Questions M1 - A-Level Maths

Find its initial speed.
Find the ball's flight time and range, $R \mathrm {~m}$.
(A) Show that the range is the same if the components of the initial velocity of the ball are $20 \mathrm {~ms} ^ { - 1 }$ in the horizontal direction and $15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in the vertical direction.
(B) State, justifying your answer, whether the range is the same whenever the ball is hit with the same initial speed.

OCR MEI M1 2013 June Q3

3 In this question take $\boldsymbol { g } = \mathbf { 1 0 }$.
The directions of the unit vectors $\left( \begin{array} { l } 1
0
0 \end{array} \right) , \left( \begin{array} { l } 0
1
0 \end{array} \right)$ and $\left( \begin{array} { l } 0
0
1 \end{array} \right)$ are east, north and vertically upwards.
Forces $\mathbf { p } , \mathbf { q }$ and $\mathbf { r }$ are given by $\mathbf { p } = \left( \begin{array} { r } - 1
- 1
5 \end{array} \right) \mathrm { N } , \mathbf { q } = \left( \begin{array} { r } - 1
- 4
2 \end{array} \right) \mathrm { N }$ and $\mathbf { r } = \left( \begin{array} { l } 2
5
0 \end{array} \right) \mathrm { N }$.

Find which of $\mathbf { p } , \mathbf { q }$ and $\mathbf { r }$ has the greatest magnitude.
A particle has mass 0.4 kg . The forces acting on it are $\mathbf { p } , \mathbf { q } , \mathbf { r }$ and its weight. Find the magnitude of the particle's acceleration and describe the direction of this acceleration.

OCR MEI M1 2013 June Q4

4 The directions of the unit vectors $\mathbf { i }$ and $\mathbf { j }$ are east and north.
The velocity of a particle, $\mathrm { vm } \mathrm { s } ^ { - 1 }$, at time $t \mathrm {~s}$ is given by $$\mathbf { v } = \left( 16 - t ^ { 2 } \right) \mathbf { i } + ( 31 - 8 t ) \mathbf { j } .$$ Find the time at which the particle is travelling on a bearing of $045 ^ { \circ }$ and the speed of the particle at this time.

OCR MEI M1 2013 June Q5

5 Fig. 5 shows blocks of mass 4 kg and 6 kg on a smooth horizontal table. They are connected by a light, inextensible string. As shown, a horizontal force $F \mathrm {~N}$ acts on the 4 kg block and a horizontal force of 30 N acts on the 6 kg block. The magnitude of the acceleration of the system is $2 \mathrm {~ms} ^ { - 2 }$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{83e69140-4abf-4713-85da-922ce7530e47-3_106_1107_1708_479} \captionsetup{labelformat=empty} \caption{Fig. 5}

\end{figure}

Find the two possible values of $F$.
Find the tension in the string in each case.

OCR MEI M1 2013 June Q6

6 A particle moves along a straight line through an origin O . Initially the particle is at O .
At time $t \mathrm {~s}$, its displacement from O is $x \mathrm {~m}$ and its velocity, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, is given by $$v = 24 - 18 t + 3 t ^ { 2 } .$$

Find the times, $T _ { 1 } \mathrm {~s}$ and $T _ { 2 } \mathrm {~s}$ (where $T _ { 2 } > T _ { 1 }$ ), at which the particle is stationary.
Find an expression for $x$ at time $t \mathrm {~s}$. Show that when $t = T _ { 1 } , x = 20$ and find the value of $x$ when $t = T _ { 2 }$. Section B (36 marks)
$7 \quad$ Abi and Bob are standing on the ground and are trying to raise a small object of weight 250 N to the top of a building. They are using long light ropes. Fig. 7.1 shows the initial situation. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{83e69140-4abf-4713-85da-922ce7530e47-4_773_1071_429_497} \captionsetup{labelformat=empty} \caption{Fig. 7.1}
\end{figure} Abi pulls vertically downwards on the rope A with a force $F$ N. This rope passes over a small smooth pulley and is then connected to the object. Bob pulls on another rope, B, in order to keep the object away from the side of the building. In this situation, the object is stationary and in equilibrium. The tension in rope B, which is horizontal, is 25 N . The pulley is 30 m above the object. The part of rope A between the pulley and the object makes an angle $\theta$ with the vertical.

OCR MEI M1 2013 June Q8

8 Fig. 8.1 shows a sledge of mass 40 kg . It is being pulled across a horizontal surface of deep snow by a light horizontal rope. There is a constant resistance to its motion. The tension in the rope is 120 N . \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{83e69140-4abf-4713-85da-922ce7530e47-6_122_849_456_609} \captionsetup{labelformat=empty} \caption{Fig. 8.1}

\end{figure} The sledge is initially at rest. After 10 seconds its speed is $5 \mathrm {~ms} ^ { - 1 }$.

Show that the resistance to motion is 100 N . When the speed of the sledge is $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the rope breaks. The resistance to motion remains 100 N .
Find the speed of the sledge
(A) 1.6 seconds after the rope breaks,
(B) 6 seconds after the rope breaks. The sledge is then pushed to the bottom of a ski slope. This is a plane at an angle of $15 ^ { \circ }$ to the horizontal. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{83e69140-4abf-4713-85da-922ce7530e47-6_259_853_1457_607} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
\end{figure} The sledge is attached by a light rope to a winch at the top of the slope. The rope is parallel to the slope and has a constant tension of 200 N . Fig. 8.2 shows the situation when the sledge is part of the way up the slope. The ski slope is smooth.
Show that when the sledge has moved from being at rest at the bottom of the slope to the point when its speed is $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, it has travelled a distance of 13.0 m (to 3 significant figures). When the speed of the sledge is $8 \mathrm {~ms} ^ { - 1 }$, this rope also breaks.
Find the time between the rope breaking and the sledge reaching the bottom of the slope.

Edexcel M1 Specimen Q1

A particle $P$ is moving with constant velocity $( - 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. At time $t = 6 \mathrm {~s} P$ is at the point with position vector $( - 4 \mathbf { i } - 7 \mathbf { j } ) \mathrm { m }$. Find the distance of $P$ from the origin at time $t = 2 \mathrm {~s}$.

Edexcel M1 Specimen Q2

2. Particle $P$ has mass $m \mathrm {~kg}$ and particle $Q$ has mass $3 m \mathrm {~kg}$. The particles are moving in opposite directions along a smooth horizontal plane when they collide directly. Immediately before the collision $P$ has speed $4 u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $Q$ has speed $k u \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where $k$ is a constant. 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 $u$, the magnitude of the impulse exerted on $P$ by $Q$.

Edexcel M1 Specimen Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ab3c0d61-3cab-4050-8288-6052e8404eb1-06_182_872_310_543} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A small box is pushed along a floor. The floor is modelled as a rough horizontal plane and the box is modelled as a particle. The coefficient of friction between the box and the floor is $\frac { 1 } { 2 }$. The box is pushed by a force of magnitude 100 N which acts at an angle of $30 ^ { \circ }$ with the floor, as shown in Figure 1. Given that the box moves with constant speed, find the mass of the box.

Edexcel M1 Specimen Q6

A ball is projected vertically upwards with a speed of $14.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from a point which is 49 m above horizontal ground. Modelling the ball as a particle moving freely under gravity, find
1. the greatest height, above the ground, reached by the ball,
2. the speed with which the ball first strikes the ground,
3. the total time from when the ball is projected to when it first strikes the ground.

Edexcel M1 Specimen Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ab3c0d61-3cab-4050-8288-6052e8404eb1-20_264_684_319_639} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A particle of mass 0.4 kg is held at rest on a fixed rough plane by a horizontal force of magnitude $P$ newtons. The force acts in the vertical plane containing the line of greatest slope of the inclined plane which passes through the particle. The plane is inclined to the horizontal at an angle $\alpha$, where $\tan \alpha = \frac { 3 } { 4 }$, as shown in Figure 2. The coefficient of friction between the particle and the plane is $\frac { 1 } { 3 }$.
Given that the particle is on the point of sliding up the plane, find

the magnitude of the normal reaction between the particle and the plane,
the value of $P$.

Edexcel M1 Specimen Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ab3c0d61-3cab-4050-8288-6052e8404eb1-24_862_412_310_774} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Two particles $A$ and $B$ have mass 0.4 kg and 0.3 kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed above a horizontal floor. Both particles are held, with the string taut, at a height of 1 m above the floor, as shown in Figure 3. The particles are released from rest and in the subsequent motion $B$ does not reach the pulley.

Find the tension in the string immediately after the particles are released.
Find the acceleration of $A$ immediately after the particles are released. When the particles have been moving for 0.5 s , the string breaks.
Find the further time that elapses until $B$ hits the floor.

OCR M1 2014 June Q1

1 A particle $P$ is projected vertically downwards with initial speed $3.5 \mathrm {~ms} ^ { - 1 }$ from a point $A$ which is 5 m above horizontal ground.

Find the speed of $P$ immediately before it strikes the ground. After striking the ground, $P$ rebounds and moves vertically upwards and 0.87 s after leaving the ground $P$ passes through $A$.
Calculate the speed of $P$ immediately after it leaves the ground. It is given that the mass of $P$ is 0.2 kg .
Calculate the change in the momentum of $P$ as a result of its collision with the ground.

OCR M1 2014 June Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-2_309_520_941_744} A particle rests on a smooth horizontal surface. Three horizontal forces of magnitudes $2.5 \mathrm {~N} , F \mathrm {~N}$ and 2.4 N act on the particle on bearings $\theta ^ { \circ } , 180 ^ { \circ }$ and $270 ^ { \circ }$ respectively (see diagram). The particle is in equilibrium.

Find $\theta$ and $F$. The 2.4 N force suddenly ceases to act on the particle, which has mass 0.2 kg .
Find the magnitude and direction of the acceleration of the particle.

OCR M1 2014 June Q3

3 A particle $P$ travels in a straight line. The velocity of $P$ at time $t$ seconds after it passes through a fixed point $A$ is given by $\left( 0.6 t ^ { 2 } + 3 \right) \mathrm { ms } ^ { - 1 }$. Find

the velocity of $P$ when it passes through $A$,
the displacement of $P$ from $A$ when $t = 1.5$,
the velocity of $P$ when it has acceleration $6 \mathrm {~ms} ^ { - 2 }$.

OCR M1 2014 June Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-3_136_824_260_623} Particles $P$ and $Q$ are moving towards each other with constant speeds $4 \mathrm {~ms} ^ { - 1 }$ and $2 \mathrm {~ms} ^ { - 1 }$ along the same straight line on a smooth horizontal surface (see diagram). $P$ has mass 0.2 kg and $Q$ has mass 0.3 kg . The two particles collide.

Show that $Q$ must change its direction of motion in the collision.
Given that $P$ and $Q$ move with equal speed after the collision, calculate both possible values for their speed after they collide.

OCR M1 2014 June Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-3_652_1675_959_187} A particle $P$ can move in a straight line on a horizontal surface. At time $t$ seconds the displacement of $P$ from a fixed point $A$ on the line is $x \mathrm {~m}$. The diagram shows the $( t , x )$ graph for $P$. In the interval $0 \leqslant t \leqslant 10$, either the speed of $P$ is $4 \mathrm {~ms} ^ { - 1 }$, or $P$ is at rest.

Show by calculation that $T = 1.75$.
State the velocity of $P$ when
(a) $t = 2$,
(b) $t = 8$,
(c) $t = 9$.
Calculate the distance travelled by $P$ in the interval $0 \leqslant t \leqslant 10$. For $t > 10$, the displacement of $P$ from $A$ is given by $x = 20 t - t ^ { 2 } - 96$.
Calculate the value of $t$, where $t > 10$, for which the speed of $P$ is $4 \mathrm {~ms} ^ { - 1 }$.

OCR M1 2014 June Q6

6 A particle $P$ of weight 8 N rests on a horizontal surface. A horizontal force of magnitude 3 N acts on $P$, and $P$ is in limiting equilibrium.

Calculate the coefficient of friction between $P$ and the surface.
Find the magnitude and direction of the contact force exerted by the surface on $P$.
\includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-4_190_579_580_598} The initial 3 N force continues to act on $P$ in its original direction. An additional force of magnitude $T \mathrm {~N}$, acting in the same vertical plane as the 3 N force, is now applied to $P$ at an angle of $\theta ^ { \circ }$ above the horizontal (see diagram). $P$ is again in limiting equilibrium.
(a) Given that $\theta = 0$, find $T$.
(b) Given instead that $\theta = 30$, calculate $T$.

OCR M1 2014 June Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-5_510_1091_269_479}
$A$ and $B$ are points at the upper and lower ends, respectively, of a line of greatest slope on a plane inclined at $30 ^ { \circ }$ to the horizontal. $M$ is the mid-point of $A B$. Two particles $P$ and $Q$, joined by a taut light inextensible string, are placed on the plane at $A$ and $M$ respectively. The particles are simultaneously projected with speed $0.6 \mathrm {~ms} ^ { - 1 }$ down the line of greatest slope (see diagram). The particles move down the plane with acceleration $0.9 \mathrm {~ms} ^ { - 2 }$. At the instant 2 s after projection, $P$ is at $M$ and $Q$ is at $B$. The particle $Q$ subsequently remains at rest at $B$.

Find the distance $A B$. The plane is rough between $A$ and $M$, but smooth between $M$ and $B$.
Calculate the speed of $P$ when it reaches $B$.
$P$ has mass 0.4 kg and $Q$ has mass 0.3 kg .
By considering the motion of $Q$, calculate the tension in the string while both particles are moving down the plane.
Calculate the coefficient of friction between $P$ and the plane between $A$ and $M$. \section*{END OF QUESTION PAPER}

CAIE M1 2024 June Q1

1 Two particles $P$ and $Q$ of masses 0.2 kg and 0.5 kg respectively are at rest on a smooth horizontal plane. Particle $P$ is projected with a speed $6 \mathrm {~ms} ^ { - 1 }$ directly towards $Q$. After $P$ and $Q$ collide, $P$ moves with a speed of $1 \mathrm {~ms} ^ { - 1 }$. Find the two possible speeds of $Q$ after the collision.
\includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-02_2716_35_143_2012}

CAIE M1 2024 June Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-03_721_622_296_724} A particle of mass 0.2 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point on a vertical wall. The particle is held in equilibrium by a force of magnitude $X \mathrm {~N}$, perpendicular to the string, with the string taut and making an angle of $30 ^ { \circ }$ with the wall (see diagram). Find the tension in the string and the value of $X$.

CAIE M1 2024 June Q3

3 A car travels along a straight road with constant acceleration $a \mathrm {~ms} ^ { - 2 }$, where $a > 0$. The car passes through points $A , B$ and $C$ in that order. The speed of the car at $A$ is $u \mathrm {~ms} ^ { - 1 }$ in the direction $A B$. The distance $B C$ is twice the distance $A B$. The car takes 8 seconds to travel from $A$ to $B$ and 10 seconds to travel from $B$ to $C$.

Find $u$ in terms of $a$.
Find the speed of the car at $C$ in terms of $a$.

CAIE M1 2024 June Q4

4 A particle travels in a straight line. The velocity of the particle at time $t \mathrm {~s}$ after leaving a point $O$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where $$v = k t ^ { 2 } - 4 t + 3$$ The distance travelled by the particle in the first 2 s of its motion is 6 m . You may assume that $v > 0$ in the first 2s of its motion.

Find the value of $k$.
Find the value of the minimum velocity of the particle. You do not need to show that this velocity is a minimum.

CAIE M1 2024 June Q5

5 A van of mass 4500 kg is towing a trailer of mass 750 kg down a straight hill inclined at an angle of $\theta$ to the horizontal where $\sin \theta = 0.05$. The van and the trailer are connected by a light rigid tow-bar which is parallel to the road. There are constant resistance forces of 2500 N on the van and 300 N on the trailer.

It is given that the tension in the tow-bar is 450 N . Find the acceleration of the trailer and the driving force of the van's engine.
On another occasion, the van and trailer ascend a straight hill inclined at an angle of $\alpha$ to the horizontal where $\sin \alpha = 0.09$. The driving force of the van's engine is now 9100 N , and the speed of the van at the bottom of the hill is $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The resistances to motion are unchanged.
1. Find the acceleration of the van and the tension in the tow-bar.
2. Find the speed of the van when it has travelled a distance of 375 m up the hill.

CAIE M1 2024 June Q6

6 A cyclist is travelling along a straight horizontal road. The total mass of the cyclist and her bicycle is 80 kg . There is a constant resistance force of magnitude 32 N to the cyclist's motion. At an instant when she is travelling at $7 \mathrm {~ms} ^ { - 1 }$, her acceleration is $0.1 \mathrm {~ms} ^ { - 2 }$.

Find the power output of the cyclist.
Find the steady speed that the cyclist can maintain if her power output and the resistance force are both unchanged.
\includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-08_2718_35_141_2012}
\includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-09_2724_35_136_20} The cyclist later descends a straight hill of length 32.2 m , inclined at an angle of $\sin ^ { - 1 } \left( \frac { 1 } { 20 } \right)$ to the horizontal. Her power output is now 120 W , and the resistance force now has variable magnitude such that the work done against this force in descending the hill is 1128 J . The time taken to descend the hill is 4 s .
Given that the speed of the cyclist at the top of the hill is $7.5 \mathrm {~ms} ^ { - 1 }$, find her speed at the bottom of the hill.

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