Questions - Edexcel M1 - A-Level Maths

Find the speed and direction of $C$ after the collision.
Find, in Ns, the magnitude of the impulse exerted by $B$ on $A$ in the collision.
State a modelling assumption which you have made about the trucks in your solution Immediately after the collision, a constant braking force of magnitude 250 N is applied to $C$. It comes to rest in a distance $d$ metres.
Find the value of $d$.
(4)

Edexcel M1 Specimen Q6

6. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{e590030f-0c46-42ab-80b8-3627d3c36908-5_345_1255_1329_265}

\end{figure} A particle of mass $m$ rests on a rough plane inclined at an angle $\alpha$ to the horizontal, where $\tan \alpha = \frac { 3 } { 4 }$. The particle is attached to one end of a light inextensible string which lies in a line of greatest slope of the plane and passes over a small light smooth pulley $P$ fixed at the top of the plane. The other end of the string is attached to a particle $B$ of mass $3 m$, and $B$ hangs freely below $P$, as shown in Fig. 4. The particles are released from rest with the string taut. The particle $B$ moves down with acceleration of magnitude $\frac { 1 } { 2 } g$. Find

the tension in the string,
the coefficient of friction between $A$ and the plane.

Edexcel M1 Specimen Q7

7. Two cars $A$ and $B$ are moving on straight horizontal roads with constant velocities. The velocity of $A$ is $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ due east, and the velocity of $B$ is $( 10 \mathbf { i } + 10 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$, where $\mathbf { i }$ and $\mathbf { j }$ are unit vectors directed due east and due north respectively. Initially $A$ is at the fixed origin $O$, and the position vector of $B$ is $300 \mathbf { i }$ m relative to $O$. At time $t$ seconds, the position vectors of $A$ and $B$ are $\mathbf { r }$ metres and $\mathbf { s }$ metres respectively.

Find expressions for $\mathbf { r }$ and $\mathbf { s }$ in terms of $t$.
Hence write down an expression for $\overrightarrow { A B }$ in terms of $t$.
Find the time when the bearing of $B$ from $A$ is $045 ^ { \circ }$.
Find the time when the cars are again 300 m apart. END

Edexcel M1 2014 January Q1

A truck $P$ of mass $2 M$ is moving with speed $U$ on smooth straight horizontal rails. It collides directly with another truck $Q$ of mass $3 M$ which is moving with speed $4 U$ in the opposite direction on the same rails. The trucks join so that immediately after the collision they move together. By modelling the trucks as particles, find
1. the speed of the trucks immediately after the collision,
2. the magnitude of the impulse exerted on $P$ by $Q$ in the collision.

Edexcel M1 2014 January Q2

2. A particle $P$ is moving with constant velocity ( $2 \mathbf { i } - 3 \mathbf { j }$ ) $\mathrm { m } \mathrm { s } ^ { - 1 }$.

Find the speed of $P$. The particle $P$ passes through the point $A$ and 4 seconds later passes through the point with position vector ( $\mathbf { i } - 4 \mathbf { j }$ ) m.
Find the position vector of $A$.

Edexcel M1 2014 January Q3

3. A beam $A B$ has length 15 m and mass 25 kg . The beam is smoothly supported at the point $P$, where $A P = 8 \mathrm {~m}$. A man of mass 100 kg stands on the beam at a distance of 2 m from $A$ and another man stands on the beam at a distance of 1 m from $B$. The beam is modelled as a non-uniform rod and the men are modelled as particles. The beam is in equilibrium in a horizontal position with the reaction on the beam at $P$ having magnitude 2009 N. Find the distance of the centre of mass of the beam from $A$.

Edexcel M1 2014 January Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{fade35da-8dca-4d98-a07c-ed3a173fccda-08_396_483_214_735} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A fixed rough plane is inclined to the horizontal at an angle $\alpha$, where $\tan \alpha = \frac { 3 } { 4 }$ A small box of mass $m$ is at rest on the plane. A force of magnitude $k m g$, where $k$ is a constant, is applied to the box. The line of action of the force is at angle $\alpha$ to the line of greatest slope of the plane through the box, as shown in Figure 1, and lies in the same vertical plane as this line of greatest slope. The coefficient of friction between the box and the plane is $\mu$. The box is on the point of slipping up the plane. By modelling the box as a particle, find $k$ in terms of $\mu$.

Edexcel M1 2014 January Q5

5. A racing car is moving along a straight horizontal track with constant acceleration. There are three checkpoints, $P , Q$ and $R$, on the track, where $P Q = 48 \mathrm {~m}$ and $Q R = 200 \mathrm {~m}$. The car takes 3 s to travel from $P$ to $Q$ and 5 s to travel from $Q$ to $R$. Find

the acceleration of the car,
the speed of the car as it passes $P$.

Edexcel M1 2014 January Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{fade35da-8dca-4d98-a07c-ed3a173fccda-16_398_860_210_543} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Two particles $P$ and $Q$ have masses 0.1 kg and 0.5 kg respectively. The particles are attached to the ends of a light inextensible string. Particle $P$ is held at rest on a rough horizontal table. The string lies along the table and passes over a small smooth pulley which is fixed to the edge of the table. Particle $Q$ is at rest on a smooth plane which is inclined to the horizontal at an angle $\theta$, where $\tan \theta = \frac { 4 } { 3 }$ The string lies in the vertical plane which contains the pulley and a line of greatest slope of the inclined plane, as shown in Figure 2. Particle $P$ is released from rest with the string taut. During the first 0.5 s of the motion $P$ does not reach the pulley and $Q$ moves 0.75 m down the plane.

Find the tension in the string during the first 0.5 s of the motion.
Find the coefficient of friction between $P$ and the table.
\includegraphics[max width=\textwidth, alt={}, center]{fade35da-8dca-4d98-a07c-ed3a173fccda-19_72_59_2613_1886}

Edexcel M1 2014 January Q7

7. A force $\mathbf { F }$ is given by $\mathbf { F } = ( 9 \mathbf { i } + 13 \mathbf { j } )$ N.

Find the size of the angle between the direction of $\mathbf { F }$ and the vector $\mathbf { j }$. The force $\mathbf { F }$ is the resultant of two forces $\mathbf { P }$ and $\mathbf { Q }$. The line of action of $\mathbf { P }$ is parallel to the vector ( $2 \mathbf { i } - \mathbf { j }$ ). The line of action of $\mathbf { Q }$ is parallel to the vector ( $\mathbf { i } + 3 \mathbf { j }$ ).
Find, in terms of $\mathbf { i }$ and $\mathbf { j }$,
1. the force $\mathbf { P }$,
2. the force $\mathbf { Q }$.

Edexcel M1 2014 January Q8

8. Two trains, $A$ and $B$, start together from rest, at time $t = 0$, at a station and move along parallel straight horizontal tracks. Both trains come to rest at the next station after 180 s . Train $A$ moves with constant acceleration $\frac { 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 2 }$ for 30 s , then moves at constant speed for 120 s and then moves with constant deceleration for the final 30 s . Train $B$ moves with constant acceleration for 90 s and then moves with constant deceleration for the final 90 s .

Sketch, on the same axes, the speed-time graphs for the motion of the two trains between the two stations.
Find the acceleration of train $B$ for the first half of its journey.
Find the times when the two trains are moving at the same speed.
Find the distance between the trains 96 s after they start.
\includegraphics[max width=\textwidth, alt={}, center]{fade35da-8dca-4d98-a07c-ed3a173fccda-28_43_58_2457_1893}

Edexcel M1 2015 January Q1

A railway truck $A$ of mass $m$ and a second railway truck $B$ of mass $4 m$ are moving in opposite directions on a smooth straight horizontal track when they collide directly. Immediately before the collision the speed of truck $A$ is $3 u$ and the speed of truck $B$ is $2 u$. In the collision the trucks join together. Modelling the trucks as particles, find
1. the speed of $A$ immediately after the collision,
2. the direction of motion of $A$ immediately after the collision,
3. the magnitude of the impulse exerted by $A$ on $B$ in the collision.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aaa8b297-347c-4a9b-a2c2-c4bd70d56912-03_534_1065_118_445} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A block of mass 50 kg lies on a rough plane which is inclined to the horizontal at an angle $\alpha$, where $\tan \alpha = \frac { 7 } { 24 }$. The block is held at rest by a vertical rope, as shown in Figure 1, and is on the point of sliding down the plane. The block is modelled as a particle and the rope is modelled as a light inextensible string. Given that the friction force acting on the block has magnitude 65.8 N, find
the tension in the rope,
the coefficient of friction between the block and the plane.

Edexcel M1 2015 January Q3

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

A particle $P$ is moving with constant velocity $( - 6 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. At time $t = 0 , P$ passes through the point with position vector $( 21 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }$, relative to a fixed origin $O$.

Find the direction of motion of $P$, giving your answer as a bearing to the nearest degree.
Write down the position vector of $P$ at time $t$ seconds.
Find the time at which $P$ is north-west of $O$.

Edexcel M1 2015 January Q4

4. The points $P$ and $Q$ are at the same height $h$ metres above horizontal ground. A small stone is dropped from rest from $P$. Half a second later a second small stone is thrown vertically downwards from $Q$ with speed $7.35 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Given that the stones hit the ground at the same time, find the value of $h$.

Edexcel M1 2015 January Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{aaa8b297-347c-4a9b-a2c2-c4bd70d56912-07_237_657_264_703} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A particle $P$ of mass 2 kg is pushed up a line of greatest slope of a rough plane by a horizontal force of magnitude $X$ newtons, as shown in Figure 2. The force acts in the vertical plane which contains $P$ and a line of greatest slope of the plane. The plane is inclined to the horizontal at an angle $\alpha$, where $\tan \alpha = \frac { 3 } { 4 }$
The coefficient of friction between $P$ and the plane is 0.5
Given that the acceleration of $P$ is $1.45 \mathrm {~m} \mathrm {~s} ^ { - 2 }$, find the value of $X$.

Edexcel M1 2015 January Q6

6. A uniform rod $A C$, of weight $W$ and length $3 l$, rests horizontally on two supports, one at $A$ and one at $B$, where $A B = 2 l$. A particle of weight $2 W$ is placed on the rod at a distance $x$ from $A$. The rod remains horizontal and in equilibrium.

Find the greatest possible value of $x$. The magnitude of the reaction of the support at $A$ is $R$. Due to a weakness in the support at $A$, the greatest possible value of $R$ is $2 W$,
find the least possible value of $x$.

Edexcel M1 2015 January Q7

7. A train travels along a straight horizontal track between two stations $A$ and $B$. The train starts from rest at $A$ and moves with constant acceleration until it reaches its maximum speed of $108 \mathrm {~km} \mathrm {~h} ^ { - 1 }$. The train then travels at this speed before it moves with constant deceleration coming to rest at $B$. The journey from $A$ to $B$ takes 8 minutes.

Change $108 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ into $\mathrm { m } \mathrm { s } ^ { - 1 }$.
Sketch a speed-time graph for the motion of the train between the two stations $A$ and $B$. Given that the distance between the two stations is 12 km and that the time spent decelerating is three times the time spent accelerating,
find the acceleration, in $\mathrm { m } \mathrm { s } ^ { - 2 }$, of the train.

Edexcel M1 2015 January Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{aaa8b297-347c-4a9b-a2c2-c4bd70d56912-13_668_901_262_566} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A particle $A$ of mass $3 m$ is held at rest on a rough horizontal table. The particle is attached to one end of a light inextensible string. The string passes over a small smooth pulley $P$ which is fixed at the edge of the table. The other end of the string is attached to a particle $B$ of mass $2 m$, which hangs freely, vertically below $P$. The system is released from rest, with the string taut, when $A$ is 1.3 m from $P$ and $B$ is 1 m above the horizontal floor, as shown in Figure 3. Given that $B$ hits the floor 2 s after release and does not rebound,

find the acceleration of $A$ during the first two seconds,
find the coefficient of friction between $A$ and the table,
determine whether $A$ reaches the pulley.
\includegraphics[max width=\textwidth, alt={}, center]{aaa8b297-347c-4a9b-a2c2-c4bd70d56912-14_106_59_2478_1833}

Edexcel M1 2016 January Q1

A truck of mass 2400 kg is pulling a trailer of mass $M \mathrm {~kg}$ along a straight horizontal road. The tow bar, connecting the truck to the trailer, is horizontal and parallel to the direction of motion. The tow bar is modelled as being light and inextensible. The resistance forces acting on the truck and the trailer are constant and of magnitude 400 N and 200 N respectively. The acceleration of the truck is $0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ and the tension in the tow bar is 600 N.
1. Find the magnitude of the driving force of the truck.
2. Find the value of $M$.
3. Explain how you have used the fact that the tow bar is inextensible in your calculations.

Edexcel M1 2016 January Q2

Two particles $P$ and $Q$ are moving in opposite directions along the same horizontal straight line. Particle $P$ is moving due east and particle $Q$ is moving due west. Particle $P$ has mass $2 m$ and particle $Q$ has mass $3 m$. The particles collide directly. Immediately before the collision, the speed of $P$ is $4 u$ and the speed of $Q$ is $u$. The magnitude of the impulse in the collision is $\frac { 33 } { 5 } m u$.
1. Find the speed and direction of motion of $P$ immediately after the collision.
2. Find the speed and direction of motion of $Q$ immediately after the collision.
  VILIV SIMI NI III IM I ON OC
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\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{054e11cb-9416-40c0-9dde-8d12818bab3f-04_268_862_123_543} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A boy is pulling a sledge of mass 8 kg in a straight line at a constant speed across rough horizontal ground by means of a rope. The rope is inclined at $30 ^ { \circ }$ to the ground, as shown in Figure 1. The coefficient of friction between the sledge and the ground is $\frac { 1 } { 5 }$. By modelling the sledge as a particle and the rope as a light inextensible string, find the tension in the rope.

Edexcel M1 2016 January Q4

4. A small stone is projected vertically upwards from the point $O$ and moves freely under gravity. The point $A$ is 3.6 m vertically above $O$. When the stone first reaches $A$, the stone is moving upwards with speed $11.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The stone is modelled as a particle.

Find the maximum height above $O$ reached by the stone.
Find the total time between the instant when the stone was projected from $O$ and the instant when it returns to $O$.
Sketch a velocity-time graph to represent the motion of the stone from the instant when it passes through $A$ moving upwards to the instant when it returns to $O$. Show, on the axes, the coordinates of the points where your graph meets the axes.

Edexcel M1 2016 January Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{054e11cb-9416-40c0-9dde-8d12818bab3f-07_531_1182_114_383} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A non-uniform rod $A B$ has length 4 m and weight 120 N . The centre of mass of the rod is at the point $G$ where $A G = 2.2 \mathrm {~m}$. The rod is suspended in a horizontal position by two vertical light inextensible strings, one at each end, as shown in Figure 2. A particle of weight 40 N is placed on the rod at the point $P$, where $A P = x$ metres. The rod remains horizontal and in equilibrium.

Find, in terms of $x$,
1. the tension in the string at $A$,
2. the tension in the string at $B$. Either string will break if the tension in it exceeds 84 N .
Find the range of possible values of $x$.

Edexcel M1 2016 January Q6

6. [In this question $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin.] At 2 pm , the position vector of ship $P$ is $( 5 \mathbf { i } - 3 \mathbf { j } ) \mathrm { km }$ and the position vector of ship $Q$ is $( 7 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }$.

Find the distance between $P$ and $Q$ at 2 pm . Ship $P$ is moving with constant velocity $( 2 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$ and ship $Q$ is moving with constant velocity $( - 3 \mathbf { i } - 15 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$.
Find the position vector of $P$ at time $t$ hours after 2 pm .
Find the position vector of $Q$ at time $t$ hours after 2 pm .
Show that $Q$ will meet $P$ and find the time at which they meet.
Find the position vector of the point at which they meet.

Edexcel M1 2016 January Q7

7. $$P ( 2 \mathrm {~kg} )$$ \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{054e11cb-9416-40c0-9dde-8d12818bab3f-11_529_899_269_525} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A particle $P$ of mass 2 kg is attached to one end of a light inextensible string. A particle $Q$ of mass 5 kg is attached to the other end of the string. The string passes over a small smooth light pulley. The pulley is fixed at a point on the intersection of a rough horizontal table and a fixed smooth inclined plane. The string lies along the table and also lies in a vertical plane which contains a line of greatest slope of the inclined plane. This plane is inclined to the horizontal at an angle $\alpha$, where $\tan \alpha = \frac { 3 } { 4 }$. Particle $P$ is at rest on the table, a distance $d$ metres from the pulley. Particle $Q$ is on the inclined plane with the string taut, as shown in Figure 3. The coefficient of friction between $P$ and the table is $\frac { 1 } { 4 }$. The system is released from rest and $P$ slides along the table towards the pulley.
Assuming that $P$ has not reached the pulley and that $Q$ remains on the inclined plane,

write down an equation of motion for $P$,
write down an equation of motion for $Q$,
1. find the acceleration of $P$,
2. find the tension in the string. When $P$ has moved a distance 0.5 m from its initial position, the string breaks. Given that $P$ comes to rest just as it reaches the pulley,
find the value of $d$.

Edexcel M1 2017 January Q1

A train moves along a straight horizontal track between two stations $R$ and $S$. Initially the train is at rest at $R$. The train accelerates uniformly at $\frac { 1 } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 }$ from rest at $R$ until it is moving with speed $15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. For the next 200 seconds the train maintains a constant speed of $15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The train then decelerates uniformly at $\frac { 1 } { 4 } \mathrm {~m} \mathrm {~s} ^ { - 2 }$ until it comes to rest at $S$.

Find

the time taken by the train to travel from $R$ to $S$,
the distance from $R$ to $S$,
the average speed of the train during the journey from $R$ to $S$.

Questions — Edexcel M1 (599 questions)

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