and height 4 cm . The cylinder $C$ is attached to $S$ so that the plane face of $S$ coincides with a plane face of $C$, to form the paperweight $P$, shown in Figure 5. The density of the material used to make $S$ is three times the density of the material used to make $C$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9b995178-a4be-4d5a-95f8-6c2978ff01b3-16_572_456_1617_758} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} The plane face of $P$ rests in equilibrium on a desk lid that is inclined at an angle $\theta ^ { \circ }$ to the horizontal. The lid is sufficiently rough to prevent $P$ from slipping. Given that $P$ is on the point of toppling,
find the value of $\theta$.

Edexcel FM2 2019 June Q6

13 marks Challenging +1.2

The points $A$ and $B$ lie on a smooth horizontal surface with $A B = 4.5 \mathrm {~m}$.

A light elastic string has natural length 1.5 m and modulus of elasticity 15 N . One end of the string is attached to $A$ and the other end of the string is attached to $B$. A particle, $P$, of mass 0.2 kg , is attached to the stretched string so that $A P B$ is a straight line and $A P = 1.5 \mathrm {~m}$. The particle rests in equilibrium on the surface. The particle is now moved directly towards $A$ and is held on the surface so $A P B$ is a straight line with $A P = 1 \mathrm {~m}$. The particle is released from rest.

Prove that $P$ moves with simple harmonic motion.
Find
1. the maximum speed of $P$ during the motion,
2. the maximum acceleration of $P$ during the motion.
Find the total time, in each complete oscillation of $P$, for which the speed of $P$ is greater than $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Edexcel FM2 2019 June Q7

12 marks Standard +0.8

A particle, $P$, of mass $m$ is attached to one end of a light rod of length $L$. The other end of the rod is attached to a fixed point $O$ so that the rod is free to rotate in a vertical plane about $O$. The particle is held with the rod horizontal and is then projected vertically downwards with speed $u$. The particle first comes to instantaneous rest at the point $A$.
1. Explain why the acceleration of $P$ at $A$ is perpendicular to $O A$.
At the instant when $P$ is at the point $A$ the acceleration of $P$ is in a direction making an angle $\theta$ with the horizontal. Given that $u ^ { 2 } = \frac { 2 g L } { 3 }$,
find
1. the magnitude of the acceleration of $P$ at the point $A$,
2. the size of $\theta$.
Find, in terms of $m$ and $g$, the magnitude of the tension in the rod at the instant when $P$ is at its lowest point.

Edexcel FM2 2020 June Q1

7 marks Standard +0.3

Three particles of masses $3 m$, $4 m$ and $2 m$ are placed at the points $( - 2,2 ) , ( 3,1 )$ and ( $p , p$ ) respectively.

The value of $p$ is such that the distance of the centre of mass of the three particles from the point ( 0,0 ) is as small as possible. Find the value of $p$.

Edexcel FM2 2020 June Q2

10 marks Standard +0.8

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{962c2b40-3c45-4eed-a0af-a59068bda0e1-04_506_590_255_429} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{962c2b40-3c45-4eed-a0af-a59068bda0e1-04_296_327_456_1311} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A uniform plane figure $R$, shown shaded in Figure 1, is bounded by the $x$-axis, the line with equation $x = \ln 5$, the curve with equation $y = 8 \mathrm { e } ^ { - x }$ and the line with equation $x = \ln 2$. The unit of length on each axis is one metre. The area of $R$ is $2.4 \mathrm {~m} ^ { 2 }$
The centre of mass of $R$ is at the point with coordinates $( \bar { x } , \bar { y } )$.

Use algebraic integration to show that $\bar { y } = 1.4$ Figure 2 shows a uniform lamina $A B C D$, which is the same size and shape as $R$. The lamina is freely suspended from $C$ and hangs in equilibrium with $C B$ at an angle $\theta ^ { \circ }$ to the downward vertical.
Find the value of $\theta$

Edexcel FM2 2020 June Q3

10 marks Challenging +1.2

A particle $P$ of mass 0.5 kg is moving along the positive $x$-axis in the direction of $x$ increasing. At time $t$ seconds $( t \geqslant 0 ) , P$ is $x$ metres from the origin $O$ and the speed of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The resultant force acting on $P$ is directed towards $O$ and has magnitude $k v ^ { 2 } \mathrm {~N}$, where $k$ is a positive constant.

When $x = 1 , v = 4$ and when $x = 2 , v = 2$

Show that $v = a b ^ { x }$, where $a$ and $b$ are constants to be found. The time taken for the speed of $P$ to decrease from $4 \mathrm {~ms} ^ { - 1 }$ to $2 \mathrm {~ms} ^ { - 1 }$ is $T$ seconds.
Show that $T = \frac { 1 } { 4 \ln 2 }$

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