Questions - Edexcel M1 - A-Level Maths

Show that one possible value of $q$ is ${ } ^ { - } 1$ and find the other possible value of $q$. Given that $q = { } ^ { - } 1$, and that the pedestrian started walking at the point with position vector $( 6 \mathbf { i } - \mathbf { j } ) \mathrm { m }$,
find the length of time for which the pedestrian is less than 5 m from $O$.

Edexcel M1 Q5

5. A sledgehammer of mass 12 kg is being used to drive a wooden post of mass 4 kg into the ground. A labourer moves the sledgehammer from rest at a point 0.5 m vertically above the post with constant acceleration $16 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ directed towards the post.

Find the velocity with which the sledgehammer hits the post. When the sledgehammer hits the post, they both move together with common speed, $V$.
Show that $V = 3 \mathrm {~ms} ^ { - 1 }$. As the sledgehammer hits the post, the labourer relaxes his grip and applies no further force. The sledgehammer and post are brought to rest by the action of a resistive force from the ground of magnitude 1500 N .
Find, in centimetres, the total distance that the sledgehammer and the post travel together before coming to rest.

Edexcel M1 Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6fb27fe5-055a-4701-bd80-e66ebd57292a-4_252_726_194_561} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} Figure 2 shows a picnic bench of mass 20 kg which consists of a horizontal plank of wood of length 2 m resting on two supports, each of which is 0.6 m from the centre of the plank. Luigi sits on the bench at its midpoint and his mother Maria sits at one end. Their masses are 40 kg and 75 kg respectively. By modelling the bench as a uniform rod and Luigi and Maria as particles,

find the reaction at each of the two supports. Luigi moves to sit closer to his mother.
Find how close Luigi can get to his mother before the reaction at one of the supports becomes zero.
Explain the significance of a zero reaction at one of the supports.

Edexcel M1 Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6fb27fe5-055a-4701-bd80-e66ebd57292a-5_417_1016_237_440} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} Figure 3 shows a particle of mass 4 kg resting on the surface of a rough plane inclined at an angle of $30 ^ { \circ }$ to the horizontal. It is connected by a light inextensible string passing over a smooth pulley at the top of the plane, to a particle of mass 5 kg which hangs freely. The coefficient of friction between the 4 kg mass and the plane is $\mu$ and when the system is released from rest the 4 kg mass starts to move up the slope.

Show that the acceleration of the system is $\frac { 1 } { 9 } ( 3 - 2 \mu \sqrt { 3 } ) \mathrm { g } \mathrm { ms } ^ { - 2 }$.
Hence, find the maximum value of $\mu$. Given that $\mu = \frac { 1 } { 2 }$,
find the tension in the string in terms of $g$,
show that the magnitude of the force on the pulley is given by $\frac { 5 } { 3 } ( 2 \sqrt { 3 } + 1 ) \mathrm { g }$. END

Edexcel M1 Q1

The resultant of two forces $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ is $( - 2 \mathbf { i } + 9 \mathbf { j } ) \mathrm { N }$.

Given that $\mathbf { F } _ { \mathbf { 1 } } = ( 2 p \mathbf { i } - 3 q \mathbf { j } ) \mathrm { N }$ and $\mathbf { F } _ { \mathbf { 2 } } = ( 5 q \mathbf { i } + 4 p \mathbf { j } ) \mathrm { N }$, calculate the values of $p$ and $q$.
(5 marks)

Edexcel M1 Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{10b4d660-3980-4204-b18d-5240dea61a45-2_321_666_584_534} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} Figure 1 shows a toy lorry being pulled by a piece of string, up a ramp inclined at an angle of $25 ^ { \circ }$ to the horizontal. When the string is pulled with a force of 20 N parallel to the line of greatest slope of the ramp, the lorry is on the point of moving up the ramp. In a simple model of the situation, the ramp is considered to be smooth.

Draw a diagram showing all the forces acting on the lorry.
Find the weight of the lorry and the magnitude of the reaction between the lorry and the ramp, giving your answers to an appropriate degree of accuracy.
Write down any modelling assumptions that you have made about
1. the lorry,
2. the string. In a more refined model, the ramp is assumed to be rough.
State the effect that this would have on your answers to part (b).

Edexcel M1 Q3

3. A cannon of mass 600 kg lies on a rough horizontal surface and is used to fire a 3 kg shell horizontally at $200 \mathrm {~ms} ^ { - 1 }$.

Find the impulse which the shell exerts on the cannon.
Find the speed with which the cannon recoils. Given that the coefficient of friction between the cannon and the surface is 0.75 ,
calculate, to the nearest centimetre, the distance that the cannon travels before coming to rest.

Edexcel M1 Q4

4. The position of an aeroplane flying in a straight horizontal line at constant speed is plotted on a radar screen. At 2 p.m. the position vector of the aeroplane is $( 80 \mathbf { i } + 5 \mathbf { j } )$, where $\mathbf { i }$ and $\mathbf { j }$ are unit vectors directed east and north respectively relative to a fixed origin, $O$, on the screen. Ten minutes later the position of the aeroplane on the screen is $( 32 \mathbf { i } + 19 \mathbf { j } )$. Each unit on the screen represents 1 km .

Find the position vector of the aeroplane at 2:30 p.m.
Find the speed of the aeroplane in $\mathrm { km } \mathrm { h } ^ { - 1 }$.
Find, correct to the nearest degree, the bearing on which the aeroplane is flying.

Edexcel M1 Q5

5. A car on a straight test track starts from rest and accelerates to a speed of $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in 6 seconds. The car maintains this speed for a further 50 seconds before decelerating to rest. In a simple model of this motion, the acceleration and deceleration are assumed to be uniform and the magnitude of the deceleration to be 1.5 times that of the acceleration.

Show that the total time for which the car is moving is 60 seconds.
Sketch a velocity-time graph for this journey. Given that the total distance travelled is 1320 metres,
find $V$. In a more sophisticated model, the acceleration is assumed to be inversely proportional to the velocity of the car.
Explain how the acceleration would vary during the first six seconds under this model.
(2 marks)

Edexcel M1 Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{10b4d660-3980-4204-b18d-5240dea61a45-4_250_1036_1251_422} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} Figure 2 shows a bench of length 3 m being used in a gymnasium.
The bench rests horizontally on two identical supports which are 2.2 m apart and equidistant from the middle of the bench.

Explain why it is reasonable to model the bench as a uniform rod. When a gymnast of mass 55 kg stands on the bench 0.1 m from one end, the bench is on the point of tilting.
Find the mass of the bench. The first gymnast dismounts and a second gymnast of mass 33 kg steps onto the bench at a distance of 0.4 m from its centre.
Show that the magnitudes of the reaction forces on the two supports are in the ratio $5 : 3$.
(6 marks)

Edexcel M1 Q7

7. A car of mass 1250 kg tows a caravan of mass 850 kg up a hill inclined at an angle $\alpha$ to the horizontal where $\sin \alpha = \frac { 1 } { 14 }$. The total resistance to motion experienced by the car is 400 N , and by the caravan is 500 N . Given that the driving force of the engine is 3 kN ,

show that the acceleration of the system is $0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }$,
find the tension in the towbar linking the car and the caravan. Starting from rest, the car accelerates uniformly for 540 m until it reaches a speed of $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at the top of the hill.
Find v. At the top of the hill the road becomes level and the driver maintains the speed at which the car and caravan reached the top of the hill.
Assuming that the resistance to motion on each part of the system is unchanged, find the percentage reduction in the driving force of the engine required to achieve this.

Edexcel M1 Q1

At time $t = 0$, a particle of mass 2 kg has velocity $( 8 \mathbf { i } + \lambda \mathbf { j } ) \mathrm { ms } ^ { - 1 }$ where $\mathbf { i }$ and $\mathbf { j }$ are horizontal perpendicular unit vectors and $\lambda > 0$.

Given that the speed of the particle at time $t = 0$ is $17 \mathrm {~m} \mathrm {~s} ^ { - 1 }$,

find the value of $\lambda$. The particle experiences a constant retarding force $\mathbf { F }$ so that when $t = 5$, it has velocity $( 3 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$.
Show that $\mathbf { F }$ can be written in the form $\mu ( \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }$ where $\mu$ is a constant which you should find.
(5 marks)

Edexcel M1 Q2

2. A monk uses a small brush to clean the stone floor of a monastery by pushing the brush with a force of $P$ Newtons at an angle of $60 ^ { \circ }$ to the vertical. He moves the brush at a constant speed. The mass of the brush is 0.5 kg and the coefficient of friction between the brush and the floor is $\frac { 1 } { \sqrt { 3 } }$. The brush is modelled as a particle and air resistance is ignored.

Show that $P = \frac { g } { 2 }$ Newtons.
Explain why it is reasonable to ignore air resistance in this situation.

Edexcel M1 Q3

3. A small van of mass 1500 kg is used to tow a car of mass 750 kg by means of a rope of length 9 m joined to both vehicles. The van sets off with the rope slack and reaches a speed of $2 \mathrm {~ms} ^ { - 1 }$ just before the rope becomes taut and jerks the car into motion. Immediately after the rope becomes taut, the van and car travel with common speed $V \mathrm {~ms} ^ { - 1 }$.

Show that $V = \frac { 4 } { 3 }$.
Calculate the magnitude of the impulse on the car when the rope tightens. The van and car eventually reach a steady speed of $18 \mathrm {~ms} ^ { - 1 }$ with the rope taut when a child runs out into the road, 30 m in front of the van. The van driver brakes sharply and decelerates uniformly to rest in a distance of 27 m . It takes the driver of the car 1 second to react to the van starting to brake. He then brakes and the car decelerates uniformly at $f \mathrm {~m} \mathrm {~s} ^ { - 2 }$, coming to rest before colliding with the van.
Find the set of possible values of $f$.
(5 marks)

Edexcel M1 Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4fe54579-ac77-46f9-85e1-2e95963d6b3e-3_467_348_201_708} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} Figure 1 shows a weight $A$ of mass 6 kg connected by a light, inextensible string which passes over a smooth, fixed pulley to a box $B$ of mass 5 kg . There is an object $C$ of mass 3 kg resting on the horizontal floor of box $B$. The system is released from rest. Find, giving your answers in terms of $g$,

the acceleration of the system,
the force on the pulley.
Show that the reaction between $C$ and the floor of $B$ is $\frac { 18 } { 7 } \mathrm {~g}$ newtons.

Edexcel M1 Q5

5. Two flies $P$ and $Q$, are crawling vertically up a wall. At time $t = 0$, the flies are at the same height above the ground, with $P$ crawling at a steady speed of $4 \mathrm { cms } ^ { - 1 }$.
$Q$ starts from rest at time $t = 0$ and accelerates uniformly to a speed of $6 \mathrm {~cm} \mathrm {~s} ^ { - 1 }$ in 6 seconds. Fly $Q$ then maintains this speed.

Find the value of $t$ when the two flies are moving at the same speed.
Sketch on the same diagram, speed-time graphs to illustrate the motion of the two flies. Given that the distance of the two flies from the top of the wall at time $t = 0$ is $x \mathrm {~cm}$ and that $Q$ reaches the top of the wall first,
show that $x > 36$.

Edexcel M1 Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4fe54579-ac77-46f9-85e1-2e95963d6b3e-4_288_1275_201_410} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} Figure 2 shows a uniform plank $A B$ of length 8 m and mass 50 kg suspended horizontally by two light vertical inextensible strings attached at either end of the plank. The maximum tension that either string can support is 40 gN . A rock of mass $M \mathrm {~kg}$ is placed on the plank at $A$ and rolled along the plank to $B$ without either string breaking.

Explain, with the aid of a sketch-graph, how the tension in the string at $A$ varies with $x$, the distance of the rock from $A$.
Show that $M \leq 15$. The first rock is removed and a second rock of mass 20 kg is placed on the plank.
Find the fraction of the plank on which the rock can be placed without one of the strings breaking.

Edexcel M1 Q7

7. At 6 a.m. a cargo ship has position vector $( 7 \mathbf { i } + 56 \mathbf { j } ) \mathrm { km }$ relative to a fixed origin $O$ on the coast and moves with constant velocity $( 9 \mathbf { i } - 6 \mathbf { j } ) \mathrm { kmh } ^ { - 1 }$. A ferry sails from $O$ at 6 a.m. and moves with constant velocity $( 12 \mathbf { i } + 18 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$. The unit vectors $\mathbf { i }$ and $\mathbf { j }$ are directed due east and due north respectively.

Show that the position vector of the cargo ship $t$ hours after 6 a.m. is given by $$[ ( 7 + 9 t ) \mathbf { i } + ( 56 - 6 t ) \mathbf { j } ] \mathrm { km }$$ and find the position vector of the ferry in terms of $t$.
Show that if both vessels maintain their course and speed, they will collide and find the time and position vector at which this occurs.
(6 marks)
At 8 a.m. the captain of the ferry realises that a collision is imminent and changes course so that the ferry now has velocity $( 21 \mathbf { i } + 6 \mathbf { j } ) \mathrm { kmh } ^ { - 1 }$.
Find the distance between the two ships at the time when they would have collided.

Edexcel M1 Q1

In a safety test, a car of mass 800 kg is driven directly at a wall at a constant speed of $15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The constant force exerted by the wall on the car in bringing it to rest is 60 kN .
1. Calculate the magnitude of the impulse exerted by the wall on the car.
2. Find the time it takes for the car to come to rest.
3. Show that the deceleration of the car is $75 \mathrm {~ms} ^ { - 2 }$.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2def617f-3c25-4458-8500-8e20ba7c1e53-2_531_661_678_539} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows an aerial view of a revolving door consisting of 4 panels, each of width 1.2 m and set at $90 ^ { \circ }$ intervals, which are free to rotate about a fixed central column, $O$. The revolving door is situated outside a lecture theatre and four students are trying to push the door. Two of the students are pushing panels $O A$ and $O D$ clockwise (as viewed from above) with horizontal forces of 70 N and 90 N respectively, whilst the other two are pushing panels $O B$ and $O C$ anti-clockwise with horizontal forces of 80 N and 60 N respectively.
Calculate the total moment about $O$ when the four students are pushing the panels at their outer edge, 1.2 m from $O$.
(3 marks)
The student at $C$ moves her hand 0.2 m closer to $O$ and the student at $D$ moves his hand $x \mathrm {~m}$ closer to $O$. Given that the students all push in the same directions and with the same forces as in part (a), and that the door is in equilibrium,
find the value of $x$.

Edexcel M1 Q3

3. During a cricket match, the batsman hits the ball and begins running with constant velocity $4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ to try and score a run. When the batsman is at the fixed origin $O$, the ball is thrown by a member of the opposing team with velocity $\left( { } ^ { - } 8 \mathbf { i } + 24 \mathbf { j } \right) \mathrm { ms } ^ { - 1 }$ from the point with position vector $( 30 \mathbf { i } - 60 \mathbf { j } ) \mathrm { m }$, where $\mathbf { i }$ and $\mathbf { j }$ are horizontal perpendicular unit vectors. At time $t$ seconds after the ball is thrown, the position vectors of the batsman and the ball are $\mathbf { r }$ metres and s metres respectively. In a model of the situation, the ball is assumed to travel horizontally and air resistance is considered to be negligible.

Find expressions for $\mathbf { r }$ and $\mathbf { s }$ in terms of $t$.
Show that the ball hits the batsman and find the position vector of the batsman when this occurs.
Write down two reasons why the assumptions used in these calculations are unlikely to provide a realistic model.
(2 marks)

Edexcel M1 Q4

4. In a physics experiment, two balls $A$ and $B$, of mass $4 m$ and $3 m$ respectively, are travelling towards one another on a straight horizontal track. Both balls are travelling with speed $2 \mathrm {~ms} ^ { - 1 }$ immediately before they collide. As a result of the impact, $A$ is brought to rest and the direction of motion of $B$ is reversed.
Modelling the track as smooth and the balls as particles,

find the speed of $B$ immediately after the collision. A student notices that after the collision, $B$ comes to rest 0.2 m from $A$.
Show that the coefficient of friction between $B$ and the track is 0.113 , correct to 3 decimal places.

Edexcel M1 Q5

5. A cyclist is riding up a hill inclined at an angle of $5 ^ { \circ }$ to the horizontal. She produces a driving force of 50 N and experiences resistive forces which total 20 N . Given that the combined mass of the cyclist and her bicycle is 70 kg ,

find, correct to 2 decimal places, the magnitude of the deceleration of the cyclist. When the cyclist reaches the top of the hill, her speed is $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. She subsequently accelerates uniformly so that in the fifth second after she has reached the top of the hill, she travels 12 m .
Find her speed at the end of the fifth second.
(8 marks)

Edexcel M1 Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2def617f-3c25-4458-8500-8e20ba7c1e53-4_451_734_964_607} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} Figure 2 shows a particle $A$ of mass 5 kg , lying on a smooth horizontal table which is 0.9 m above the floor. A light inextensible string of length 0.7 m connects $A$ to a particle $B$ of mass 2 kg . The string passes over a smooth pulley which is fixed to the edge of the table and $B$ hangs vertically 0.4 m below the pulley. When the system is released from rest,

show that the magnitude of the force exerted on the pulley is $\frac { 10 \sqrt { 2 } } { 7 } \mathrm {~g} \mathrm {~N}$,
find the speed with which $A$ hits the pulley. When $A$ hits the pulley, the string breaks and $B$ subsequently falls freely under gravity.
Find the speed with which $B$ hits the ground.

Edexcel M1 Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2def617f-3c25-4458-8500-8e20ba7c1e53-5_355_682_237_550} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} Figure 3 shows a block of mass 25 kg held in equilibrium on a plane inclined at an angle of $35 ^ { \circ }$ to the horizontal by means of a string which is at an angle of $15 ^ { \circ }$ to the line of greatest slope of the plane. In an initial model of the situation, the plane is assumed to be smooth. Giving your answers correct to 3 significant figures,

show that the tension in the string is 145 N ,
find the magnitude of the reaction between the plane and the block. In a more refined model, the plane is assumed to be rough.
Given that the tension in the string can be increased to 200 N before the block begins to move up the slope,
find, correct to 3 significant figures, the magnitude of the frictional force and state the direction in which it acts.
(4 marks)
Without performing any further calculations, state whether the reaction calculated in part (b) will increase, decrease or remain the same in the refined model. Give a reason for your answer.

Edexcel M1 Q1

Two particles $P$ and $Q$, of mass $m$ and $k m$ respectively, are travelling in opposite directions on a straight horizontal path with speeds $3 u$ and $2 u$ respectively. $P$ and $Q$ collide and, as a result, the direction of motion of both particles is reversed and their speeds are halved.
1. Find the value of $k$.
2. Write down an expression in terms of $m$ and $u$ for the magnitude of the impulse which $P$ exerts on $Q$ during the collision.
  (3 marks)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{54642aff-2042-494e-ba4a-8332bd47a751-2_222_1170_790_372} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a plank $A B$ of mass 40 kg and length 6 m , which rests on supports at each of its ends. The plank is wedge-shaped, being thicker at end $A$ than at end $B$. A woman of mass 60 kg stands on the plank at a distance of 2 m from $B$.
Suggest suitable modelling assumptions which can be made about
1. the plank,
2. the woman. Given that the reactions at each support are of equal magnitude,
find the magnitude of the reaction on the support at $A$,
calculate the distance of the centre of mass of the plank from $A$.

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