Questions M1 - A-Level Maths

1. write down an equation of motion for $A$,
2. write down an equation of motion for $B$,
find, in terms of $g$, the acceleration of $A$,
find the magnitude of the force exerted on the pulley by the string.
State how you have used the information that $P$ is a smooth pulley.

Edexcel M1 2021 October Q8

12 marks Standard +0.3

8. [In this question $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin.] At 7 am a ship leaves a port and moves with constant velocity. The position vector of the port is $( - 2 \mathbf { i } + 9 \mathbf { j } ) \mathrm { km }$. At 7.36 am the ship is at the point with position vector $( 4 \mathbf { i } + 6 \mathbf { j } ) \mathrm { km }$.

Show that the velocity of the ship is $( 10 \mathbf { i } - 5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$
Find the position vector of the ship $t$ hours after leaving port. At 8.48 am a passenger on the ship notices that a lighthouse is due east of the ship. At 9 am the same passenger notices that the lighthouse is now north east of the ship.
Find the position vector of the lighthouse.
Find the position vector of the ship when it is due south of the lighthouse.

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Edexcel M1 2022 October Q1

5 marks Easy -1.2

A railway truck $S$ of mass 20 tonnes is moving along a straight horizontal track when it collides with another railway truck $T$ of mass 30 tonnes which is at rest. Immediately before the collision the speed of $S$ is $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
As a result of the collision, the two railway trucks join together.
Find
1. the common speed of the railway trucks immediately after the collision,
2. the magnitude of the impulse exerted on $S$ in the collision, stating the units of your answer.

Edexcel M1 2022 October Q2

6 marks Standard +0.3

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2633b149-96db-4b80-96c2-e3e6bfbee174-04_515_1282_269_331} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A uniform rod $A B$ has length $2 a$ and mass $M$. The rod is held in equilibrium in a horizontal position by two vertical light strings which are attached to the rod at $C$ and $D$, where $A C = \frac { 2 } { 5 } a$ and $D B = \frac { 3 } { 5 } a$, as shown in Figure 1. A particle $P$ is placed on the rod at $B$.
The rod remains horizontal and in equilibrium.

Find, in terms of $M$, the largest possible mass of the particle $P$ Given that the mass of $P$ is $\frac { 1 } { 2 } M$
find, in terms of $M$ and $g$, the tension in the string that is attached to the rod at $C$.

Edexcel M1 2022 October Q3

11 marks Standard +0.3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2633b149-96db-4b80-96c2-e3e6bfbee174-08_301_636_287_657} \captionsetup{labelformat=empty} \caption{Figure 2}

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

Show that when $X = 14.7$ there is no frictional force acting on $P$ The coefficient of friction between $P$ and the plane is 0.5
Find the smallest possible value of $X$.
VIAV SIHI NI IIIIM I I N OC
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VILV SIMI NI III M I I N OC \includegraphics[max width=\textwidth, alt={}, center]{2633b149-96db-4b80-96c2-e3e6bfbee174-11_88_63_2631_1886}

Edexcel M1 2022 October Q4

6 marks Moderate -0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2633b149-96db-4b80-96c2-e3e6bfbee174-12_543_264_296_842} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Two children, Alan and Bhavana, are standing on the horizontal floor of a lift, as shown in Figure 3. The lift has mass 250 kg . The lift is raised vertically upwards with constant acceleration by a vertical cable which is attached to the top of the lift. The cable is modelled as being light and inextensible. While the lift is accelerating upwards, the tension in the cable is 3616 N . As the lift accelerates upwards, the floor of the lift exerts a force of magnitude 565 N on Alan and a force of magnitude 226 N on Bhavana. Air resistance is modelled as being negligible and Alan and Bhavana are modelled as particles.

By considering the forces acting on the lift only, find the acceleration of the lift.
Find the mass of Alan.

Edexcel M1 2022 October Q5

9 marks Standard +0.3

5. A small ball is projected vertically upwards with speed $29.4 \mathrm {~ms} ^ { - 1 }$ from a point $A$ which is 19.6 m above horizontal ground. The ball is modelled as a particle moving freely under gravity until it hits the ground. It is assumed that the ball does not rebound.

Find the distance travelled by the ball while its speed is less than $14.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
Find the time for which the ball is moving with a speed of more than $29.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
Sketch a speed-time graph for the motion of the ball from the instant when it is projected from $A$ to the instant when it hits the ground. Show clearly where your graph meets the axes.
Q
7

Edexcel M1 2022 October Q7

13 marks Standard +0.3

7
6. [In this question, $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors.] A particle $A$ of mass 0.5 kg is at rest on a smooth horizontal plane.
At time $t = 0$, two forces, $\mathbf { F } _ { 1 } = ( - 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }$ and $\mathbf { F } _ { 2 } = ( p \mathbf { i } + q \mathbf { j } ) \mathrm { N }$, where $p$ and $q$ are constants, are applied to $A$. Given that $A$ moves in the direction of the vector $( \mathbf { i } - 2 \mathbf { j } )$,

show that $2 p + q - 4 = 0$ Given that $p = 5$
find the speed of $A$ at time $t = 4$ seconds.
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2633b149-96db-4b80-96c2-e3e6bfbee174-24_451_851_310_493} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A particle $P$ of mass $m$ is attached to one end of a light inextensible string. Another particle $Q$, also of mass $m$, is attached to the other end of the string. The string passes over a small smooth pulley which is fixed at the edge of a rough horizontal table. Particle $Q$ is held at rest on the table and particle $P$ hangs vertically below the pulley with the string taut, as shown in Figure 4. The pulley, $P$ and $Q$ all lie in the same vertical plane.
The coefficient of friction between $Q$ and the table is $\mu$, where $\mu < 1$
Particle $Q$ is released from rest.
The tension in the string before $Q$ hits the pulley is $k m g$, where $k$ is a constant.
Find $k$ in terms of $\mu$. Given that $Q$ is initially a distance $d$ from the pulley,
find, in terms of $d , g$ and $\mu$, the time taken by $Q$, after release, to reach the pulley.
Describe what would happen if $\mu \geqslant 1$, giving a reason for your answer.

Edexcel M1 2022 October Q8

16 marks Standard +0.3

8. [In this question, $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin $O$.] Two ships, $A$ and $B$, are moving with constant velocities.
The velocity of $A$ is $( 3 \mathbf { i } + 12 \mathbf { j } ) \mathrm { kmh } ^ { - 1 }$ and the velocity of $B$ is $( p \mathbf { i } + q \mathbf { j } ) \mathrm { kmh } ^ { - 1 }$

Find the speed of $A$. The ships are modelled as particles.
At 12 noon, $A$ is at the point with position vector $( - 9 \mathbf { i } + 6 \mathbf { j } ) \mathrm { km }$ and $B$ is at the point with position vector $( 16 \mathbf { i } + 6 \mathbf { j } ) \mathrm { km }$. At time $t$ hours after 12 noon, $$\overrightarrow { A B } = [ ( 25 - 12 t ) \mathbf { i } - 9 t \mathbf { j } ] \mathrm { km }$$
Find the value of $p$ and the value of $q$.
Find the bearing of $A$ from $B$ when the ships are 15 km apart, giving your answer to the nearest degree.
\includegraphics[max width=\textwidth, alt={}, center]{2633b149-96db-4b80-96c2-e3e6bfbee174-32_120_150_2508_1804}
\includegraphics[max width=\textwidth, alt={}, center]{2633b149-96db-4b80-96c2-e3e6bfbee174-32_143_191_2633_1779}

Edexcel M1 2023 October Q1

5 marks Moderate -0.3

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{017cc2b0-9ec3-45ff-94c0-9d989badfd5d-02_529_1362_246_349} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a beam $A B$ with weight 24 N and length 6 m .
The beam is suspended by two light vertical ropes. The ropes are attached to the points $C$ and $D$ on the beam where $A C = x$ metres and $C D = 2 \mathrm {~m}$. The tension in the rope attached to the beam at $C$ is double the tension in the rope attached to the beam at $D$. The beam is modelled as a uniform rod, resting horizontally in equilibrium.
Find

the tension in the rope attached to the beam at $D$.
the value of $x$.

Edexcel M1 2023 October Q2

10 marks Standard +0.2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{017cc2b0-9ec3-45ff-94c0-9d989badfd5d-04_677_1620_294_169} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Two fixed points, $A$ and $B$, are on a straight horizontal road.
The acceleration-time graph in Figure 2 represents the motion of a car travelling along the road as it moves from $A$ to $B$. At time $t = 0$, the car passes through $A$ with speed $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
At time $t = 20 \mathrm {~s}$, the car passes through $B$ with speed $v \mathrm {~ms} ^ { - 1 }$

Show that $v = 18$
Sketch a speed-time graph for the motion of the car from $A$ to $B$.
Find the distance $A B$.

Edexcel M1 2023 October Q3

10 marks Moderate -0.8

A hammer is used to hit a tent peg into soft ground.

The hammer has mass 1.8 kg and the tent peg has mass 0.2 kg .
The hammer and tent peg are both modelled as particles and the impact is modelled as a direct collision. Immediately before the impact, the tent peg is stationary and the hammer is moving vertically downwards with speed $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ Immediately after the impact, the hammer and tent peg move together, vertically downwards, with the same speed $v \mathrm {~ms} ^ { - 1 }$

Find the value of $v$
Find the magnitude of the impulse exerted on the tent peg by the hammer, stating the units of your answer. The ground exerts a constant vertical resistive force of magnitude $R$ newtons, bringing the hammer and tent peg to rest after they travel a distance of 12 cm .
Find the value of $R$.

Edexcel M1 2023 October Q4

10 marks Standard +0.3

\hspace{0pt} [In this question $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors directed due east and due north respectively.]

A particle $P$ moves with constant acceleration $( - \lambda \mathbf { i } + 2 \lambda \mathbf { j } ) \mathrm { ms } ^ { - 2 }$, where $\lambda$ is a positive constant. At time $t = 0$, the velocity of $P$ is $( 5 \mathbf { i } - 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$

Find the velocity of $P$ when $t = 5 \mathrm {~s}$, giving your answer in terms of $\mathbf { i } , \mathbf { j }$ and $\lambda$. The speed of $P$ when $t = 5 \mathrm {~s}$ is $13 \mathrm {~ms} ^ { - 1 }$
Show that $$25 \lambda ^ { 2 } - 42 \lambda - 16 = 0$$
Find the direction of motion of $P$ when $t = 4 \mathrm {~s}$, giving your answer as a bearing to the nearest degree.

Edexcel M1 2023 October Q5

12 marks Standard +0.3

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{017cc2b0-9ec3-45ff-94c0-9d989badfd5d-16_757_460_246_804} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A small ring of mass 0.2 kg is attached to one end of a light inextensible string.
The ring is threaded onto a fixed rough vertical rod.
The string is taut and makes an angle $\theta$ with the rod, as shown in Figure 3, where $\tan \theta = \frac { 12 } { 5 }$ Given that the ring is in equilibrium and that the tension in the string is 10 N ,

find the magnitude of the frictional force acting on the ring,
state the direction of the frictional force acting on the ring. The coefficient of friction between the ring and the rod is $\frac { 1 } { 4 }$
Given that the ring is in equilibrium, and that the tension in the string, $T$ newtons, can now vary,
1. find the minimum value of $T$
2. find the maximum value of $T$

Edexcel M1 2023 October Q6

15 marks Moderate -0.3

\hspace{0pt} [In this question $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin $O$.]

At 12:00, a ship $P$ sets sail from a harbour with position vector $( 15 \mathbf { i } + 36 \mathbf { j } ) \mathrm { km }$. At 12:30, $P$ is at the point with position vector $( 20 \mathbf { i } + 34 \mathbf { j } ) \mathrm { km }$. Given that $P$ moves with constant velocity,

show that the velocity of $P$ is $( 10 \mathbf { i } - 4 \mathbf { j } ) \mathrm { kmh } ^ { - 1 }$ At time $t$ hours after 12:00, the position vector of $P$ is $\mathbf { p } \mathrm { km }$.
Find an expression for $\mathbf { p }$ in terms of $\mathbf { i } , \mathbf { j }$ and $t$. A second ship $Q$ is also travelling at a constant velocity.
At time $t$ hours after 12:00, the position vector of $Q$ is given by $\mathbf { q } \mathrm { km }$, where $$\mathbf { q } = ( 42 - 8 t ) \mathbf { i } + ( 9 + 14 t ) \mathbf { j }$$ Ships $P$ and $Q$ are modelled as particles.
If both ships maintained their course,
1. verify that they would collide at 13:30
2. find the position vector of the point at which the collision would occur. At 12:30 $Q$ changes speed and direction to avoid the collision.
  Ship $Q$ now travels due north with a constant speed of $15 \mathrm { kmh } ^ { - 1 }$
  Ship $P$ maintains the same constant velocity throughout.
Find the exact distance between $P$ and $Q$ at 14:30

Edexcel M1 2023 October Q7

13 marks Standard +0.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{017cc2b0-9ec3-45ff-94c0-9d989badfd5d-24_339_942_244_635} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a block $A$ of mass $m$ held at rest on a rough plane.
The plane is inclined at an angle $\alpha$ to the horizontal and the coefficient of friction between the block and the plane is $\mu$. One end of a light inextensible string is now attached to $A$. The string passes over a small smooth pulley which is fixed at the top of the plane.
The other end of the string is attached to a block $B$ of mass $k m$.
Block $B$ hangs vertically below the pulley, with the string taut.
The string from $A$ to the pulley lies along a line of greatest slope of the plane.
Both $A$ and $B$ are modelled as particles.
When the system is released from rest, $A$ moves up the plane and the tension in the string is $\frac { 4 m g } { 3 }$ Given that $\mu = \frac { 1 } { 3 }$ and $\tan \alpha = \frac { 7 } { 24 }$

1. find the magnitude of the acceleration of $A$, giving your answer in terms of $g$,
2. find the value of $k$.
Find the magnitude of the resultant force exerted on the pulley by the string, giving your answer in terms of $m$ and $g$.

Edexcel M1 2018 Specimen Q2

6 marks Moderate -0.3

2. Two particles $P$ and $Q$ are moving in opposite directions along the same horizontal straight line. Particle $P$ has mass $m$ and particle $Q$ has mass $k m$. The particles collide directly. Immediately before the collision, the speed of $P$ is $u$ and the speed of $Q$ is $2 u$. As a result of the collision, the direction of motion of each particle is reversed and the speed of each particle is halved.

Find the value of $k$.
Find, in terms of $m$ and $u$ only, the magnitude of the impulse exerted on $Q$ by $P$ in the collision.

Edexcel M1 2018 Specimen Q3

10 marks Moderate -0.3

3. A block $A$ of mass 9 kg is released from rest from a point $P$ which is a height $h$ metres above horizontal soft ground. The block falls and strikes another block $B$ of mass 1.5 kg which is on the ground vertically below $P$. The speed of $A$ immediately before it strikes $B$ is $7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The blocks are modelled as particles.

Find the value of $h$. Immediately after the impact the blocks move downwards together with the same speed and both come to rest after sinking a vertical distance of 12 cm into the ground. Assuming that the resistance offered by the ground has constant magnitude $R$ newtons,
find the value of $R$.
\includegraphics[max width=\textwidth, alt={}, center]{6ab8838f-d6f8-4761-8def-1022d97d4e82-07_2252_51_315_36}

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\includegraphics[max width=\textwidth, alt={}, center]{6ab8838f-d6f8-4761-8def-1022d97d4e82-09_2249_45_318_37}

Edexcel M1 2018 Specimen Q4

10 marks Moderate -0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6ab8838f-d6f8-4761-8def-1022d97d4e82-10_238_1161_267_388} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A diving board $A B$ consists of a wooden plank of length 4 m and mass 30 kg . The plank is held at rest in a horizontal position by two supports at the points $A$ and $C$, where $A C = 0.6 \mathrm {~m}$, as shown in Figure 1. The force on the plank at $A$ acts vertically downwards and the force on the plank at $C$ acts vertically upwards. A diver of mass 50 kg is standing on the board at the end $B$. The diver is modelled as a particle and the plank is modelled as a uniform rod. The plank is in equilibrium.

Find
1. the magnitude of the force acting on the plank at $A$,
2. the magnitude of the force acting on the plank at $C$. The support at $A$ will break if subjected to a force whose magnitude is greater than 5000 N .
Find, in kg, the greatest integer mass of a diver who can stand on the board at $B$ without breaking the support at $A$.
Explain how you have used the fact that the diver is modelled as a particle.

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Edexcel M1 2018 Specimen Q5

10 marks Moderate -0.8

Two forces, $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$, act on a particle $A$.
$\mathbf { F } _ { 1 } = ( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { N }$ and $\mathbf { F } _ { 2 } = ( p \mathbf { i } + q \mathbf { j } ) \mathrm { N }$, where $p$ and $q$ are constants.
Given that the resultant of $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ is parallel to ( $\mathbf { i } + 2 \mathbf { j }$ ),
1. show that $2 p - q + 7 = 0$
Given that $q = 11$ and that the mass of $A$ is 2 kg , and that $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ are the only forces acting on $A$,
find the magnitude of the acceleration of $A$.
\includegraphics[max width=\textwidth, alt={}, center]{6ab8838f-d6f8-4761-8def-1022d97d4e82-15_2255_51_314_36}

Edexcel M1 2018 Specimen Q6

17 marks Moderate -0.8

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6ab8838f-d6f8-4761-8def-1022d97d4e82-16_264_997_269_461} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Two cars, $A$ and $B$, move on parallel straight horizontal tracks. Initially $A$ and $B$ are both at rest with $A$ at the point $P$ and $B$ at the point $Q$, as shown in Figure 2. At time $t = 0$ seconds, $A$ starts to move with constant acceleration $a \mathrm {~m} \mathrm {~s} ^ { - 2 }$ for 3.5 s , reaching a speed of $14 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Car $A$ then moves with constant speed $14 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Find the value of $a$. Car $B$ also starts to move at time $t = 0$ seconds, in the same direction as car $A$. Car $B$ moves with a constant acceleration of $3 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. At time $t = T$ seconds, $B$ overtakes $A$. At this instant $A$ is moving with constant speed.
On a diagram, sketch, on the same axes, a speed-time graph for the motion of $A$ for the interval $0 \leqslant t \leqslant T$ and a speed-time graph for the motion of $B$ for the interval $0 \leqslant t \leqslant T$.
Find the value of $T$.
Find the distance of car $B$ from the point $Q$ when $B$ overtakes $A$.
On a new diagram, sketch, on the same axes, an acceleration-time graph for the motion of $A$ for the interval $0 \leqslant t \leqslant T$ and an acceleration-time graph for the motion of $B$ for the interval $0 \leqslant t \leqslant T$. $\_\_\_\_$ VAYV SIHI NI JIIIM ION OC
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Edexcel M1 2018 Specimen Q7

15 marks Standard +0.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6ab8838f-d6f8-4761-8def-1022d97d4e82-20_568_1045_264_461} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A particle $P$ of mass 4 kg is attached to one end of a light inextensible string. A particle $Q$ of mass $m \mathrm {~kg}$ is attached to the other end of the string. The string passes over a small smooth pulley which is fixed at a point on the intersection of two fixed inclined planes. The string lies in a vertical plane that contains a line of greatest slope of each of the two inclined planes. The first plane is inclined to the horizontal at an angle $\alpha$, where $\tan \alpha = \frac { 3 } { 4 }$ and the second plane is inclined to the horizontal at an angle $\beta$, where $\tan \beta = \frac { 4 } { 3 }$. Particle $P$ is on the first plane and particle $Q$ is on the second plane with the string taut, as shown in Figure 3. The first plane is rough and the coefficient of friction between $P$ and the plane is $\frac { 1 } { 4 }$. The second plane is smooth. The system is in limiting equilibrium. Given that $P$ is on the point of slipping down the first plane,

find the value of $m$,
find the magnitude of the force exerted on the pulley by the string,
find the direction of the force exerted on the pulley by the string. \includegraphics[max width=\textwidth, alt={}, center]{6ab8838f-d6f8-4761-8def-1022d97d4e82-21_2258_50_314_37}

VIAV SIHI NI BIIIM ION OC VGHV SIHI NI GHIYM ION OC VJ4V SIHI NI JIIYM ION OC
\includegraphics[max width=\textwidth, alt={}, center]{6ab8838f-d6f8-4761-8def-1022d97d4e82-23_2258_50_314_37}

\includegraphics[max width=\textwidth, alt={}]{6ab8838f-d6f8-4761-8def-1022d97d4e82-24_2655_1830_105_121}

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