Questions - A-Level Maths

AQA M1 2010 January Q7

14 marks Moderate -0.8

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

10 marks Standard +0.3

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 $T$.

AQA M1 2007 June Q1

7 marks Easy -1.2

1 A ball is released from rest at a height $h$ metres above ground level. The ball hits the ground 1.5 seconds after it is released. Assume that the ball is a particle that does not experience any air resistance.

Show that the speed of the ball is $14.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when it hits the ground.
Find $h$.
Find the distance that the ball has fallen when its speed is $5 \mathrm {~ms} ^ { - 1 }$.

AQA M1 2007 June Q2

5 marks Moderate -0.8

2 Two particles, $A$ and $B$, are moving on a smooth horizontal surface. Particle $A$ has mass 2 kg and velocity $\left[ \begin{array} { r } 3 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$. Particle $B$ has mass 3 kg and velocity $\left[ \begin{array} { r } - 4 \\ 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$. The two particles collide, and they coalesce during the collision.

Find the velocity of the combined particles after the collision.
Find the speed of the combined particles after the collision.

AQA M1 2007 June Q3

10 marks Moderate -0.8

3 A sign, of mass 2 kg , is suspended from the ceiling of a supermarket by two light strings. It hangs in equilibrium with each string making an angle of $35 ^ { \circ }$ to the vertical, as shown in the diagram. Model the sign as a particle. \includegraphics[max width=\textwidth, alt={}, center]{81f3753c-f148-44be-8b35-0a8e531016dd-2_424_385_1790_824}

By resolving forces horizontally, show that the tension is the same in each string.
Find the tension in each string.
If the tension in a string exceeds 40 N , the string will break. Find the mass of the heaviest sign that could be suspended as shown in the diagram.

AQA M1 2007 June Q4

9 marks Moderate -0.3

4 A car, of mass 1200 kg , is connected by a tow rope to a truck, of mass 2800 kg . The truck tows the car in a straight line along a horizontal road. Assume that the tow rope is horizontal. A horizontal driving force of magnitude 3000 N acts on the truck. A horizontal resistance force of magnitude 800 N acts on the car. The car and truck accelerate at $0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. \includegraphics[max width=\textwidth, alt={}, center]{81f3753c-f148-44be-8b35-0a8e531016dd-3_177_1002_580_513}

Find the tension in the tow rope.
Show that the magnitude of the horizontal resistance force acting on the truck is 600 N .
In fact, the tow rope is not horizontal. Assume that the resistance forces and the driving force are unchanged. Is the tension in the tow rope greater or less than in part (a)? Explain why.

AQA M1 2007 June Q5

5 marks Moderate -0.3

5 An aeroplane flies in air that is moving due east at a speed of $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The velocity of the aeroplane relative to the air is $150 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ due north. The aeroplane actually travels on a bearing of $030 ^ { \circ }$.

Show that $V = 86.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, correct to three significant figures.
Find the magnitude of the resultant velocity of the aeroplane.

AQA M1 2007 June Q6

15 marks Moderate -0.8

6 A box, of mass 3 kg , is placed on a slope inclined at an angle of $30 ^ { \circ }$ to the horizontal. The box slides down the slope. Assume that air resistance can be ignored.

A simple model assumes that the slope is smooth.
1. Draw a diagram to show the forces acting on the box.
2. Show that the acceleration of the box is $4.9 \mathrm {~ms} ^ { - 2 }$.
A revised model assumes that the slope is rough. The box slides down the slope from rest, travelling 5 metres in 2 seconds.
1. Show that the acceleration of the box is $2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
2. Find the magnitude of the friction force acting on the box.
3. Find the coefficient of friction between the box and the slope.
4. In reality, air resistance affects the motion of the box. Explain how its acceleration would change if you took this into account.

AQA M1 2007 June Q7

12 marks Moderate -0.3

7 An arrow is fired from a point $A$ with a velocity of $25 \mathrm {~ms} ^ { - 1 }$, at an angle of $40 ^ { \circ }$ above the horizontal. The arrow hits a target at the point $B$ which is at the same level as the point $A$, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{81f3753c-f148-44be-8b35-0a8e531016dd-4_195_1093_1594_511}

State two assumptions that you should make in order to model the motion of the arrow.
(2 marks)
Show that the time that it takes for the arrow to travel from $A$ to $B$ is 3.28 seconds, correct to three significant figures.
Find the distance between the points $A$ and $B$.
State the magnitude and direction of the velocity of the arrow when it hits the target.
Find the minimum speed of the arrow during its flight.

AQA M1 2007 June Q8

12 marks Moderate -0.8

8 A boat is initially at the origin, heading due east at $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. It then experiences a constant acceleration of $( - 0.2 \mathbf { i } + 0.25 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }$. The unit vectors $\mathbf { i }$ and $\mathbf { j }$ are directed east and north respectively.

State the initial velocity of the boat as a vector.
Find an expression for the velocity of the boat $t$ seconds after it has started to accelerate.
Find the value of $t$ when the boat is travelling due north.
Find the bearing of the boat from the origin when the boat is travelling due north.

OCR M1 Q1

7 marks Standard +0.3

1 \includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-02_508_501_274_822} A light inextensible string has its ends attached to two fixed points $A$ and $B$. The point $A$ is vertically above $B$. A smooth ring $R$ of mass $m \mathrm {~kg}$ is threaded on the string and is pulled by a force of magnitude 1.6 N acting upwards at $45 ^ { \circ }$ to the horizontal. The section $A R$ of the string makes an angle of $30 ^ { \circ }$ with the downward vertical and the section $B R$ is horizontal (see diagram). The ring is in equilibrium with the string taut.

Give a reason why the tension in the part $A R$ of the string is the same as that in the part $B R$.
Show that the tension in the string is 0.754 N , correct to 3 significant figures.
Find the value of $m$.

OCR M1 Q2

7 marks Moderate -0.3

2 \includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-02_643_289_1475_927} Particles $A$ and $B$, of masses 0.2 kg and 0.3 kg respectively, are attached to the ends of a light inextensible string. Particle $A$ is held at rest at a fixed point and $B$ hangs vertically below $A$. Particle $A$ is now released. As the particles fall the air resistance acting on $A$ is 0.4 N and the air resistance acting on $B$ is 0.25 N (see diagram). The downward acceleration of each of the particles is $a \mathrm {~ms} ^ { - 2 }$ and the tension in the string is $T \mathrm {~N}$.

Write down two equations in $a$ and $T$ obtained by applying Newton's second law to $A$ and to $B$.
Find the values of $a$ and $T$. \section*{June 2005}

OCR M1 Q5

17 marks Standard +0.3

5 \includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-03_697_579_1238_781} Two small rings $A$ and $B$ are attached to opposite ends of a light inextensible string. The rings are threaded on a rough wire which is fixed vertically. $A$ is above $B$. A horizontal force is applied to a point $P$ of the string. Both parts $A P$ and $B P$ of the string are taut. The system is in equilibrium with angle $B A P = \alpha$ and angle $A B P = \beta$ (see diagram). The weight of $A$ is 2 N and the tensions in the parts $A P$ and $B P$ of the string are 7 N and $T \mathrm {~N}$ respectively. It is given that $\cos \alpha = 0.28$ and $\sin \alpha = 0.96$, and that $A$ is in limiting equilibrium.

Find the coefficient of friction between the wire and the ring $A$.
By considering the forces acting at $P$, show that $T \cos \beta = 1.96$.
Given that there is no frictional force acting on $B$, find the mass of $B$. \section*{June 2005}

OCR M1 Q7

17 marks Standard +0.3

7 \includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-04_632_1121_936_511} A car $P$ starts from rest and travels along a straight road for 600 s . The ( $t , v$ ) graph for the journey is shown in the diagram. This graph consists of three straight line segments. Find

the distance travelled by $P$,
the deceleration of $P$ during the interval $500 < t < 600$. Another car $Q$ starts from rest at the same instant as $P$ and travels in the same direction along the same road for 600 s . At time $t \mathrm {~s}$ after starting the velocity of $Q$ is $\left( 600 t ^ { 2 } - t ^ { 3 } \right) \times 10 ^ { - 6 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find an expression in terms of $t$ for the acceleration of $Q$.
Find how much less $Q$ 's deceleration is than $P$ 's when $t = 550$.
Show that $Q$ has its maximum velocity when $t = 400$.
Find how much further $Q$ has travelled than $P$ when $t = 400$. \section*{Jan 2006} 1 \includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-05_547_455_269_845} Particles $P$ and $Q$, of masses 0.3 kg and 0.4 kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The system is in motion with the string taut and with each of the particles moving vertically. The downward acceleration of $P$ is $a \mathrm {~ms} ^ { - 2 }$ (see diagram).
Show that $a = - 1.4$. Initially $P$ and $Q$ are at the same horizontal level. $P$ 's initial velocity is vertically downwards and has magnitude $2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Assuming that $P$ does not reach the floor and that $Q$ does not reach the pulley, find the time taken for $P$ to return to its initial position. 2 \includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-05_616_809_1562_667} An object of mass 0.08 kg is attached to one end of a light inextensible string. The other end of the string is attached to the underside of the roof inside a furniture van. The van is moving horizontally with constant acceleration $1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. The string makes a constant angle $\alpha$ with the downward vertical and the tension in the string is $T \mathrm {~N}$ (see diagram).
By applying Newton's second law horizontally to the object, find the value of $T \sin \alpha$.
Find the value of $T$. 3 A motorcyclist starts from rest at a point $O$ and travels in a straight line. His velocity after $t$ seconds is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, for $0 \leqslant t \leqslant T$, where $v = 7.2 t - 0.45 t ^ { 2 }$. The motorcyclist's acceleration is zero when $t = T$.
Find the value of $T$.
Show that $v = 28.8$ when $t = T$. For $t \geqslant T$ the motorcyclist travels in the same direction as before, but with constant speed $28.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find the displacement of the motorcyclist from $O$ when $t = 31$. 4 \includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-06_225_623_813_762} A block of mass 2 kg is at rest on a rough horizontal plane, acted on by a force of magnitude 12 N at an angle of $15 ^ { \circ }$ upwards from the horizontal (see diagram).
Find the frictional component of the contact force exerted on the block by the plane.
Show that the normal component of the contact force exerted on the block by the plane has magnitude 16.5 N , correct to 3 significant figures. It is given that the block is on the point of sliding.
Find the coefficient of friction between the block and the plane. The force of magnitude 12 N is now replaced by a horizontal force of magnitude 20 N . The block starts to move.
Find the acceleration of the block. 5 A man drives a car on a horizontal straight road. At $t = 0$, where the time $t$ is in seconds, the car runs out of petrol. At this instant the car is moving at $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The car decelerates uniformly, coming to rest when $t = 8$. The man then walks back along the road at $0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ until he reaches a petrol station a distance of 420 m from his car. After his arrival at the petrol station it takes him 250 s to obtain a can of petrol. He is then given a lift back to his car on a motorcycle. The motorcycle starts from rest and accelerates uniformly until its speed is $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$; it then decelerates uniformly, coming to rest at the stationary car at time $t = T$.
Sketch the shape of the $( t , v )$ graph for the man for $0 \leqslant t \leqslant T$. [Your sketch need not be drawn to scale; numerical values need not be shown.]
Find the deceleration of the car for $0 < t < 8$.
Find the value of $T$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-07_540_542_267_799} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A smooth ring $R$ of weight $W \mathrm {~N}$ is threaded on a light inextensible string. The ends of the string are attached to fixed points $A$ and $B$, where $A$ is vertically above $B$. A horizontal force of magnitude $P \mathrm {~N}$ acts on $R$. The system is in equilibrium with the string taut; $A R$ makes an angle $\alpha$ with the downward vertical and $B R$ makes an angle $\beta$ with the upward vertical (see Fig.1).
By considering the vertical components of the forces acting on $R$, show that $\alpha < \beta$.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-07_537_559_1302_833} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} It is given that when $P = 14 , A R = 0.4 \mathrm {~m} , B R = 0.3 \mathrm {~m}$ and the distance of $R$ from the vertical line $A B$ is 0.24 m (see Fig. 2). Find
(a) the tension in the string,
(b) the value of $W$.
For the case when $P = 0$,
(a) describe the position of $R$,
(b) state the tension in the string. \includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-08_624_1077_267_536} $P Q$ is a line of greatest slope, of length 4 m , on a smooth plane inclined at $30 ^ { \circ }$ to the horizontal. Particles $A$ and $B$, of masses 0.15 kg and 0.5 kg respectively, move along $P Q$ with $A$ below $B$. The particles are both moving upwards, $A$ with speed $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $B$ with speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, when they collide at the mid-point of $P Q$ (see diagram). Particle $A$ is instantaneously at rest immediately after the collision.
Show that $B$ does not reach $Q$ in the subsequent motion.
Find the time interval between the instant of $A$ 's arrival at $P$ and the instant of $B$ 's arrival at $P$. \section*{June 2006} 1 Each of two wagons has an unloaded mass of 1200 kg . One of the wagons carries a load of mass $m \mathrm {~kg}$ and the other wagon is unloaded. The wagons are moving towards each other on the same rails, each with speed $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, when they collide. Immediately after the collision the loaded wagon is at rest and the speed of the unloaded wagon is $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find the value of $m$. 2 \includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-09_622_716_541_715} Forces of magnitudes 6.5 N and 2.5 N act at a point in the directions shown. The resultant of the two forces has magnitude $R \mathrm {~N}$ and acts at right angles to the force of magnitude 2.5 N (see diagram).
Show that $\theta = 22.6 ^ { \circ }$, correct to 3 significant figures.
Find the value of $R$. 3 A man travels 360 m along a straight road. He walks for the first 120 m at $1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, runs the next 180 m at $4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, and then walks the final 60 m at $1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The man's displacement from his starting point after $t$ seconds is $x$ metres.
Sketch the $( t , x )$ graph for the journey, showing the values of $t$ for which $x = 120,300$ and 360 . A woman jogs the same 360 m route at constant speed, starting at the same instant as the man and finishing at the same instant as the man.
Draw a dotted line on your $( t , x )$ graph to represent the woman's journey.
Calculate the value of $t$ at which the man overtakes the woman. June 2006
4 A cyclist travels along a straight road. Her velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, at time $t$ seconds after starting from a point $O$, is given by $$\begin{aligned} & v = 2 \quad \text { for } 0 \leqslant t \leqslant 10 , \\ & v = 0.03 t ^ { 2 } - 0.3 t + 2 \quad \text { for } t \geqslant 10 . \end{aligned}$$
Find the displacement of the cyclist from $O$ when $t = 10$.
Show that, for $t \geqslant 10$, the displacement of the cyclist from $O$ is given by the expression $0.01 t ^ { 3 } - 0.15 t ^ { 2 } + 2 t + 5$.
Find the time when the acceleration of the cyclist is $0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. Hence find the displacement of the cyclist from $O$ when her acceleration is $0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. 5 A block of mass $m \mathrm {~kg}$ is at rest on a horizontal plane. The coefficient of friction between the block and the plane is 0.2 .
When a horizontal force of magnitude 5 N acts on the block, the block is on the point of slipping. Find the value of $m$.
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-10_312_711_1244_758} When a force of magnitude $P \mathrm {~N}$ acts downwards on the block at an angle $\alpha$ to the horizontal, as shown in the diagram, the frictional force on the block has magnitude 6 N and the block is again on the point of slipping. Find
(a) the value of $\alpha$ in degrees,
(b) the value of $P$. June 2006
6 \includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-11_317_999_264_575} A train of total mass 80000 kg consists of an engine $E$ and two trucks $A$ and $B$. The engine $E$ and truck $A$ are connected by a rigid coupling $X$, and trucks $A$ and $B$ are connected by another rigid coupling $Y$. The couplings are light and horizontal. The train is moving along a straight horizontal track. The resistances to motion acting on $E , A$ and $B$ are $10500 \mathrm {~N} , 3000 \mathrm {~N}$ and 1500 N respectively (see diagram).
By modelling the whole train as a single particle, show that it is decelerating when the driving force of the engine is less than 15000 N .
Show that, when the magnitude of the driving force is 35000 N , the acceleration of the train is $0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
Hence find the mass of $E$, given that the tension in the coupling $X$ is 8500 N when the magnitude of the driving force is 35000 N . The driving force is replaced by a braking force of magnitude 15000 N acting on the engine. The force exerted by the coupling $Y$ is zero.
Find the mass of $B$.
Show that the coupling $X$ exerts a forward force of magnitude 1500 N on the engine. 7 A particle of mass 0.1 kg is at rest at a point $A$ on a rough plane inclined at $15 ^ { \circ }$ to the horizontal. The particle is given an initial velocity of $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and starts to move up a line of greatest slope of the plane. The particle comes to instantaneous rest after 1.5 s .
Find the coefficient of friction between the particle and the plane.
Show that, after coming to instantaneous rest, the particle moves down the plane.
Find the speed with which the particle passes through $A$ during its downward motion. 1 A trailer of mass 600 kg is attached to a car of mass 1100 kg by a light rigid horizontal tow-bar. The car and trailer are travelling along a horizontal straight road with acceleration $0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
Given that the force exerted on the trailer by the tow-bar is 700 N , find the resistance to motion of the trailer.
Given also that the driving force of the car is 2100 N , find the resistance to motion of the car. 2 \includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-12_595_791_671_676} Three horizontal forces of magnitudes $15 \mathrm {~N} , 11 \mathrm {~N}$ and 13 N act on a particle $P$ in the directions shown in the diagram. The angles $\alpha$ and $\beta$ are such that $\sin \alpha = 0.28 , \cos \alpha = 0.96 , \sin \beta = 0.8$ and $\cos \beta = 0.6$.
Show that the component, in the $y$-direction, of the resultant of the three forces is zero.
Find the magnitude of the resultant of the three forces.
State the direction of the resultant of the three forces. 3 \includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-12_351_716_1804_715} A block $B$ of mass 0.4 kg and a particle $P$ of mass 0.3 kg are connected by a light inextensible string. The string passes over a smooth pulley at the edge of a rough horizontal table. $B$ is in contact with the table and the part of the string between $B$ and the pulley is horizontal. $P$ hangs freely below the pulley (see diagram).
The system is in limiting equilibrium with the string taut and $P$ on the point of moving downwards. Find the coefficient of friction between $B$ and the table.
A horizontal force of magnitude $X \mathrm {~N}$, acting directly away from the pulley, is now applied to $B$. The system is again in limiting equilibrium with the string taut, and with $P$ now on the point of moving upwards. Find the value of $X$. 4 \includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-13_225_1155_264_497} Three uniform spheres $L , M$ and $N$ have masses $0.8 \mathrm {~kg} , 0.6 \mathrm {~kg}$ and 0.7 kg respectively. The spheres are moving in a straight line on a smooth horizontal table, with $M$ between $L$ and $N$. The sphere $L$ is moving towards $M$ with speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the spheres $M$ and $N$ are moving towards $L$ with speeds $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively (see diagram).
$L$ collides with $M$. As a result of this collision the direction of motion of $M$ is reversed, and its speed remains $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find the speed of $L$ after the collision.
$M$ then collides with $N$.
(a) Find the total momentum of $M$ and $N$ in the direction of $M$ 's motion before this collision takes place, and deduce that the direction of motion of $N$ is reversed as a result of this collision.
(b) Given that $M$ is at rest immediately after this collision, find the speed of $N$ immediately after this collision. 5 A particle starts from rest at a point $A$ at time $t = 0$, where $t$ is in seconds. The particle moves in a straight line. For $0 \leqslant t \leqslant 4$ the acceleration is $1.8 t \mathrm {~m} \mathrm {~s} ^ { - 2 }$, and for $4 \leqslant t \leqslant 7$ the particle has constant acceleration $7.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
Find an expression for the velocity of the particle in terms of $t$, valid for $0 \leqslant t \leqslant 4$.
Show that the displacement of the particle from $A$ is 19.2 m when $t = 4$.
Find the displacement of the particle from $A$ when $t = 7$. \section*{Jan 2007} 6 \includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-14_558_1373_267_386} The diagram shows the ( $t , v$ ) graph for the motion of a hoist used to deliver materials to different levels at a building site. The hoist moves vertically. The graph consists of straight line segments. In the first stage the hoist travels upwards from ground level for 25 s , coming to rest 8 m above ground level.
Find the greatest speed reached by the hoist during this stage. The second stage consists of a 40 s wait at the level reached during the first stage. In the third stage the hoist continues upwards until it comes to rest 40 m above ground level, arriving 135 s after leaving ground level. The hoist accelerates at $0.02 \mathrm {~ms} ^ { - 2 }$ for the first 40 s of the third stage, reaching a speed of $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find
the value of $V$,
the length of time during the third stage for which the hoist is moving at constant speed,
the deceleration of the hoist in the final part of the third stage. 7 A particle $P$ of mass 0.5 kg moves upwards along a line of greatest slope of a rough plane inclined at an angle of $40 ^ { \circ }$ to the horizontal. $P$ reaches its highest point and then moves back down the plane. The coefficient of friction between $P$ and the plane is 0.6 .
Show that the magnitude of the frictional force acting on $P$ is 2.25 N , correct to 3 significant figures.
Find the acceleration of $P$ when it is moving
(a) up the plane,
(b) down the plane.
When $P$ is moving up the plane, it passes through a point $A$ with speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
(a) Find the length of time before $P$ reaches its highest point.
(b) Find the total length of time for $P$ to travel from the point $A$ to its highest point and back to $A$.

AQA M2 Q1

Moderate -0.3

1 A uniform beam, $A B$, has mass 20 kg and length 7 metres. A rope is attached to the beam at $A$. A second rope is attached to the beam at the point $C$, which is 2 metres from $B$. Both of the ropes are vertical. The beam is in equilibrium in a horizontal position, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-003_298_906_756_552} Find the tensions in the two ropes.

AQA M2 Q2

Moderate -0.8

2 A particle, of mass 2 kg , is attached to one end of a light inextensible string. The other end is fixed to the point $O$. The particle is set into motion, so that it describes a horizontal circle of radius 0.6 metres, with the string at an angle of $30 ^ { \circ }$ to the vertical. The centre of the circle is vertically below $O$. \includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-003_346_340_1580_842}

Show that the tension in the string is 22.6 N , correct to three significant figures.
Find the speed of the particle.

AQA M2 Q3

Moderate -0.8

3 A particle moves in a straight line and at time $t$ has velocity $v$, where $$v = 2 t - 12 \mathrm { e } ^ { - t } , \quad t \geqslant 0$$

1. Find an expression for the acceleration of the particle at time $t$.
2. State the range of values of the acceleration of the particle.
When $t = 0$, the particle is at the origin. Find an expression for the displacement of the particle from the origin at time $t$.
(4 marks)

AQA M2 Q4

Standard +0.3

4 A car has a maximum speed of $42 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when it is moving on a horizontal road. When the speed of the car is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, it experiences a resistance force of magnitude $30 v$ newtons.

Show that the maximum power of the car is 52920 W .
The car has mass 1200 kg . It travels, from rest, up a slope inclined at $5 ^ { \circ }$ to the horizontal.
1. Show that, when the car is travelling at its maximum speed $\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }$ up the slope, $$V ^ { 2 } + 392 \sin 5 ^ { \circ } V - 1764 = 0$$
2. Hence find $V$.

AQA M2 Q5

Standard +0.3

5 A car, of mass 1600 kg , is travelling along a straight horizontal road at a speed of $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when the driving force is removed. The car then freewheels and experiences a resistance force. The resistance force has magnitude $40 v$ newtons, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the speed of the car after it has been freewheeling for $t$ seconds. Find an expression for $v$ in terms of $t$.

AQA M2 Q6

Standard +0.3

6 A particle $P$, of mass $m \mathrm {~kg}$, is placed at the point $Q$ on the top of a smooth upturned hemisphere of radius 3 metres and centre $O$. The plane face of the hemisphere is fixed to a horizontal table. The particle is set into motion with an initial horizontal velocity of $2 \mathrm {~ms} ^ { - 1 }$. When the particle is on the surface of the hemisphere, the angle between $O P$ and $O Q$ is $\theta$ and the particle has speed $v \mathrm {~ms} ^ { - 1 }$. \includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-005_419_1013_607_511}

Show that $v ^ { 2 } = 4 + 6 g ( 1 - \cos \theta )$.
Find the value of $\theta$ when the particle leaves the hemisphere.

AQA M2 Q7

Standard +0.3

7 A particle, of mass 10 kg , is attached to one end of a light elastic string of natural length 0.4 metres and modulus of elasticity 100 N . The other end of the string is fixed to the point $O$.

Find the length of the elastic string when the particle hangs in equilibrium directly below $O$.
The particle is pulled down and held at a point $P$, which is 1 metre vertically below $O$. Show that the elastic potential energy of the string when the particle is in this position is 45 J .
The particle is released from rest at the point $P$. In the subsequent motion, the particle has speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when it is $x$ metres below $\boldsymbol { O }$.
1. Show that, while the string is taut, $$v ^ { 2 } = 39.6 x - 25 x ^ { 2 } - 14.6$$
2. Find the value of $x$ when the particle comes to rest for the first time after being released, given that the string is still taut.

AQA M2 Q8

Standard +0.3

8 Two small blocks, $A$ and $B$, of masses 0.8 kg and 1.2 kg respectively, are stuck together. A spring has natural length 0.5 metres and modulus of elasticity 49 N . One end of the spring is attached to the top of the block $A$ and the other end of the spring is attached to a fixed point $O$.

The system hangs in equilibrium with the blocks stuck together, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-017_385_239_669_881} Find the extension of the spring.
Show that the elastic potential energy of the spring when the system is in equilibrium is 1.96 J .
The system is hanging in this equilibrium position when block $B$ falls off and block $A$ begins to move vertically upwards. Block $A$ next comes to rest when the spring is compressed by $x$ metres.
1. Show that $x$ satisfies the equation $$x ^ { 2 } + 0.16 x - 0.008 = 0$$
2. Find the value of $x$.

AQA M2 2007 January Q1

8 marks Moderate -0.8

1 A child, of mass 35 kg , slides down a slide in a water park. The child, starting from rest, slides from the point $A$ to the point $B$, which is 10 metres vertically below the level of $A$, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-2_259_595_685_705}

In a simple model, all resistance forces are ignored. Use an energy method to find the speed of the child at $B$.
State one resistance force that has been ignored in answering part (a).
In fact, when the child slides down the slide, she reaches $B$ with a speed of $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Given that the slide is 20 metres long and the sum of the resistance forces has a constant magnitude of $F$ newtons, use an energy method to find the value of $F$.
(4 marks)

AQA M2 2007 January Q2

6 marks Moderate -0.8

2 A hotel sign consists of a uniform rectangular lamina of weight $W$. The sign is suspended in equilibrium in a vertical plane by two vertical light chains attached to the sign at the points $A$ and $B$, as shown in the diagram. The edge containing $A$ and $B$ is horizontal. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-2_289_529_1859_726} The tensions in the chains attached at $A$ and $B$ are $T _ { A }$ and $T _ { B }$ respectively.

Draw a diagram to show the forces acting on the sign.
Find $T _ { A }$ and $T _ { B }$ in terms of $W$.
Explain how you have used the fact that the lamina is uniform in answering part (b).

AQA M2 2007 January Q3

6 marks Moderate -0.3

3 A light inextensible string has length $2 a$. One end of the string is attached to a fixed point $O$ and a particle of mass $m$ is attached to the other end. Initially, the particle is held at the point $A$ with the string taut and horizontal. The particle is then released from rest and moves in a circular path. Subsequently, it passes through the point $B$, which is directly below $O$. The points $O , A$ and $B$ are as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-3_426_437_575_772}

Show that the speed of the particle at $B$ is $2 \sqrt { a g }$.
Find the tension in the string as the particle passes through $B$. Give your answer in terms of $m$ and $g$.

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