Questions - Edexcel M1 - A-Level Maths

Show that the velocity vector of $S$ is $( 20 \mathbf { i } - 48 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$ and find the velocity vector of $T$.
Find expressions for the position vectors of $S$ and $T$ at time $t$ minutes after 9.00 a.m.
Show that the bearing of $T$ from $S$ remains constant during the motion, and find this bearing.
Show that if the trains continue at the given speeds they will collide.

Edexcel M1 Q1

\begin{enumerate} \item Two particles, $P$ and $Q$, of mass 2 kg and 1.5 kg respectively are at rest on a smooth, horizontal surface. They are connected by a light, inelastic string which is initially slack. Particle $P$ is projected away from $Q$ with a speed of $7 \mathrm {~ms} ^ { - 1 }$.

Find the common speed of the particles after the string becomes taut.
Calculate the impulse in the string when it jerks tight. \item Particle $A$ has velocity $( 8 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and particle $B$ has velocity $( 15 \mathbf { i } - 8 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$ where $\mathbf { i }$ and $\mathbf { j }$ are perpendicular, horizontal unit vectors.

Edexcel M1 Q4

4. A cyclist and her bicycle have a combined mass of 78 kg . While riding on level ground and using her greatest driving force, she is able to accelerate uniformly from rest to $10 \mathrm {~ms} ^ { - 1 }$ in 15 seconds against constant resistive forces that total 60 N .

Show that her maximum driving force is 112 N . The cyclist begins to ascend a hill, inclined at an angle $\alpha$ to the horizontal, riding with her maximum driving force and against the same resistive forces. In this case, she is able to maintain a steady speed.
Find the angle $\alpha$, giving your answer to the nearest degree.
Comment on the assumption that the resistive force remains constant
1. in the case when the cyclist is accelerating,
2. in the case when she is maintaining a steady speed.
  (2 marks)

Edexcel M1 Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2108a1be-0214-42c4-9cb4-8622cc0fa496-3_318_832_1165_452} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} Figure 2 shows a large block of mass 50 kg being pulled on rough horizontal ground by means of a rope attached to the block. The tension in the rope is 200 N and it makes an angle of $40 ^ { \circ }$ with the horizontal. Under these conditions, the block is on the point of moving. Modelling the block as a particle,

show that the coefficient of friction between the block and the ground is 0.424 correct to 3 significant figures.
(6 marks)
The angle with the horizontal at which the rope is being pulled is reduced to $30 ^ { \circ }$. Ignoring air resistance and assuming that the tension in the rope and the coefficient of friction remain unchanged,
find the acceleration of the block.
(6 marks) Turn over

Edexcel M1 Q6

6. Anila is practising catching tennis balls. She uses a mobile computer-controlled machine which fires tennis balls vertically upwards from a height of 2.5 metres above the ground. Once it has fired a ball, the machine is programmed to move position rapidly to allow Anila time to get into a suitable position to catch the ball. The machine fires a ball at $24 \mathrm {~ms} ^ { - 1 }$ vertically upwards and Anila catches the ball just before it touches the ground.

Draw a speed-time graph for the motion of the ball from the time it is fired by the machine to the instant before Anila catches it.
Find, to the nearest centimetre, the maximum height which the ball reaches above the ground.
Calculate the speed at which the ball is travelling when Anila catches it.
Calculate the length of time that the ball is in the air.

Edexcel M1 Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2108a1be-0214-42c4-9cb4-8622cc0fa496-5_531_1061_283_468} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} Figure 3 shows a particle $X$ of mass 3 kg on a smooth plane inclined at an angle $30 ^ { \circ }$ to the horizontal, and a particle $Y$ of mass 2 kg on a smooth plane inclined at an angle $60 ^ { \circ }$ to the horizontal. The two particles are connected by a light, inextensible string of length 2.5 metres passing over a smooth pulley at $C$ which is the highest point of the two planes. Initially, $Y$ is at a point just below $C$ touching the pulley with the string taut. When the particles are released from rest they travel along the lines of greatest slope, $A C$ in the case of $X$ and $B C$ in the case of $Y$, of their respective planes. $A$ and $B$ are the points where the planes meet the horizontal ground and $A B = 4$ metres.

Show that the initial acceleration of the system is given by $\frac { g } { 10 } ( 2 \sqrt { 3 } - 3 ) \mathrm { ms } ^ { - 2 }$.
By finding the tension in the string, or otherwise, find the magnitude of the force exerted on the pulley and the angle that this force makes with the vertical.
Find, correct to 2 decimal places, the speed with which $Y$ hits the ground.

Edexcel M1 Q1

A constant force, $\mathbf { F }$, acts on a particle, $P$, of mass 5 kg causing its velocity to change from $\left( { } ^ { - } 2 \mathbf { i } + \mathbf { j } \right) \mathrm { ms } ^ { - 1 }$ to $( 4 \mathbf { i } - 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$ in 2 seconds.
1. Find, in the form $a \mathbf { i } + b \mathbf { j }$, the acceleration of $P$.
2. Show that the magnitude of $\mathbf { F }$ is 25 N and find, to the nearest degree, the acute angle between the line of action of $\mathbf { F }$ and the vector $\mathbf { j }$.
  (5 marks)
3. A particle $A$ of mass $3 m$ is moving along a straight line with constant speed $u \mathrm {~ms} ^ { - 1 }$. It collides with a particle $B$ of mass $2 m$ moving at the same speed but in the opposite direction. As a result of the collision, $A$ is brought to rest.
4. Show that, after the collision, $B$ has changed its direction of motion and that its speed has been halved.
Given that the magnitude of the impulse exerted by $A$ on $B$ is $9 m \mathrm { Ns }$,
find the value of $u$.

Edexcel M1 Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e83297d6-d90c-42be-b67a-38ba2a495cb7-2_288_798_1318_525} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} Figure 1 shows two window cleaners, Alan and Baber, of mass 60 kg and 100 kg respectively standing on a platform $P Q$ of length 3 metres and mass 20 kg . The platform is suspended by two vertical cables attached to the ends $P$ and $Q$. Alan is standing at the point $A , 1.25$ metres from $P$, Baber is standing at the point $B$ and the tension in the cable at $P$ is twice the tension in the cable at $Q$. Modelling the platform as a uniform rod and Alan and Baber as particles,

find the tension in the cable at $P$,
find the distance $B P$.
State how you have used the modelling assumptions that
1. the platform is uniform,
2. the platform is a rod.
  (2 marks)

Edexcel M1 Q4

4. A sports car is being driven along a straight test track. It passes the point $O$ at time $t = 0$ at which time it begins to decelerate uniformly. The car passes the points $L$ and $M$ at times $t = 1$ and $t = 4$ respectively. Given that $O L$ is 54 m and $L M$ is 90 m ,

find the rate of deceleration of the car. The car subsequently comes to rest at $N$.
Find the distance $M N$.
(4 marks)

Edexcel M1 Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e83297d6-d90c-42be-b67a-38ba2a495cb7-3_360_620_822_502} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} A particle $P$, of mass 2 kg , lies on a rough plane inclined at an angle of $30 ^ { \circ }$ to the horizontal. A force $H$, whose line of action is parallel to the line of greatest slope of the plane, is applied to the particle as shown in Figure 2. The coefficient of friction between the particle and the plane is $\frac { 1 } { \sqrt { 3 } }$. Given that the particle is on the point of moving up the plane,

draw a diagram showing all the forces acting on the particle,
show that the ratio of the magnitude of the frictional force to the magnitude of $H$ is equal to $1 : 2$ The force $H$ is now removed but $P$ remains at rest.
Use the principle of friction to explain how this is possible.

Edexcel M1 Q6

6. A car of mass 1.25 tonnes tows a caravan of mass 0.75 tonnes along a straight, level road. The total resistance to motion experienced by the car and the caravan is 1200 N . The car and caravan accelerate uniformly from rest to $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in 20 seconds.

Calculate the driving force produced by the car's engine. Given that the resistance to motion experienced by the car and by the caravan are in the same ratio as their masses,
find these resistances and the tension in the towbar. When the car and caravan are travelling at a steady speed of $25 \mathrm {~ms} ^ { - 1 }$, the towbar snaps. Assuming that the caravan experiences the same resistive force as before,
calculate the distance travelled by the caravan before it comes to rest,
give a reason why your answer to (c) may be unrealistic.

Edexcel M1 Q7

7. Two ramblers, Alison and Bill, are out walking. At midday, Alison is at the fixed origin $O$, and Bill is at the point with position vector $\left( { } ^ { - } 5 \mathbf { i } + 12 \mathbf { j } \right) \mathrm { km }$ relative to $O$, where $\mathbf { i }$ and $\mathbf { j }$ are perpendicular, horizontal unit vectors. They are both walking with constant velocity - Alison at $( 2 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$, and Bill at a speed of $2 \sqrt { } 10 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ in a direction parallel to the vector ( $3 \mathbf { i } + \mathbf { j }$ ).

Find the distance between the two ramblers at midday.
Show that the velocity of Bill is $( 6 \mathbf { i } + 2 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$.
Show that, at time $t$ hours after midday, the position vector of Bill relative to Alison is $$[ ( 4 t - 5 ) \mathbf { i } + ( 12 - 3 t ) \mathbf { j } ] \mathrm { km } .$$
Show that the distance, $d \mathrm {~km}$, between the two ramblers is given by $$d ^ { 2 } = 25 t ^ { 2 } - 112 t + 169$$
Using your answer to part (d), find the length of time to the nearest minute for which the distance between the Alison and Bill is less than 11 km .

Edexcel M1 Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8a0ff401-83da-4539-a9e9-68736c57df2a-2_520_1278_207_333} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} Figure 1 shows a distance-time graph for a car journey from Birmingham to Newquay which included a stop for lunch at a service station near Exeter. During the first part of the journey three-quarters of the total distance, $d$, was covered in 3 hours. After a 1 hour stop, the remaining distance was completed in 2 hours.

Calculate, in the form $k : 1$, the ratio of the average speed during the first 3 hours of the journey to the average speed during the last 2 hours of the journey.
(4 marks)
Given that the average speed of the car over the whole journey (excluding the stop) was $80 \mathrm { kmh } ^ { - 1 }$,
find the average speed of the car on the first part of the journey.
(4 marks)

Edexcel M1 Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8a0ff401-83da-4539-a9e9-68736c57df2a-2_291_613_1599_516} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} Figure 2 shows a washing line suspended at either end by vertical rigid poles. A jacket of mass 0.7 kg is suspended in equilibrium part of the way along the line. The sections of the washing line on either side of the jacket make angles of $35 ^ { \circ }$ and $40 ^ { \circ }$ with the horizontal.

Find the tension in the washing line on each side of the jacket.
Explain why, in practice, the angles are likely to be very similar in value.

Edexcel M1 Q3

3. In a simple model for the motion of a car, its velocity, $\mathbf { v }$, at time $t$ seconds, is given by $$\mathbf { v } = \left( 3 t ^ { 2 } - 2 t + 8 \right) \mathbf { i } + ( 5 t + 6 ) \mathbf { j } \mathrm { ms } ^ { - 1 }$$

Calculate the speed of the car when $t = 0$.
Find the values of $t$ for which the velocity of the car is parallel to the vector $( \mathbf { i } + \mathbf { j } )$.
Why would this model not be appropriate for large values of $t$ ?

Edexcel M1 Q4

4. The force $\mathbf { F } _ { \mathbf { 1 } } = ( 5 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }$ acts at the point $A$ on a lamina where the position vector of $A$, relative to a fixed origin $O$, is $( 3 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }$.

Calculate the magnitude and the sense of the moment of the force about $O$. Another force $\mathbf { F } _ { 2 } = ( p \mathbf { i } + q \mathbf { j } )$, acts at the point $B$ with position vector ( ${ } ^ { - } \mathbf { i } + 4 \mathbf { j }$ ) m so that the resultant moment of the two forces, $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$, about $O$ is zero. Given also that the moment of $\mathbf { F } _ { 2 }$ about $A$ is 34 Ns in a clockwise sense,
find the values of $p$ and $q$.

Edexcel M1 Q5

5. A car and a motorbike are at rest adjacent to one another at a set of traffic lights on a long, straight stretch of road. They set off simultaneously at time $t = 0$. The motorcyclist accelerates uniformly at $6 \mathrm {~ms} ^ { - 2 }$ until he reaches a speed of $30 \mathrm {~ms} ^ { - 1 }$ which he then maintains. The car driver accelerates uniformly for 9 seconds until she reaches $36 \mathrm {~ms} ^ { - 1 }$ and then remains at this speed.

Find the acceleration of the car.
Draw on the same diagram speed-time graphs to illustrate the movements of both vehicles.
Find the value of $t$ when the car again draws level with the motorcyclist.

Edexcel M1 Q6

6. Corinne and her brother Dermot are lifted by their parents onto the two ends of a rope which is slung over a large, horizontal branch. When their parents let go of them Dermot, whose mass is 54 kg , begins to descend with an acceleration of $1 \mathrm {~ms} ^ { - 2 }$. By modelling the children as a pair of particles connected by a light inextensible string, and the branch as a smooth pulley,

show that Corinne's mass is 44 kg ,
calculate the tension in the rope,
find the force on the branch. In a more sophisticated model, the branch is assumed to be rough.
Explain what effect this would have on the initial acceleration of the children.
(1 mark)

Edexcel M1 Q7

7. Two particles $A$ and $B$, of mass $3 M \mathrm {~kg}$ and $2 M \mathrm {~kg}$ respectively, are moving towards each other on a rough horizontal track. Just before they collide, $A$ has speed $3 \mathrm {~ms} ^ { - 1 }$ and $B$ has speed $5 \mathrm {~ms} ^ { - 1 }$. Immediately after the impact, the direction of motion of both particles has been reversed and they are both travelling at the same speed, $v$.

Show that $v = 1 \mathrm {~ms} ^ { - 1 }$. The magnitude of the impulse exerted on $A$ during the collision is 24 Ns.
Find the value of $M$. Given that the coefficient of friction between $A$ and the track is 0.1 ,
find the time taken from the moment of impact until $A$ comes to rest. END

Edexcel M1 Q1

A particle, $P$, of mass 5 kg moves with speed $3 \mathrm {~ms} ^ { - 1 }$ along a smooth horizontal track. It strikes a particle $Q$ of mass 2 kg which is at rest on the track. Immediately after the collision, $P$ and $Q$ move in the same direction with speeds $v$ and $2 v \mathrm {~ms} ^ { - 1 }$ respectively.
1. Calculate the value of $v$.
2. Calculate the magnitude of the impulse received by $Q$ on impact.
3. A particle $P$ moves with a constant velocity $( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$ with respect to a fixed origin $O$. It passes through the point $A$ whose position vector is $( 2 \mathbf { i } + 11 \mathbf { j } )$ m at $t = 0$.
4. Find the angle in degrees that the velocity vector of $P$ makes with the vector $\mathbf { i }$.
5. Calculate the distance of $P$ from $O$ when $t = 2$.
6. A car of mass 1250 kg is moving at constant speed up a hill, inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = \frac { 1 } { 10 }$. The driving force produced by the engine is 1800 N .
7. Calculate the resistance to motion which the car experiences.
At the top of the hill, the road becomes horizontal.
Find the initial acceleration of the car.

Edexcel M1 Q4

4. A non-uniform plank $A B$ of mass 20 kg and length 6 m is supported at both ends so that it is horizontal. When a woman of mass 60 kg stands on the plank at a distance of 2 m from $B$, the magnitude of the reaction at $A$ is $35 g \mathrm {~N}$.

Suggest a suitable model for
1. the plank,
2. the woman.
Calculate the magnitude of the reaction at $B$, giving your answer in terms of $g$.
Explain briefly, in the context of the problem, the term 'non-uniform'.
Find the distance of the centre of mass of the plank from $A$.

Edexcel M1 Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{bf4361b8-acd2-4133-81b7-2f68d018486f-3_99_1036_242_379} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} The points $A , O$ and $B$ lie on a straight horizontal track as shown in Figure 1. $A$ is 20 m from $O$ and $B$ is on the other side of $O$ at a distance $x \mathrm {~m}$ from $O$. At time $t = 0$, a particle $P$ starts from rest at $O$ and moves towards $B$ with uniform acceleration of $3 \mathrm {~ms} ^ { - 2 }$. At the same instant, another particle $Q$, which is at the point $A$, is moving with a velocity of $3 \mathrm {~ms} ^ { - 1 }$ in the direction of $O$ with uniform acceleration of $4 \mathrm {~ms} ^ { - 2 }$ in the same direction. Given that the $Q$ collides with $P$ at $B$, find the value of $x$.

Edexcel M1 Q6

6. A sledge of mass 4 kg rests in limiting equilibrium on a rough slope inclined at an angle $10 ^ { \circ }$ to the horizontal. By modelling the sledge as a particle,

show that the coefficient of friction, $\mu$, between the sledge and the ground is 0.176 correct to 3 significant figures.
(6 marks)
The sledge is placed on a steeper part of the slope which is inclined at an angle $30 ^ { \circ }$ to the horizontal. The value of $\mu$ remains unchanged.
Find the minimum extra force required along the line of greatest slope to prevent the sledge from slipping down the hill.

Edexcel M1 Q7

7. Whilst looking over the edge of a vertical cliff, 122.5 metres in height, Jim dislodges a stone. The stone falls freely from rest towards the sea below. Ignoring the effect of air resistance,

calculate the time it would take for the stone to reach the sea,
find the speed with which the stone would hit the water. Two seconds after the stone begins to fall, Jim throws a tennis ball downwards at the stone. The tennis ball's initial speed is $u \mathrm {~ms} ^ { - 1 }$ and it hits the stone before they both reach the water.
Find the minimum value of $u$.
If you had taken air resistance into account in your calculations, what effect would this have had on your answer to part (c)? Explain your answer.
(2 marks)

Edexcel M1 Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{bf4361b8-acd2-4133-81b7-2f68d018486f-4_478_529_1142_589} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} Figure 2 shows two particles $P$ and $Q$, of mass 3 kg and 2 kg respectively, attached to the ends of a light, inextensible string which passes over a smooth, fixed pulley. The system is released from rest with $P$ and $Q$ at the same level 1.5 metres above the ground and 2 metres below the pulley.

Show that the initial acceleration of the system is $\frac { g } { 5 } \mathrm {~ms} ^ { - 2 }$.
Find the tension in the string.
Find the speed with which $P$ hits the ground. When $P$ hits the ground, it does not rebound.
What is the closest that $Q$ gets to the pulley.

Questions — Edexcel M1 (599 questions)

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