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Edexcel FM1 2021 June Q7

9 marks Standard +0.8

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

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7f077b82-6b39-4cb5-8574-bfa308c88df3-24_543_789_294_639} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 represents the plan view of part of a smooth horizontal floor, where $A B$ is a fixed smooth vertical wall. The direction of $\overrightarrow { A B }$ is in the direction of the vector $( \mathbf { i } + \mathbf { j } )$ A small ball of mass 0.25 kg is moving on the floor when it strikes the wall $A B$.
Immediately before its impact with the wall $A B$, the velocity of the ball is $( 8 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ Immediately after its impact with the wall $A B$, the velocity of the ball is $\mathbf { v m s } ^ { - 1 }$ The coefficient of restitution between the ball and the wall is $\frac { 1 } { 3 }$ By modelling the ball as a particle,

show that $\mathbf { v } = 4 \mathbf { i } + 6 \mathbf { j }$
Find the magnitude of the impulse received by the ball in the impact.

Edexcel FM1 2022 June Q1

8 marks Standard +0.3

A particle $A$ of mass $3 m$ and a particle $B$ of mass $m$ are moving along the same straight line on a smooth horizontal surface. The particles are moving in opposite directions towards each other when they collide directly.

Immediately before the collision, the speed of $A$ is $k u$ and the speed of $B$ is $u$. Immediately after the collision, the speed of $A$ is $v$ and the speed of $B$ is $2 v$. The magnitude of the impulse received by $B$ in the collision is $\frac { 3 } { 2 } m u$.

Find $v$ in terms of $u$ only.
Find the two possible values of $k$.

Find the size of the angle through which the direction of motion of $A$ is turned as a result of the collision.
Find, in terms of $m$ and $u$, the magnitude of the impulse received by $B$ in the collision.

Edexcel FM1 2022 June Q5

10 marks Standard +0.8

Two particles, $P$ and $Q$, are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly.
The mass of $P$ is $3 m$ and the mass of $Q$ is $4 m$.
Immediately before the collision the speed of $P$ is $2 u$ and the speed of $Q$ is $u$.
The coefficient of restitution between $P$ and $Q$ is $e$.

Show that the speed of $Q$ immediately after the collision is $\frac { u } { 7 } ( 9 e + 2 )$ After the collision with $P$, particle $Q$ collides directly with a fixed vertical wall and rebounds. The wall is perpendicular to the direction of motion of $Q$.
The coefficient of restitution between $Q$ and the wall is $\frac { 1 } { 2 }$
Find the complete range of possible values of $e$ for which there is a second collision between $P$ and $Q$.

Edexcel FM1 2022 June Q6

13 marks Standard +0.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{86a37170-046f-46e5-9c8c-06d5f98ca4fe-20_497_1337_246_365} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Two blocks, $A$ and $B$, of masses 2 kg and 4 kg respectively are attached to the ends of a light inextensible string. Initially $A$ is held on a fixed rough plane. The plane is inclined to horizontal ground at an angle $\theta$, where $\tan \theta = \frac { 3 } { 4 }$ The string passes over a small smooth light pulley $P$ that is fixed at the top of the plane. The part of the string from $A$ to $P$ is parallel to a line of greatest slope of the plane. Block $A$ is held on the plane with the distance $A P$ greater than 3 m .
Block $B$ hangs freely below $P$ at a distance of 3 m above the ground, as shown in Figure 4. The coefficient of friction between $A$ and the plane is $\mu$ Block $A$ is released from rest with the string taut.
By modelling the blocks as particles,

find the potential energy lost by the whole system as a result of $B$ falling 3 m . Given that the speed of $B$ at the instant it hits the ground is $4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and ignoring air resistance,
use the work-energy principle to find the value of $\mu$ After $B$ hits the ground, $A$ continues to move up the plane but does not reach the pulley in the subsequent motion.
Block $A$ comes to instantaneous rest after moving a total distance of ( $3 + d$ ) m from its point of release. Ignoring air resistance,
use the work-energy principle to find the value of $d$ \includegraphics[max width=\textwidth, alt={}, center]{86a37170-046f-46e5-9c8c-06d5f98ca4fe-20_2255_50_309_1981}

Edexcel FM1 2022 June Q7

12 marks Standard +0.8

A spring of natural length $a$ has one end attached to a fixed point $A$. The other end of the spring is attached to a package $P$ of mass $m$.
The package $P$ is held at rest at the point $B$, which is vertically below $A$ such that $A B = 3 a$.
After being released from rest at $B$, the package $P$ first comes to instantaneous rest at $A$. Air resistance is modelled as being negligible.

By modelling the spring as being light and modelling $P$ as a particle,

show that the modulus of elasticity of the spring is $2 m g$
1. Show that $P$ attains its maximum speed when the extension of the spring is $\frac { 1 } { 2 } a$
2. Use the principle of conservation of mechanical energy to find the maximum speed, giving your answer in terms of $a$ and $g$. In reality, the spring is not light.
State one way in which this would affect your energy equation in part (b).

Edexcel FM1 2022 June Q8

10 marks Challenging +1.2

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{86a37170-046f-46e5-9c8c-06d5f98ca4fe-28_567_1406_244_333} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} Figure 5 represents the plan view of part of a smooth horizontal floor, where $R S$ and $S T$ are smooth fixed vertical walls. The vector $\overrightarrow { R S }$ is in the direction of $\mathbf { i }$ and the vector $\overrightarrow { S T }$ is in the direction of $( 2 \mathbf { i } + \mathbf { j } )$. A small ball $B$ is projected across the floor towards $R S$. Immediately before the impact with $R S$, the velocity of $B$ is $( 6 \mathbf { i } - 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. The ball bounces off $R S$ and then hits $S T$. The ball is modelled as a particle.
Given that the coefficient of restitution between $B$ and $R S$ is $e$,

find the full range of possible values of $e$. It is now given that $e = \frac { 1 } { 4 }$ and that the coefficient of restitution between $B$ and $S T$ is $\frac { 1 } { 2 }$
Find, in terms of $\mathbf { i }$ and $\mathbf { j }$, the velocity of $B$ immediately after its impact with $S T$.

Edexcel FM1 2023 June Q1

6 marks Standard +0.3

A particle $P$ of mass 2 kg is moving with velocity $( - 4 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ when it receives an impulse $( - 6 \mathbf { i } + 42 \mathbf { j } )$ N s.

Find the speed of $P$ immediately after receiving the impulse. The angle through which the direction of motion of $P$ has been deflected by the impulse is $\alpha ^ { \circ }$
Find the value of $\alpha$

Edexcel FM1 2023 June Q2

8 marks Standard +0.3

A car of mass 1000 kg moves in a straight line along a horizontal road at a constant speed $U \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The resistance to the motion of the car is a constant force of magnitude 400 N.

The engine of the car is working at a constant rate of 16 kW .

Find the value of $U$. The car now pulls a trailer of mass 600 kg in a straight line along the road using a tow rope which is parallel to the direction of motion. The resistance to the motion of the car is again a constant force of magnitude 400 N . The resistance to the motion of the trailer is a constant force of magnitude 300 N . The engine of the car is working at a constant rate of 16 kW .
The tow rope is modelled as being light and inextensible.
Using the model,
find the tension in the tow rope at the instant when the speed of the car is $\frac { 20 } { 3 } \mathrm {~ms} ^ { - 1 }$

Edexcel FM1 2023 June Q3

10 marks Standard +0.8

A particle $P$ of mass $2 m$ is moving in a straight line with speed $3 u$ on a smooth horizontal plane. It collides directly with a particle $Q$ of mass $m$ that is moving on the plane with speed $2 u$ in the opposite direction to $P$.
The coefficient of restitution between $P$ and $Q$ is $e$, where $e > \frac { 4 } { 5 }$

Show that the speed of $Q$ immediately after the collision is $\frac { ( 4 + 10 e ) u } { 3 }$ After the collision $Q$ hits a smooth fixed vertical wall that is perpendicular to the direction of motion of $Q$. The coefficient of restitution between $Q$ and the wall is $f$.
Find, in terms of $\boldsymbol { e }$, the set of values of $f$ for which there will be a second collision between $P$ and $Q$.

Edexcel FM1 2023 June Q4

15 marks Standard +0.8

A light elastic string has natural length $2 a$ and modulus of elasticity $4 m g$. One end of the elastic string is attached to a fixed point $O$. A particle $P$ of mass $m$ is attached to the other end of the elastic string.
The particle $P$ hangs freely in equilibrium at the point $E$, which is vertically below $O$

Find the length $O E$. Particle $P$ is now pulled vertically downwards to the point $A$, where $O A = 4 a$, and released from rest. The resistance to the motion of $P$ is a constant force of magnitude $\frac { 1 } { 4 } m g$.
Find, in terms of $a$ and $g$, the speed of $P$ after it has moved a distance $a$. Particle $P$ is now held at $O$ Particle $P$ is released from rest and reaches its maximum speed at the point $B$. The resistance to the motion of $P$ is again a constant force of magnitude $\frac { 1 } { 4 } m g$.
Find the distance $O B$.

Edexcel FM1 2023 June Q5

10 marks Challenging +1.2

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0da9cd5b-6f6f-4607-bd4f-c8ae164466ae-16_758_1399_280_333} \captionsetup{labelformat=empty} \caption{Figure 1}

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

Show that $\tan \beta = \frac { ( m + M ) \tan \alpha } { ( m - e M ) }$ Given that $m = e M$,
show that the directions of motion of the two spheres immediately after the collision are perpendicular.

Edexcel FM1 2023 June Q6

12 marks Challenging +1.2

A particle $P$ of mass $m$ is falling vertically when it strikes a fixed smooth inclined plane. The plane is inclined to the horizontal at an angle $\alpha$, where $0 < \alpha \leqslant 45 ^ { \circ }$

At the instant immediately before the impact, the speed of $P$ is $u$.
At the instant immediately after the impact, $P$ is moving horizontally with speed $v$.

Show that the magnitude of the impulse exerted on the plane by $P$ is $m u \sec \alpha$ The coefficient of restitution between $P$ and the plane is $e$, where $e > 0$
Show that $v ^ { 2 } = u ^ { 2 } \left( \sin ^ { 2 } \alpha + e ^ { 2 } \cos ^ { 2 } \alpha \right)$
Show that the kinetic energy lost by $P$ in the impact is $$\frac { 1 } { 2 } m u ^ { 2 } \left( 1 - e ^ { 2 } \right) \cos ^ { 2 } \alpha$$
Hence find, in terms of $m$, $u$ and $e$ only, the kinetic energy lost by $P$ in the impact.

Edexcel FM1 2023 June Q7

14 marks Challenging +1.2

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0da9cd5b-6f6f-4607-bd4f-c8ae164466ae-24_721_1367_280_349} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A small smooth snooker ball is projected from the corner $A$ of a horizontal rectangular snooker table $A B C D$. The ball is projected so it first hits the side $D C$ at the point $P$, then hits the side $C B$ at the point $Q$ and then returns to $A$. Angle $A P D = \alpha$, Angle $Q P C = \beta$, Angle $A Q B = \gamma$ The ball moves along $A P$ with speed $U$, along $P Q$ with speed $V$ and along $Q A$ with speed $W$, as shown in Figure 2. The coefficient of restitution between the ball and side $D C$ is $e _ { 1 }$ The coefficient of restitution between the ball and side $C B$ is $e _ { 2 }$ The ball is modelled as a particle. \section*{Use the model to answer all parts of this question.}

Show that $\tan \beta = e _ { 1 } \tan \alpha$
Hence show that $e _ { 1 } \tan \alpha = e _ { 2 } \cot \gamma$
By considering (angle $A P Q$ + angle $A Q P$ ) or otherwise, show that it would be possible for the ball to return to $A$ only if $e _ { 2 } > e _ { 1 }$ If instead $e _ { 1 } = e _ { 2 }$, the ball would not return to $A$.
Given that $e _ { 1 } = e _ { 2 }$
use the result from part (b) to describe the path of the ball after it hits $C B$ at $Q$, explaining your answer.

Edexcel FM1 2024 June Q1

9 marks Moderate -0.3

\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 $\sin \alpha = \frac { 2 } { 49 }$
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.

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Edexcel FM1 2021 June Q7

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