Questions M1 - A-Level Maths

Edexcel M1 2002 June Q5

5. A particle $P$ of mass 2 kg moves in a plane under the action of a single constant force $\mathbf { F }$ newtons. At time $t$ seconds, the velocity of $P$ is $\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }$. When $t = 0 , \mathbf { v } = ( - 5 \mathbf { i } + 7 \mathbf { j } )$ and when $t = 3 , \mathbf { v } = ( \mathbf { i } - 2 \mathbf { j } )$.

Find in degrees the angle between the direction of motion of $P$ when $t = 3$ and the vector $\mathbf { j }$.
Find the acceleration of $P$.
Find the magnitude of $\mathbf { F }$.
Find in terms of $t$ the velocity of $P$.
Find the time at which $P$ is moving parallel to the vector $\mathbf { i } + \mathbf { j }$.

Edexcel M1 2002 June Q6

6. A man travels in a lift to the top of a tall office block. The lift starts from rest on the ground floor and moves vertically. It comes to rest again at the top floor, having moved a vertical distance of 27 m . The lift initially accelerates with a constant acceleration of $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ until it reaches a speed of $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. It then moves with a constant speed of $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ for $T$ seconds. Finally it decelerates with a constant deceleration for 2.5 s before coming to rest at the top floor.

Sketch a speed-time graph for the motion of the lift.
Hence, or otherwise, find the value of $T$.
Sketch an acceleration-time graph for the motion of the lift. The mass of the man is 80 kg and the mass of the lift is 120 kg . The lift is pulled up by means of a vertical cable attached to the top of the lift. By modelling the cable as light and inextensible, find
the tension in the cable when the lift is accelerating,
(3)
the magnitude of the force exerted by the lift on the man during the last 2.5 s of the motion.
(3)

Edexcel M1 2002 June Q7

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{d23dfb3f-2969-4bca-a369-1d51e6ba0052-6_376_979_391_678}

\end{figure} Particles $A$ and $B$, of mass $2 m$ and $m$ respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley fixed at the edge of a rough horizontal table. Particle $A$ is held on the table, while $B$ rests on a smooth plane inclined at $30 ^ { \circ }$ to the horizontal, as shown in Fig. 3. The string is in the same vertical plane as a line of greatest slope of the inclined plane. The coefficient of friction between $A$ and the table is $\mu$. The particle $A$ is released from rest and begins to move. By writing down an equation of motion for each particle,

show that, while both particles move with the string taut. Each particle has an acceleration of magnitude $\frac { 1 } { 6 } ( 1 - 4 \mu ) g$. When each particle has moved a distance $h$, the string breaks. The particle $A$ comes to rest before reaching the pulley. Given that $\mu = 0.2$,
find, in terms of $h$, the total distance moved by $A$. For the model described above,
state two physical factors, apart from air resistance, which could be taken into account to make the model more realistic.

AQA M1 Q1

1 A particle $A$ moves across a smooth horizontal surface in a straight line. The particle $A$ has mass 2 kg and speed $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. A particle $B$, which has mass 3 kg , is at rest on the surface. The particle $A$ collides with the particle $B$.
\includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-003_147_506_644_733}

If, after the collision, $A$ is at rest and $B$ moves away from $A$, find the speed of $B$.
If, after the collision, $A$ and $B$ move away from each other with speeds $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $4 v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively, as shown in the diagram below, find the value of $v$.
\includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-003_138_506_1144_730}

AQA M1 Q4

4 Water flows in a constant direction at a constant speed of $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$. A boat travels in the water at a speed of $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ relative to the water.

The direction in which the boat travels relative to the water is perpendicular to the direction of motion of the water. The resultant velocity of the boat is $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $74 ^ { \circ }$ to the direction of motion of the water, as shown in the diagram. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_120_164_662_488} \captionsetup{labelformat=empty} \caption{Velocity of the water}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_126_186_667_890} \captionsetup{labelformat=empty} \caption{Velocity of the boat relative to the water}
\end{figure}
1. Find $V$.
2. Show that $u = 3.44$, correct to three significant figures.
The boat changes course so that it travels relative to the water at an angle of $45 ^ { \circ }$ to the direction of motion of the water. The resultant velocity of the boat is now of magnitude $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The velocity of the water is unchanged, as shown in the diagram below. $$\xrightarrow { 3.44 \mathrm {~m} \mathrm {~s} ^ { - 1 } }$$
\includegraphics[max width=\textwidth, alt={}]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_132_273_1493_895}
Velocity of the boat relative to the water
\includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_232_355_1498_1384} Find the value of $v$.
(4 marks)

AQA M1 Q5

5 A golf ball is projected from a point $O$ with initial velocity $V$ at an angle $\alpha$ to the horizontal. The ball first hits the ground at a point $A$ which is at the same horizontal level as $O$, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-005_227_602_484_735} It is given that $V \cos \alpha = 6 u$ and $V \sin \alpha = 2.5 u$.

Show that the time taken for the ball to travel from $O$ to $A$ is $\frac { 5 u } { g }$.
Find, in terms of $g$ and $u$, the distance $O A$.
Find $V$, in terms of $u$.
State, in terms of $u$, the least speed of the ball during its flight from $O$ to $A$.

AQA M1 Q6

6 A van moves from rest on a straight horizontal road.

In a simple model, the first 30 seconds of the motion are represented by three separate stages, each lasting 10 seconds and each with a constant acceleration. During the first stage, the van accelerates from rest to a velocity of $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
During the second stage, the van accelerates from $4 \mathrm {~ms} ^ { - 1 }$ to $12 \mathrm {~ms} ^ { - 1 }$.
During the third stage, the van accelerates from $12 \mathrm {~ms} ^ { - 1 }$ to $16 \mathrm {~ms} ^ { - 1 }$.
1. Sketch a velocity-time graph to represent the motion of the van during the first 30 seconds of its motion.
2. Find the total distance that the van travels during the 30 seconds.
3. Find the average speed of the van during the 30 seconds.
4. Find the greatest acceleration of the van during the 30 seconds.
In another model of the 30 seconds of the motion, the acceleration of the van is assumed to vary during the first and third stages of the motion, but to be constant during the second stage, as shown in the velocity-time graph below.
\includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-006_554_1138_1432_539} The velocity of the van takes the same values at the beginning and the end of each stage of the motion as in part (a).
1. State, with a reason, whether the distance travelled by the van during the first 10 seconds of the motion in this model is greater or less than the distance travelled during the same time interval in the model in part (a).
2. Give one reason why this model represents the motion of the van more realistically than the model in part (a).

AQA M1 Q7

7 A builder ties two identical buckets, $P$ and $Q$, to the ends of a light inextensible rope. He hangs the rope over a smooth beam so that the buckets hang in equilibrium, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-007_360_296_502_904} The buckets are each of mass 0.6 kg .

1. State the magnitude of the tension in the rope.
2. State the magnitude and direction of the force exerted on the beam by the rope.
The bucket $Q$ is held at rest while a stone, of mass 0.2 kg , is placed inside it. The system is then released from rest and, in the subsequent motion, bucket $Q$ moves vertically downwards with the stone inside.
1. By forming an equation of motion for each bucket, show that the magnitude of the tension in the rope during the motion is 6.72 newtons, correct to three significant figures.
2. State the magnitude of the force exerted on the beam by the rope while the motion takes place.

AQA M1 Q8

8 A rough slope is inclined at an angle of $25 ^ { \circ }$ to the horizontal. A box of weight 80 newtons is on the slope. A rope is attached to the box and is parallel to the slope. The tension in the rope is of magnitude $T$ newtons. The diagram shows the slope, the box and the rope.
\includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-008_307_469_500_840}

The box is held in equilibrium by the rope.
1. Show that the normal reaction force between the box and the slope is 72.5 newtons, correct to three significant figures.
2. The coefficient of friction between the box and the slope is 0.32 . Find the magnitude of the maximum value of the frictional force which can act on the box.
3. Find the least possible tension in the rope to prevent the box from moving down the slope.
4. Find the greatest possible tension in the rope.
5. Show that the mass of the box is approximately 8.16 kg .
The rope is now released and the box slides down the slope. Find the acceleration of the box. General Certificate of Education
June 2006
Advanced Subsidiary Examination ASSESSMENT and
REALIFIEATIONS
ALLIANCE Tuesday 6 June 20061.30 pm to 2.45 pm \section*{For this paper you must have:}
- an 8-page answer book
- the blue AQA booklet of formulae and statistical tables
You may use a graphics calculator. Time allowed: 1 hour 15 minutes \section*{Instructions}
- Use blue or black ink or ball-point pen. Pencil should only be used for drawing.
- Write the information required on the front of your answer book. The Examining Body for this paper is AQA. The Paper Reference is MM1A/W.
- Answer all questions.
- Show all necessary working; otherwise marks for method may be lost.
- The final answer to questions requiring the use of calculators should be given to three significant figures, unless stated otherwise.
- Take $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$, unless stated otherwise.
\section*{Information}
- The maximum mark for this paper is 60 .
- The marks for questions are shown in brackets.
- Unit Mechanics 1A has a written paper and coursework.
\section*{Advice}
- Unless stated otherwise, you may quote formulae, without proof, from the booklet.
Answer all questions. 1 A small stone is dropped from a high bridge and falls vertically.
Find the distance that the stone falls during the first 4 seconds of its motion. (3 marks)
Find the speed of the stone when it has been falling for 4 seconds. 2 A car travels along a straight horizontal road. The motion of the car can be modelled as three separate stages. During the first stage, the car accelerates uniformly from rest to a velocity of $10 \mathrm {~ms} ^ { - 1 }$ in 6 seconds. During the second stage, the car travels with a constant velocity of $10 \mathrm {~ms} ^ { - 1 }$ for a further 4 seconds. During the third stage of the motion, the car travels with a uniform retardation of magnitude $0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ until it comes to rest.
Show that the time taken for the third stage of the motion is 12.5 seconds.
Sketch a velocity-time graph for the car during the three stages of the motion.
Find the total distance travelled by the car during the motion. 3 A stone rests in equilibrium on a rough plane inclined at an angle of $16 ^ { \circ }$ to the horizontal, as shown in the diagram. The mass of the stone is 0.5 kg .
\includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-010_222_528_1978_731}
Draw a diagram to show the forces acting on the stone.
Show that the magnitude of the frictional force acting on the stone is 1.35 newtons, correct to three significant figures.
Find the magnitude of the normal reaction force between the stone and the plane.
Hence find an inequality for the value of $\mu$, the coefficient of friction between the stone and the plane. 4 A block $P$ is attached to a can $Q$ by a light inextensible string. The string hangs over a smooth peg so that $P$ and $Q$ hang freely, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-011_246_259_470_886} The block $P$ and the can $Q$ each has mass 0.2 kg . The can $Q$ contains a small stone of mass 0.1 kg . The system is released from rest and the can $Q$ and the stone move vertically downwards.
By forming two equations of motion, show that the magnitude of the acceleration of $P$ and $Q$ is $1.96 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
Find the magnitude of the reaction force between the can and the stone. 5 The points $A$ and $B$ have position vectors $( 3 \mathbf { i } + 2 \mathbf { j } )$ metres and $( 6 \mathbf { i } - 4 \mathbf { j } )$ metres respectively. The vectors $\mathbf { i }$ and $\mathbf { j }$ are in a horizontal plane.
A particle moves from $A$ to $B$ with constant velocity $( \mathbf { i } - 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$. Calculate the time that the particle takes to move from $A$ to $B$.
The particle then moves from $B$ to a point $C$ with a constant acceleration of $2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 2 }$. It takes 4 seconds to move from $B$ to $C$.
1. Find the position vector of $C$.
2. Find the distance $A C$.

AQA M1 2006 January Q1

1 A particle $A$ moves across a smooth horizontal surface in a straight line. The particle $A$ has mass 2 kg and speed $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. A particle $B$, which has mass 3 kg , is at rest on the surface. The particle $A$ collides with the particle $B$.
\includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-2_147_506_644_733}

If, after the collision, $A$ is at rest and $B$ moves away from $A$, find the speed of $B$.
If, after the collision, $A$ and $B$ move away from each other with speeds $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $4 v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively, as shown in the diagram below, find the value of $v$.
\includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-2_138_506_1144_730}

AQA M1 2006 January Q2

2 A particle $P$ moves with acceleration $( - 3 \mathbf { i } + 12 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }$. Initially the velocity of $P$ is $4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Find the velocity of $P$ at time $t$ seconds.
Find the speed of $P$ when $t = 0.5$.

AQA M1 2006 January Q3

3

A small stone is dropped from a height of 25 metres above the ground.
1. Find the time taken for the stone to reach the ground.
2. Find the speed of the stone as it reaches the ground.
A large package is dropped from the same height as the stone. Explain briefly why the time taken for the package to reach the ground is likely to be different from that for the stone.
(2 marks)

AQA M1 2006 January Q4

4 Water flows in a constant direction at a constant speed of $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$. A boat travels in the water at a speed of $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ relative to the water.

The direction in which the boat travels relative to the water is perpendicular to the direction of motion of the water. The resultant velocity of the boat is $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $74 ^ { \circ }$ to the direction of motion of the water, as shown in the diagram. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c220e6c4-2676-4022-8301-7d720dc082b2-3_120_164_662_488} \captionsetup{labelformat=empty} \caption{Velocity of the water}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c220e6c4-2676-4022-8301-7d720dc082b2-3_126_186_667_890} \captionsetup{labelformat=empty} \caption{Velocity of the boat relative to the water}
\end{figure}
1. Find $V$.
2. Show that $u = 3.44$, correct to three significant figures.
The boat changes course so that it travels relative to the water at an angle of $45 ^ { \circ }$ to the direction of motion of the water. The resultant velocity of the boat is now of magnitude $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The velocity of the water is unchanged, as shown in the diagram below. $$\xrightarrow { 3.44 \mathrm {~m} \mathrm {~s} ^ { - 1 } }$$
\includegraphics[max width=\textwidth, alt={}]{c220e6c4-2676-4022-8301-7d720dc082b2-3_132_273_1493_895}
Velocity of the boat relative to the water
\includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-3_232_355_1498_1384} Find the value of $v$.
(4 marks)

AQA M1 2006 January Q5

5 A golf ball is projected from a point $O$ with initial velocity $V$ at an angle $\alpha$ to the horizontal. The ball first hits the ground at a point $A$ which is at the same horizontal level as $O$, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-4_227_602_484_735} It is given that $V \cos \alpha = 6 u$ and $V \sin \alpha = 2.5 u$.

Show that the time taken for the ball to travel from $O$ to $A$ is $\frac { 5 u } { g }$.
Find, in terms of $g$ and $u$, the distance $O A$.
Find $V$, in terms of $u$.
State, in terms of $u$, the least speed of the ball during its flight from $O$ to $A$.

AQA M1 2006 January Q6

6 A van moves from rest on a straight horizontal road.

In a simple model, the first 30 seconds of the motion are represented by three separate stages, each lasting 10 seconds and each with a constant acceleration. During the first stage, the van accelerates from rest to a velocity of $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
During the second stage, the van accelerates from $4 \mathrm {~ms} ^ { - 1 }$ to $12 \mathrm {~ms} ^ { - 1 }$.
During the third stage, the van accelerates from $12 \mathrm {~ms} ^ { - 1 }$ to $16 \mathrm {~ms} ^ { - 1 }$.
1. Sketch a velocity-time graph to represent the motion of the van during the first 30 seconds of its motion.
2. Find the total distance that the van travels during the 30 seconds.
3. Find the average speed of the van during the 30 seconds.
4. Find the greatest acceleration of the van during the 30 seconds.
In another model of the 30 seconds of the motion, the acceleration of the van is assumed to vary during the first and third stages of the motion, but to be constant during the second stage, as shown in the velocity-time graph below.
\includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-5_554_1138_1432_539} The velocity of the van takes the same values at the beginning and the end of each stage of the motion as in part (a).
1. State, with a reason, whether the distance travelled by the van during the first 10 seconds of the motion in this model is greater or less than the distance travelled during the same time interval in the model in part (a).
2. Give one reason why this model represents the motion of the van more realistically than the model in part (a).

AQA M1 2006 January Q7

7 A builder ties two identical buckets, $P$ and $Q$, to the ends of a light inextensible rope. He hangs the rope over a smooth beam so that the buckets hang in equilibrium, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-6_360_296_502_904} The buckets are each of mass 0.6 kg .

1. State the magnitude of the tension in the rope.
2. State the magnitude and direction of the force exerted on the beam by the rope.
The bucket $Q$ is held at rest while a stone, of mass 0.2 kg , is placed inside it. The system is then released from rest and, in the subsequent motion, bucket $Q$ moves vertically downwards with the stone inside.
1. By forming an equation of motion for each bucket, show that the magnitude of the tension in the rope during the motion is 6.72 newtons, correct to three significant figures.
2. State the magnitude of the force exerted on the beam by the rope while the motion takes place.

AQA M1 2006 January Q8

8 A rough slope is inclined at an angle of $25 ^ { \circ }$ to the horizontal. A box of weight 80 newtons is on the slope. A rope is attached to the box and is parallel to the slope. The tension in the rope is of magnitude $T$ newtons. The diagram shows the slope, the box and the rope.
\includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-7_307_469_500_840}

The box is held in equilibrium by the rope.
1. Show that the normal reaction force between the box and the slope is 72.5 newtons, correct to three significant figures.
2. The coefficient of friction between the box and the slope is 0.32 . Find the magnitude of the maximum value of the frictional force which can act on the box.
3. Find the least possible tension in the rope to prevent the box from moving down the slope.
4. Find the greatest possible tension in the rope.
5. Show that the mass of the box is approximately 8.16 kg .
The rope is now released and the box slides down the slope. Find the acceleration of the box.

AQA M1 2010 January Q1

1 Two particles, $A$ and $B$, are travelling in the same direction along a straight line on a smooth horizontal surface. Particle $A$ has mass 3 kg and particle $B$ has mass 7 kg . Particle $A$ has a speed of $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and particle $B$ has a speed of $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-2_186_835_653_593} Particle $A$ and particle $B$ collide and coalesce to form a single particle. Find the speed of this single particle after the collision.

AQA M1 2010 January Q2

2 A sprinter accelerates from rest at a constant rate for the first 10 metres of a 100 -metre race. He takes 2.5 seconds to run the first 10 metres.

Find the acceleration of the sprinter during the first 2.5 seconds of the race.
Show that the speed of the sprinter at the end of the first 2.5 seconds of the race is $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
The sprinter completes the 100 -metre race, travelling the remaining 90 metres at a constant speed of $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find the total time taken for the sprinter to travel the 100 metres.
Calculate the average speed of the sprinter during the 100 -metre race.

AQA M1 2010 January Q3

3 A particle of mass 3 kg is on a smooth slope inclined at $60 ^ { \circ }$ to the horizontal. The particle is held at rest by a force of $T$ newtons parallel to the slope, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-2_337_284_2023_879}

Draw a diagram to show all the forces acting on the particle.
Show that the magnitude of the normal reaction acting on the particle is 14.7 newtons.
Find $T$.

AQA M1 2010 January Q4

4 A ball is released from rest at a height of 15 metres above ground level.

Find the speed of the ball when it hits the ground, assuming that no air resistance acts on the ball.
In fact, air resistance does act on the ball. Assume that the air resistance force has a constant magnitude of 0.9 newtons. The ball has a mass of 0.5 kg .
1. Draw a diagram to show the forces acting on the ball, including the magnitudes of the forces acting.
2. Show that the acceleration of the ball is $8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
3. Find the speed at which the ball hits the ground.
4. Explain why the assumption that the air resistance force is constant may not be valid.

AQA M1 2010 January Q5

5 The constant forces $\mathbf { F } _ { 1 } = ( 8 \mathbf { i } + 12 \mathbf { j } )$ newtons and $\mathbf { F } _ { 2 } = ( 4 \mathbf { i } - 4 \mathbf { j } )$ newtons act on a particle. No other forces act on the particle.

Find the resultant force acting on the particle.
Given that the mass of the particle is 4 kg , show that the acceleration of the particle is $( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }$.
At time $t$ seconds, the velocity of the particle is $\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }$.
1. When $t = 20 , \mathbf { v } = 40 \mathbf { i } + 32 \mathbf { j }$. Show that $\mathbf { v } = - 20 \mathbf { i } - 8 \mathbf { j }$ when $t = 0$.
2. Write down an expression for $\mathbf { v }$ at time $t$.
3. Find the times when the speed of the particle is $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

AQA M1 2010 January Q6

6 A small train at an amusement park consists of an engine and two carriages connected to each other by light horizontal rods, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-4_190_1038_420_493} The engine has mass 2000 kg and each carriage has mass 500 kg . The train moves along a straight horizontal track. A resistance force of magnitude 400 newtons acts on the engine, and resistance forces of magnitude 300 newtons act on each carriage. The train is accelerating at $0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.

Draw a diagram to show the horizontal forces acting on Carriage 2.
Show that the magnitude of the force that the rod exerts on Carriage 2 is 550 newtons.
Find the magnitude of the force that the rod attached to the engine exerts on Carriage 1.
A forward driving force of magnitude $P$ newtons acts on the engine. Find $P$.

AQA M1 2010 January Q7

7 A ball is projected horizontally with speed $\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }$ at a height of 5 metres above horizontal ground. When the ball has travelled a horizontal distance of 15 metres, it hits the ground.
\includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-4_433_1296_1674_338}

Show that the time it takes for the ball to travel to the point where it hits the ground is 1.01 seconds, correct to three significant figures.
Find $V$.
Find the speed of the ball when it hits the ground.
Find the angle between the velocity of the ball and the horizontal when the ball hits the ground. Give your answer to the nearest degree.
State two assumptions that you have made about the ball while it is moving.

AQA M1 2010 January Q8

8 A crate, of mass 200 kg , is initially at rest on a rough horizontal surface. A smooth ring is attached to the crate. A light inextensible rope is passed through the ring, and each end of the rope is attached to a tractor. The lower part of the rope is horizontal and the upper part is at an angle of $20 ^ { \circ }$ to the horizontal, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-5_344_1186_518_420} When the tractor moves forward, the crate accelerates at $0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. The coefficient of friction between the crate and the surface is 0.4 . Assume that the tension, $T$ newtons, is the same in both parts of the rope.

Draw and label a diagram to show the forces acting on the crate.
Express the normal reaction between the surface and the crate in terms of $T$.
Find $T$.

Questions M1 (1912 questions)

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