Questions - Edexcel M1 - A-Level Maths

the car is modelled as a constant force of magnitude $2 R$ newtons
the trailer is modelled as a constant force of magnitude $R$ newtons

The car and the trailer are modelled as particles.
The tow bar between the car and trailer is modelled as a light rod that is parallel to the direction of motion. Using the model,

show that the value of $R$ is 60
find the tension in the tow bar. When the car and trailer are moving at a speed of $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the tow bar breaks.
Given that the non-gravitational resistance to motion of the trailer remains unchanged,
use the model to find the further distance moved by the trailer before it first comes to rest.

Edexcel M1 2023 June Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f2737a11-4a15-41e9-9f87-31a705a8948b-22_792_841_246_612} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} A square floor space $A B C D$, with centre $O$, is modelled as a flat horizontal surface measuring 50 m by 50 m , as shown in Figure 5 .
The horizontal unit vectors $\mathbf { i }$ and $\mathbf { j }$ are in the direction of $\overrightarrow { A B }$ and $\overrightarrow { A D }$ respectively.
All position vectors are given relative to $O$.
A small robot $R$ is programmed to travel across the floor at a constant velocity.

At time $t = 0 , R$ is at the point with position vector ( $- 2 \mathbf { i } + \mathbf { j }$ ) m
At time $t = 11 \mathrm {~s} , R$ is at the point with position vector $( 9 \mathbf { i } + 23 \mathbf { j } ) \mathrm { m }$
At time $t$ seconds, the position vector of $R$ is $\mathbf { r }$ metres
1. Find, in terms of $t$, i and j, an expression for $\mathbf { r }$

A second robot $S$ is at the point $C$.

At time $t = 0 , S$ leaves $C$ and moves with constant velocity $( - \mathbf { i } - \mathbf { j } ) \mathrm { ms } ^ { - 1 }$
At time $t$ seconds, the position vector of $S$ is $\mathbf { s }$ metres
Write down, in terms of $t$, i and $\mathbf { j }$, an expression for $\mathbf { s }$
Show that

$$\overrightarrow { S R } = [ ( 2 t - 27 ) \mathbf { i } + ( 3 t - 24 ) \mathbf { j } ] \mathbf { m }$$

Find the time when the distance between $R$ and $S$ is a minimum.

Edexcel M1 2024 June Q1

Two particles, $A$ and $B$, have masses $m$ and $3 m$ respectively. The particles are connected by a light inextensible string. Initially $A$ and $B$ are at rest on a smooth horizontal plane with the string slack.

Particle $A$ is then projected along the plane away from $B$ with speed $U$.
Given that the common speed of the particles immediately after the string becomes taut is $S$

find $S$ in terms of $U$.
Find, in terms of $m$ and $U$, the magnitude of the impulse exerted on $A$ immediately after the string becomes taut.

Edexcel M1 2024 June Q2

Two forces, $\mathbf { P }$ and $\mathbf { Q }$, act on a particle.

$\mathbf { P }$ has magnitude 10 N and acts due west
Q has magnitude 8 N and acts on a bearing of $330 ^ { \circ }$

Given that $\mathbf { F } = \mathbf { P } + \mathbf { Q }$, find the magnitude of $\mathbf { F }$.

Edexcel M1 2024 June Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7a65555e-1bb2-4947-8e70-50f267017bfd-06_472_208_343_1102} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Two particles, $P$ and $Q$, have masses $4 m$ and $2 m$ respectively. The particles are connected by a light inextensible string. A second light inextensible string has one end attached to $Q$. Both strings are taut and vertical, as shown in Figure 1. The particles are accelerating vertically downwards.
Given that the tension in the string connecting the two particles is $3 m g$, find, in terms of $m$ and $g$, the tension in the upper string.

Edexcel M1 2024 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7a65555e-1bb2-4947-8e70-50f267017bfd-08_417_1745_378_258} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A non-uniform rod $A B$ has length 6.5 m and mass 1.2 kg . The centre of mass of the rod is 3 m from $A$. The rod rests on a horizontal step and overhangs the end of the step $C$ by 1.5 m , as shown in Figure 2. The rod is perpendicular to the edge of the step.
A particle of mass 4 kg is placed on the rod at $B$ and another particle, whose mass is $M \mathrm {~kg}$, is placed on the rod at $D$, where $A D = 0.5 \mathrm {~m}$. The rod remains in equilibrium in a horizontal position.

Find the smallest possible value of $M$. The particle at $B$ and the particle at $D$ are now removed.
A new particle is placed on the rod at the point $E$, where $E B = 0.9 \mathrm {~m}$.
The rod remains in equilibrium in a horizontal position but is on the point of tilting about $C$.
Find the magnitude of the force acting on the rod at $C$.

Edexcel M1 2024 June Q5

A parachute is used to deliver a box of supplies. The parachute is attached to the box.

the parachute and box are dropped from rest from a helicopter that is hovering at a height of 520 m above the ground
the parachute and box fall vertically and freely under gravity for 5 seconds, then the parachute opens
from the instant the parachute opens, it provides a resistance to motion of magnitude 3200 N
the parachute and box continue to fall vertically downwards after the parachute opens
the parachute and box are modelled throughout the motion as a particle $P$ of mass 250 kg
1. Find the distance fallen by $P$ in the first 5 seconds.
2. Find the speed with which $P$ lands on the ground.
3. Find the total time from the instant when $P$ is dropped from the helicopter to the instant when $P$ lands on the ground.
4. Sketch a speed-time graph for the motion of $P$ from the instant when $P$ is dropped from the helicopter to the instant when $P$ lands on the ground.

VJYV SIHI NI JIIYM ION OC

Vayv sthin NI JLIYM ION OA

VJYV SIHI NI JAIVM ION OC

Edexcel M1 2024 June Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7a65555e-1bb2-4947-8e70-50f267017bfd-16_272_1391_336_436} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A box of mass $m$ lies on a rough horizontal plane. The box is pulled along the plane in a straight line at constant speed by a light rope. The rope is inclined at an angle $\theta$ to the plane, as shown in Figure 3.
The coefficient of friction between the box and the plane is $\frac { 1 } { 3 }$
The box is modelled as a particle.
Given that $\tan \theta = \frac { 3 } { 4 }$

find, in terms of $m$ and $g$, the tension in the rope. The rope is now removed and the box is placed at rest on the plane.
The box is then projected horizontally along the plane with speed $u$.
The box is again modelled as a particle.
When the box has moved a distance $d$ along the plane, the speed of the box is $\frac { 1 } { 2 } u$.
Find $d$ in terms of $u$ and $g$.
VJYV SIHI NI JIIIM ION OC vauv sthin NI JLHMA LON OO V34V SIHI NI IIIIM ION OC

Edexcel M1 2024 June Q7

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

Two speedboats, $A$ and $B$, are each moving with constant velocity.

the velocity of $A$ is $40 \mathrm { kmh } ^ { - 1 }$ due east
the velocity of $B$ is $20 \mathrm { kmh } ^ { - 1 }$ on a bearing of angle $\alpha \left( 0 ^ { \circ } < \alpha < 90 ^ { \circ } \right)$, where $\tan \alpha = \frac { 4 } { 3 }$ The boats are modelled as particles.
1. Find, in terms of $\mathbf { i }$ and $\mathbf { j }$, the velocity of $B$ in $\mathrm { km } \mathrm { h } ^ { - 1 }$

At noon

the position vector of $A$ is $20 \mathbf { j } \mathrm {~km}$
the position vector of $B$ is $( 10 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }$

At time $t$ hours after noon

the position vector of $A$ is $\mathbf { r k m }$, where $\mathbf { r } = 20 \mathbf { j } + 40 t \mathbf { i }$
the position vector of $B$ is $\mathbf { s }$ km
Find an expression for $\mathbf { s }$ in terms of $t , \mathbf { i }$ and $\mathbf { j }$.
Show that at time $t$ hours after noon,

$$\overrightarrow { A B } = [ ( 10 - 24 t ) \mathbf { i } + ( 12 t - 15 ) \mathbf { j } ] \mathrm { km }$$

Show that the boats will never collide.

Find the distance between the boats when the bearing of $B$ from $A$ is $225 ^ { \circ }$

Edexcel M1 2024 June Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7a65555e-1bb2-4947-8e70-50f267017bfd-24_442_1167_341_548} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} One end of a light inextensible string is attached to a particle $A$ of mass $2 m$. The other end of the string is attached to a particle $B$ of mass $3 m$. Particle $A$ is held at rest on a rough plane which is inclined to horizontal ground at an angle $\alpha$, where $\tan \alpha = \frac { 5 } { 12 }$ The string passes over a small smooth pulley $P$ which is fixed at the top of the plane. Particle $B$ hangs vertically below $P$ with the string taut, at a height $h$ above the ground, as shown in Figure 4. The part of the string between $A$ and $P$ lies along a line of greatest slope of the plane. The two particles, the string and the pulley all lie in the same vertical plane.
The coefficient of friction between $A$ and the plane is $\frac { 11 } { 36 }$
The particle $A$ is released from rest and begins to move up the plane.

Show that the frictional force acting on $A$ as it moves up the plane is $\frac { 22 m g } { 39 }$
Write down an equation of motion for $B$.
Show that the acceleration of $A$ immediately after its release is $\frac { 1 } { 3 } g$ In the subsequent motion, $A$ comes to rest before it reaches the pulley.
Find, in terms of $h$, the total distance travelled by $A$ from when it was released from rest to when it first comes to rest again.
VJYV SIHI NI JIIIM ION OC vauv sthin NI BLIYM ION OC V34V SIHI NI IIIIMM ION OC

VJYV SIHI NI JIIIM ION OC vauv sthin NI BLIYM ION OO V34V SIHI NI IIIIMM ION OC

Edexcel M1 2016 October Q1

Two particles, $P$ and $Q$, have masses $2 m$ and $3 m$ respectively. They are moving towards each other, in opposite directions, along the same straight line, on a smooth horizontal plane. The particles collide. Immediately before they collide the speed of $P$ is $2 u$ and the speed of $Q$ is $u$. In the collision the magnitude of the impulse exerted on $P$ by $Q$ is $5 m u$.
1. Find the speed of $P$ immediately after the collision.
2. State whether the direction of motion of $P$ has been reversed by the collision.
3. Find the speed of $Q$ immediately after the collision.

Edexcel M1 2016 October Q2

2. [In this question $\mathbf { i }$ and $\mathbf { j }$ are perpendicular unit vectors in a horizontal plane.] Three forces, $( - 10 \mathbf { i } + a \mathbf { j } ) \mathrm { N } , ( b \mathbf { i } - 5 \mathbf { j } ) \mathrm { N }$ and $( 2 a \mathbf { i } + 7 \mathbf { j } ) \mathrm { N }$, where $a$ and $b$ are constants, act on a particle $P$ of mass 3 kg . The acceleration of $P$ is $( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }$

Find the value of $a$ and the value of $b$. At time $t = 0$ seconds the speed of $P$ is $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and at time $t = 4$ seconds the velocity of $P$ is $( 20 \mathbf { i } + 20 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$
Find the value of $u$.

Edexcel M1 2016 October Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6978be48-561b-49a0-a297-c8886ca66c19-06_267_1092_254_428} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A plank $A B$ has length 8 m and mass 12 kg . The plank rests on two supports. One support is at $C$, where $A C = 3 \mathrm {~m}$ and the other support is at $D$, where $A D = x$ metres. A block of mass 3 kg is placed on the plank at $B$, as shown in Figure 1. The plank rests in equilibrium in a horizontal position. The magnitude of the force exerted on the plank by the support at $D$ is twice the magnitude of the force exerted on the plank by the support at $C$. The plank is modelled as a uniform rod and the block is modelled as a particle. Find the value of $x$.

Edexcel M1 2016 October Q4

\hspace{0pt} [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 $O$ ]

A particle $P$ is moving with velocity $( \mathbf { i } - 2 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$. At time $t = 0$ hours, the position vector of $P$ is $( - 5 \mathbf { i } + 9 \mathbf { j } ) \mathrm { km }$. At time $t$ hours, the position vector of $P$ is $\mathbf { p } \mathrm { km }$.

Find an expression for $\mathbf { p }$ in terms of $t$. The point $A$ has position vector ( $3 \mathbf { i } + 2 \mathbf { j }$ ) km.
Find the position vector of $P$ when $P$ is due west of $A$. Another particle $Q$ is moving with velocity $[ ( 2 b - 1 ) \mathbf { i } + ( 5 - 2 b ) \mathbf { j } ] \mathrm { km } \mathrm { h } ^ { - 1 }$ where $b$ is a constant. Given that the particles are moving along parallel lines,
find the value of $b$.

Edexcel M1 2016 October Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6978be48-561b-49a0-a297-c8886ca66c19-10_419_933_123_525} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A particle $P$ of mass 0.5 kg is at rest on a rough plane which is inclined to the horizontal at $30 ^ { \circ }$. The particle is held in equilibrium by a force of magnitude 8 N , acting at an angle of $40 ^ { \circ }$ to the plane, as shown in Figure 2. The line of action of the force lies in the vertical plane containing $P$ and a line of greatest slope of the plane. The coefficient of friction between $P$ and the plane is $\mu$. Given that $P$ is on the point of sliding up the plane, find the value of $\mu$.

Edexcel M1 2016 October Q6

6. Two cars $A$ and $B$ are moving in the same direction along a straight horizontal road. Car $A$ is moving with uniform acceleration $0.4 \mathrm {~ms} ^ { - 2 }$ and car $B$ is moving with uniform acceleration $0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. At the instant when $B$ is 200 m behind $A$, the speed of $A$ is $35 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the speed of $B$ is $44 \mathrm {~ms} ^ { - 1 }$. Find the speed of $B$ when it overtakes $A$.
(9)

Edexcel M1 2016 October Q7

7. A train moves on a straight horizontal track between two stations $A$ and $B$. The train starts from rest at $A$ and moves with constant acceleration $1 \mathrm {~ms} ^ { - 2 }$ until it reaches a speed of $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The train maintains this speed of $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$ for the next $T$ seconds before slowing down with constant deceleration $0.5 \mathrm {~ms} ^ { - 2 }$, coming to rest at $B$. The journey from $A$ to $B$ takes 180 s and the distance between the stations is 4800 m .

Sketch a speed-time graph for the motion of the train from $A$ to $B$.
Show that $T = 180 - 3 V$.
Find the value of $V$.

Edexcel M1 2016 October Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6978be48-561b-49a0-a297-c8886ca66c19-20_312_1068_230_438} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Two particles $P$ and $Q$ have masses 2 kg and 3 kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a smooth light pulley which is fixed at the top of a rough plane. The plane is inclined to horizontal ground at an angle $\alpha$, where tan $\alpha = \frac { 3 } { 4 }$. Initially $P$ is held at rest on the inclined plane with the part of the string from $P$ to the pulley parallel to a line of greatest slope of the plane. The particle $Q$ hangs freely below the pulley at a height of 0.5 m above the ground, as shown in Figure 3. The coefficient of friction between $P$ and the plane is $\mu$. The system is released from rest, with the string taut, and $Q$ strikes the ground before $P$ reaches the pulley. The speed of $Q$ at the instant when it strikes the ground is $1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

For the motion before $Q$ strikes the ground, find the tension in the string.
Find the value of $\mu$.
END

Edexcel M1 2017 October Q1

A suitcase of mass 40 kg is being dragged in a straight line along a rough horizontal floor at constant speed using a thin strap. The strap is inclined at $20 ^ { \circ }$ above the horizontal. The coefficient of friction between the suitcase and the floor is $\frac { 3 } { 4 }$. The strap is modelled as a light inextensible string and the suitcase is modelled as a particle. Find the tension in the strap.

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0fccbe27-ff7a-4c63-bb08-770b138696b7-03_442_1296_114_324} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A metal girder $A B$, of weight 1080 N and length 6 m , rests in equilibrium in a horizontal position on two supports, one at $C$ and one at $D$, where $A C = 0.5 \mathrm {~m}$ and $B D = 2 \mathrm {~m}$, as shown in Figure 1. A boy of weight 400 N stands on the girder at $B$ and the girder remains horizontal and in equilibrium. The boy is modelled as a particle and the girder is modelled as a uniform rod.

Find
1. the magnitude of the reaction on the girder at $C$,
2. the magnitude of the reaction on the girder at $D$.
  (6) The boy now stands at a point $E$ on the girder, where $A E = x$ metres, and the girder remains horizontal and in equilibrium. Given that the magnitude of the reaction on the girder at $D$ is now 520 N greater than the magnitude of the reaction on the girder at $C$,
find the value of $x$.

Edexcel M1 2017 October Q3

Two particles $P$ and $Q$ have masses $4 m$ and $m$ respectively. They are moving in opposite directions towards each other along the same straight line on a smooth horizontal plane and collide directly. Immediately before the collision the speed of $P$ is $2 u$ and the speed of $Q$ is $4 u$. In the collision, the particles join together to form a single particle.

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

Edexcel M1 2017 October Q4

4. Two forces $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ act on a particle. The force $\mathbf { F } _ { 1 }$ has magnitude 8 N and acts due east. The resultant of $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ is a force of magnitude 14 N acting in a direction whose bearing is $120 ^ { \circ }$. Find

the magnitude of $\mathbf { F } _ { 2 }$,
the direction of $\mathbf { F } _ { 2 }$, giving your answer as a bearing to the nearest degree.

Edexcel M1 2017 October Q5

A small ball is projected vertically upwards from a point $O$ with speed $14.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The point $O$ is 2.5 m above the ground. The motion of the ball is modelled as that of a particle moving freely under gravity.

Find

the maximum height above the ground reached by the ball,
the time taken for the ball to first reach a height of 1 m above the ground,
the speed of the ball at the instant before it strikes the ground for the first time.

Edexcel M1 2017 October Q6

An athlete goes for a run along a straight horizontal road. Starting from rest, she accelerates at $0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ up to a speed of $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$. She then maintains this constant speed of $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$ before finally decelerating at $0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ back to rest. She covers a total distance of 1500 m in 270 s .
1. Sketch a speed-time graph to represent the athlete's run.
2. Show that she accelerates for $\frac { 5 V } { 3 }$ seconds.
3. Show that $V ^ { 2 } - k V + 450 = 0$, where $k$ is a constant to be found.
4. Find the value of $V$, justifying your answer.

Edexcel M1 2017 October Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0fccbe27-ff7a-4c63-bb08-770b138696b7-13_349_1347_248_303} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows two particles $A$ and $B$, of masses $3 m$ and $4 m$ respectively, attached to the ends of a light inextensible string. Initially $A$ is held at rest on the surface of a fixed rough inclined plane. The plane is inclined to the horizontal at an angle $\alpha$ where $\tan \alpha = \frac { 3 } { 4 }$. The coefficient of friction between $A$ and the plane is $\frac { 1 } { 4 }$. The string passes over a small smooth light pulley $P$ which is fixed at the top of the plane. The part of the string from $A$ to $P$ is parallel to a line of greatest slope of the plane. The particle $B$ hangs freely and is vertically below $P$. The system is released from rest with the string taut and with $B$ at a height of 1.75 m above the ground. In the subsequent motion, $A$ does not hit the pulley. For the period before $B$ hits the ground,

write down an equation of motion for each particle.
Hence show that the acceleration of $B$ is $\frac { 8 } { 35 } \mathrm {~g}$.
Explain how you have used the fact that the string is inextensible in your calculation. When $B$ hits the ground, $B$ does not rebound and comes immediately to rest.
Find the distance travelled by $A$ from the instant when the system is released to the instant when $A$ first comes to rest.

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