Questions M1 - A-Level Maths

Particle $P$ has mass $3 m$ and particle $Q$ has mass $m$. The particles are moving towards each other in opposite directions along the same straight line on a smooth horizontal plane. The particles collide directly. Immediately before the collision the speed of $P$ is $u$ and the speed of $Q$ is $3 u$. In the collision, the magnitude of the impulse exerted by $Q$ on $P$ is $5 m u$.
1. Find the speed of $P$ immediately after the collision.
2. Find the speed of $Q$ immediately after the collision.

Edexcel M1 2018 June Q2

10 marks Standard +0.3

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c0993853-dd8f-4d14-aeed-b71ad60df09c-04_360_1037_260_456} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A uniform wooden beam $A B$, of mass 20 kg and length 4 m , rests in equilibrium in a horizontal position on two supports. One support is at $C$, where $A C = 1.6 \mathrm {~m}$, and the other support is at $D$, where $D B = 0.4 \mathrm {~m}$. A boy of mass 60 kg stands on the beam at the point $P$, where $A P = 3 \mathrm {~m}$, as shown in Figure 1. The beam remains in equilibrium in a horizontal position. By modelling the boy as a particle and the beam as a uniform rod,

1. find, in terms of $g$, the magnitude of the force exerted on the beam by the support at $C$,
2. find, in terms of $g$, the magnitude of the force exerted on the beam by the support at $D$. The boy now starts to walk slowly along the beam towards the end $A$.
Find the greatest distance he can walk from $P$ without the beam tilting.

Edexcel M1 2018 June Q3

10 marks Moderate -0.3

3. A cyclist starts from rest at the point $O$ on a straight horizontal road. The cyclist moves along the road with constant acceleration $2 \mathrm {~ms} ^ { - 2 }$ for 4 seconds and then continues to move along the road at constant speed. At the instant when the cyclist stops accelerating, a motorcyclist starts from rest at the point $O$ and moves along the road with constant acceleration $4 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ in the same direction as the cyclist. The motorcyclist has been moving for $T$ seconds when she overtakes the cyclist.

Sketch, on the same axes, a speed-time graph for the motion of the cyclist and a speed-time graph for the motion of the motorcyclist, to the time when the motorcyclist overtakes the cyclist.
Find, giving your answer to 1 decimal place, the value of $T$.

Edexcel M1 2018 June Q4

13 marks Standard +0.3

4. A rough plane is inclined at an angle $\alpha$ to the horizontal, where $\tan \alpha = \frac { 3 } { 4 }$. A particle of mass 2 kg is projected with speed $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from a point $O$ on the plane, up a line of greatest slope of the plane. The coefficient of friction between the particle and the plane is 0.25

Find the magnitude of the frictional force acting on the particle as it moves up the plane. The particle comes to instantaneous rest at the point $A$.
Find the distance $O A$. The particle now moves down the plane from $A$.
Find the speed of $P$ as it passes through $O$.

Edexcel M1 2018 June Q5

15 marks Moderate -0.3

5. [In this question $\mathbf { i }$ and $\mathbf { j }$ are perpendicular horizontal unit vectors and position vectors are given relative to a fixed origin $O$.] A particle $P$ is moving in a straight line with constant velocity. At 9 am, the position vector of $P$ is $( 7 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }$ and at 9.20 am , the position vector of $P$ is $6 \mathbf { i } \mathrm {~km}$. At time $t$ hours after 9 am , the position vector of $P$ is $\mathbf { r } _ { P } \mathrm {~km}$.

Find, in $\mathrm { kmh } ^ { - 1 }$, the speed of $P$.
Show that $\mathbf { r } _ { P } = ( 7 - 3 t ) \mathbf { i } + ( 5 - 15 t ) \mathbf { j }$.
Find the value of $t$ when $\mathbf { r } _ { P }$ is parallel to $16 \mathbf { i } + 5 \mathbf { j }$. The position vector of another particle $Q$, at time $t$ hours after 9 am , is $\mathbf { r } _ { Q } \mathrm {~km}$, where $\mathbf { r } _ { Q } = ( 5 + 2 t ) \mathbf { i } + ( - 3 + 5 t ) \mathbf { j }$
Show that $P$ and $Q$ will collide and find the position vector of the point of collision.

Edexcel M1 2018 June Q6

9 marks Standard +0.3

6. A car pulls a trailer along a straight horizontal road using a light inextensible towbar. The mass of the car is $M \mathrm {~kg}$, the mass of the trailer is 600 kg and the towbar is horizontal and parallel to the direction of motion. There is a resistance to motion of magnitude 200 N acting on the car and a resistance to motion of magnitude 100 N acting on the trailer. The driver of the car spots a hazard ahead. Instantly he reduces the force produced by the engine of the car to zero and applies the brakes of the car. The brakes produce a braking force on the car of magnitude 6500 N and the car and the trailer have a constant deceleration of magnitude $4 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ Given that the resistances to motion on the car and trailer are unchanged and that the car comes to rest after travelling 40.5 m from the point where the brakes were applied, find

the thrust in the towbar while the car is braking,
the value of $M$,
the time it takes for the car to stop after the brakes are applied.

Edexcel M1 2018 June Q7

12 marks Moderate -0.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c0993853-dd8f-4d14-aeed-b71ad60df09c-24_206_1040_356_443} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A washing line $A B C D$ is fixed at the points $A$ and $D$. There are two heavy items of clothing hanging on the washing line, one fixed at $B$ and the other fixed at $C$. The washing line is modelled as a light inextensible string, the item at $B$ is modelled as a particle of mass 3 kg and the item at $C$ is modelled as a particle of mass $M \mathrm {~kg}$. The section $A B$ makes an angle $\alpha$ with the horizontal, where $\tan \alpha = \frac { 3 } { 4 }$, the section $B C$ is horizontal and the section $C D$ makes an angle $\beta$ with the horizontal, where $\tan \beta = \frac { 12 } { 5 }$, as shown in Figure 2. The system is in equilibrium.

Find the tension in $A B$.
Find the tension in BC.
Find the value of $M$.

END

Edexcel M1 2020 June Q1

7 marks Moderate -0.5

Two particles, $P$ and $Q$, with masses $m$ and $2 m$ respectively, are moving in the same direction along the same straight line when they collide directly. Immediately before they collide, $P$ is moving with speed $4 u$ and $Q$ is moving with speed $u$. Immediately after they collide, both particles are moving in the same direction and the speed of $Q$ is four times the speed of $P$.
1. Find the speed of $Q$ immediately after the collision.
2. Find the magnitude of the impulse exerted by $Q$ on $P$ in the collision.
3. State clearly the direction of this impulse.

Edexcel M1 2020 June Q2

14 marks Moderate -0.8

2. A small ball is thrown vertically upwards with speed $14.7 \mathrm {~ms} ^ { - 1 }$ from a point that is 19.6 m above horizontal ground. The ball is modelled as a particle moving freely under gravity. Find

the total time from when the ball is thrown to when it first hits the ground,
the speed of the ball immediately before it first hits the ground,
the total distance travelled by the ball from when it is thrown to when it first hits the ground.
Sketch a velocity-time graph for the motion of the ball from when it is thrown to when it first hits the ground. State the coordinates of the start point and the coordinates of the end point of your graph.
DO NOT WRITEIN THIS AREA

Edexcel M1 2020 June Q3

8 marks Standard +0.3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{05cf68a3-1ba4-487f-9edd-48a246f4194f-08_259_597_214_678} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A particle of mass 10 kg is placed on a fixed rough inclined plane. The plane is inclined to the horizontal at an angle $\alpha$, where $\tan \alpha = \frac { 3 } { 4 }$. The particle is held in equilibrium by a force of magnitude $P$ newtons, which acts up the plane, as shown in Figure 1. The line of action of the force lies in a vertical plane that contains a line of greatest slope of the plane. The coefficient of friction between the particle and the plane is $\frac { 1 } { 2 }$.

Find the normal reaction between the particle and the plane.
Find the greatest possible value of $P$.
Find the least possible value of $P$. DO NOT WRITEIN THIS AREA

VIXV SIHIANI III IM IONOO VIAV SIHI NI JYHAM ION OO VI4V SIHI NI JLIYM ION OO

Edexcel M1 2020 June Q4

8 marks Standard +0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{05cf68a3-1ba4-487f-9edd-48a246f4194f-12_536_1253_127_349} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A non-uniform beam $A B$ has length 8 m and mass $M \mathrm {~kg}$. The centre of mass of the beam is $d$ metres from $A$. The beam is supported in equilibrium in a horizontal position by two vertical light ropes. One rope is attached to the beam at $C$, where $A C = 2.5 \mathrm {~m}$ and the other rope is attached to the beam at $D$, where $D B = 2 \mathrm {~m}$, as shown in Figure 2. A gymnast, of mass 64 kg , stands on the beam at the point $X$, where $A X = 1.875 \mathrm {~m}$, and the beam remains in equilibrium in a horizontal position but is now on the point of tilting about $C$. The gymnast then dismounts from the beam. A second gymnast, of mass 48 kg , now stands on the beam at the point $Y$, where $Y B = 0.5 \mathrm {~m}$, and the beam remains in equilibrium in a horizontal position but is now on the point of tilting about $D$. The beam is modelled as a non-uniform rod and the gymnasts are modelled as particles. Find the value of $M$.

VIXV SIHIANI III IM IONOO

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Edexcel M1 2020 June Q5

13 marks Moderate -0.8

5. A particle $P$ is moving in a plane with constant acceleration. The velocity, $\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }$, of $P$ at time $t$ seconds is given by $$\mathbf { v } = ( 7 - 5 t ) \mathbf { i } + ( 12 t - 20 ) \mathbf { j }$$

Find the speed of $P$ when $t = 2$
Find, to the nearest degree, the size of the angle between the direction of motion of $P$ and the vector $\mathbf { j }$, when $t = 2$ The constant acceleration of $P$ is a m s-2
Find $\mathbf { a }$ in terms of $\mathbf { i }$ and $\mathbf { j }$
Find the value of $t$ when $P$ is moving in the direction of the vector $( - 5 \mathbf { i } + 8 \mathbf { j } )$

Edexcel M1 2020 June Q6

8 marks Standard +0.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{05cf68a3-1ba4-487f-9edd-48a246f4194f-20_328_1082_127_438} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A railway engine of mass 1500 kg is attached to a railway truck of mass 500 kg by a straight rigid coupling. The engine pushes the truck up a straight track, which is inclined to the horizontal at an angle $\alpha$, where $\sin \alpha = \frac { 7 } { 25 }$. The coupling is parallel to the track and parallel to the direction of motion, as shown in Figure 3. The engine produces a constant driving force of magnitude $D$ newtons. The engine and the truck experience constant resistances to motion, from non-gravitational forces, of magnitude 1200 N and 500 N respectively. The thrust in the coupling is 2000 N . The coupling is modelled as a light rod.

Find the acceleration of the engine and the truck.
Find the value of $D$.

Edexcel M1 2020 June Q7

9 marks Moderate -0.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{05cf68a3-1ba4-487f-9edd-48a246f4194f-24_534_426_127_760} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} One end of a light inextensible string is attached to a particle $A$ of mass $5 m$. The other end of the string is attached to a particle $B$ of mass $3 m$. The string passes over a small, smooth, light fixed pulley. Particle $A$ is held at rest with the string taut and the hanging parts of the string vertical, as shown in Figure 4. Particle A is released.

Find, in terms of $m$ and $g$, the magnitude of the force exerted on the pulley by the string while $A$ is falling and before $B$ hits the pulley.
State how, in your solution to part (a), you have used the fact that the pulley is smooth.

Edexcel M1 2020 June Q8

8 marks Moderate -0.3

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{05cf68a3-1ba4-487f-9edd-48a246f4194f-28_766_1587_278_182} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} The acceleration-time graph shown in Figure 5 represents part of a journey made by a car along a straight horizontal road. The car accelerated from rest at time $t = 0$

Find the distance travelled by the car during the first 4 s of its journey.
Find the total distance travelled by the car during the first 26s of its journey.

VIXV SIHIANI III IM IONOO VIAV SIHI NI JYHAM ION OO VI4V SIHI NI JLIYM ION OO

END

Edexcel M1 2021 June Q1

7 marks Moderate -0.3

A particle $P$ has mass $3 m$ and a particle $Q$ has mass $5 m$. The particles are moving towards each other in opposite directions along the same straight line on a smooth horizontal surface. The particles collide directly.

Immediately before the collision the speed of $P$ is $k u$, where $k$ is a constant, and the speed of $Q$ is $2 u$. Immediately after the collision the speed of $P$ is $u$ and the speed of $Q$ is $3 u$.
The direction of motion of $Q$ is reversed by the collision.

Find, in terms of $m$ and $u$, the magnitude of the impulse exerted on $Q$ by $P$ in the collision.
Find the two possible values of $k$.
\includegraphics[max width=\textwidth, alt={}, center]{5a2cf693-d966-4787-8778-ecc8a79a6265-03_2647_1837_118_114}

Edexcel M1 2021 June Q2

8 marks Standard +0.8

2. A car moves along a straight horizontal road with constant acceleration $a \mathrm {~ms} ^ { - 2 }$
where $a > 0$ The car is modelled as a particle. At time $t = 0$, the car passes point $A$ and is moving with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ In the first three seconds after passing $A$ the car travels 20 m . In the fourth second after passing $A$ the car travels 10 m . The speed of the car as it passes point $B$ is $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ Find the time taken for the car to travel from $A$ to $B$.
(8)

Edexcel M1 2021 June Q3

9 marks Moderate -0.3

3. [In this question $\mathbf { i }$ and $\mathbf { j }$ are perpendicular horizontal unit vectors.] Three forces, $\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }$ and $\mathbf { F } _ { 3 }$, are given by $$\mathbf { F } _ { 1 } = ( 5 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N } \quad \mathbf { F } _ { 2 } = ( - 3 \mathbf { i } + \mathbf { j } ) \mathrm { N } \quad \mathbf { F } _ { 3 } = ( a \mathbf { i } + b \mathbf { j } ) \mathrm { N }$$ where $a$ and $b$ are constants.
The forces $\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }$ and $\mathbf { F } _ { 3 }$ act on a particle $P$ of mass 4 kg .
Given that $P$ rests in equilibrium on a smooth horizontal surface under the action of these three forces,

find the size of the angle between the direction of $\mathbf { F } _ { 3 }$ and the direction of $- \mathbf { j }$. The force $\mathbf { F } _ { 3 }$ is now removed and replaced by the force $\mathbf { F } _ { 4 }$ given by $\mathbf { F } _ { 4 } = \lambda ( \mathbf { i } + 3 \mathbf { j } )$ N, where $\lambda$ is a positive constant. When the three forces $\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }$ and $\mathbf { F } _ { 4 }$ act on $P$, the acceleration of $P$ has magnitude $3.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
Find the value of $\lambda$.

Edexcel M1 2021 June Q4

6 marks Moderate -0.8

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5a2cf693-d966-4787-8778-ecc8a79a6265-12_647_396_251_776} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a large bucket used by a crane on a building site to move materials between the ground and the top of the building. The mass of the bucket is 15 kg . The bucket is attached to a vertical cable with the bottom of the bucket horizontal. The cable is modelled as light and inextensible. When the bucket is on the ground, a bag of cement of mass 25 kg is placed in the bucket. The bucket with the bag of cement moves vertically upwards with constant acceleration $0.2 \mathrm {~ms} ^ { - 2 }$. Air resistance is modelled as being negligible.

Find the tension in the cable. At the top of the building, the bag of cement is removed. A box of tools of mass 12 kg is now placed in the bucket. Later on the bucket with the box of tools is moving vertically downwards with constant deceleration $0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. Air resistance is again modelled as being negligible.
Find the magnitude of the normal reaction between the bucket and the box of tools.

Edexcel M1 2021 June Q5

9 marks Standard +0.3

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

A particle $P$ is moving with constant acceleration. At 2 pm , the velocity of $P$ is $( 3 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$ and at 2.30 pm the velocity of $P$ is $( \mathbf { i } + 7 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$ At time $T$ hours after $2 \mathrm { pm } , P$ is moving in the direction of the vector $( - \mathbf { i } + 2 \mathbf { j } )$

Find the value of $T$. Another particle, $Q$, has velocity $\mathbf { v } _ { Q } \mathrm {~km} \mathrm {~h} ^ { - 1 }$ at time $t$ hours after 2 pm , where $$\mathbf { v } _ { Q } = ( - 4 - 2 t ) \mathbf { i } + ( \mu + 3 t ) \mathbf { j }$$ and $\mu$ is a constant. Given that there is an instant when the velocity of $P$ is equal to the velocity of $Q$,
find the value of $\mu$.

Edexcel M1 2021 June Q6

13 marks Standard +0.3

A fixed rough plane is inclined at an angle $\theta$ to the horizontal, where $\tan \theta = \frac { 5 } { 12 }$

A particle of mass 6 kg is projected with speed $5 \mathrm {~ms} ^ { - 1 }$ from a point $A$ on the plane, up a line of greatest slope of the plane. The coefficient of friction between the particle and the plane is $\frac { 1 } { 4 }$

Find the magnitude of the frictional force acting on the particle as it moves up the plane. The particle comes to instantaneous rest at the point $B$.
Find the distance $A B$. The particle now slides down the plane from $B$. At the instant when the particle passes through the point $C$ on the plane, the speed of the particle is again $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
Find the distance $B C$. \includegraphics[max width=\textwidth, alt={}, center]{5a2cf693-d966-4787-8778-ecc8a79a6265-23_2647_1835_118_116}

Edexcel M1 2021 June Q7

10 marks Standard +0.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5a2cf693-d966-4787-8778-ecc8a79a6265-24_191_1136_255_406} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A non-uniform beam $A B$, of mass 60 kg and length $8 a$ metres, rests in equilibrium in a horizontal position on two vertical supports. One support is at $C$, where $A C = a$ metres and the other support is at $D$, where $D B = 2 a$ metres, as shown in Figure 2. The magnitude of the normal reaction between the beam and the support at $D$ is three times the magnitude of the normal reaction between the beam and the support at $C$. By modelling the beam as a non-uniform rod whose centre of mass is at a distance $x$ metres from $A$,

find an expression for $x$ in terms of $a$. A box of mass $M \mathrm {~kg}$ is placed on the beam at $E$, where $A E = 2 a$ metres.
The beam remains in equilibrium in a horizontal position.
The magnitude of the normal reaction between the beam and the support at $C$ is now equal to the magnitude of the normal reaction between the beam and the support at $D$. By modelling the box as a particle,
find the value of $M$.

Edexcel M1 2021 June Q8

13 marks Standard +0.3

8. Two trams, tram $A$ and tram $B$, run on parallel straight horizontal tracks. Initially the two trams are at rest in the depot and level with each other. At time $t = 0 , \operatorname { tram } A$ starts to move. Tram $A$ moves with constant acceleration $2 \mathrm {~ms} ^ { - 2 }$ for 5 seconds and then continues to move along the track at constant speed. At time $t = 20$ seconds, tram $B$ starts from rest and moves in the same direction as tram $A$. Tram $B$ moves with constant acceleration $3 \mathrm {~ms} ^ { - 2 }$ for 4 seconds and then continues to move along the track at constant speed. The trams are modelled as particles.

Sketch, on the same axes, a speed-time graph for the motion of tram $A$ and a speed-time graph for the motion of tram $B$, from $t = 0$ to the instant when tram $B$ overtakes $\operatorname { tram } A$. At the instant when the two trams are moving with the same speed, $\operatorname { tram } A$ is $d$ metres in front of tram $B$.
Find the value of $d$.
Find the distance of the trams from the depot at the instant when tram $B$ overtakes $\operatorname { tram } A$.
\includegraphics[max width=\textwidth, alt={}, center]{5a2cf693-d966-4787-8778-ecc8a79a6265-32_2647_1835_118_116}
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Edexcel M1 2022 June Q1

4 marks Moderate -0.8

Two particles, $P$ and $Q$, are moving towards each other in opposite directions along the same straight line when they collide directly. Immediately before the collision the speed of $Q$ is $2 u$. The mass of $Q$ is $3 m$ and the magnitude of the impulse exerted by $P$ on $Q$ in the collision is $4 m u$.

Find

the speed of $Q$ immediately after the collision,
the direction of motion of $Q$ immediately after the collision.

Edexcel M1 2022 June Q2

7 marks Moderate -0.3

2. A motorbike is moving with constant acceleration along a straight horizontal road. The motorbike passes a point $P$ and 10 seconds later passes a point $Q$. The speed of the motorbike as it passes $Q$ is $28 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
Given that $P Q = 220 \mathrm {~m}$,

find the acceleration of the motorbike,
find the distance travelled by the motorbike during the fifth second after passing $P$ VILV SIHI NI IIII M I ON OC
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