Questions - AQA M3 - A-Level Maths

Show that $x$ and $y$ satisfy the equation $$y = x \tan 30 ^ { \circ } - \frac { g x ^ { 2 } } { 800 \cos ^ { 2 } 30 ^ { \circ } }$$
On its downward flight the ball hits the horizontal crossbar of the goal at a point which is 2.5 m above the ground. Using the equation given in part (a), find the horizontal distance from $O$ to the goal.
(4 marks)
\includegraphics[max width=\textwidth, alt={}, center]{fc5bfc4b-68bb-4a23-874b-87e9558dc990-04_330_1411_1902_303}
State two modelling assumptions that you have made.

AQA M3 Q6

6 Two smooth billiard balls $A$ and $B$, of identical size and equal mass, move towards each other on a horizontal surface and collide. Just before the collision, $A$ has velocity $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a direction inclined at $30 ^ { \circ }$ to the line of centres of the balls, and $B$ has velocity $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a direction inclined at $60 ^ { \circ }$ to the line of centres, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{fc5bfc4b-68bb-4a23-874b-87e9558dc990-05_508_1420_532_294} The coefficient of restitution between the balls is $\frac { 1 } { 2 }$.

Find the speed of $B$ immediately after the collision.
Find the angle between the velocity of $B$ and the line of centres of the balls immediately after the collision.

AQA M3 Q7

57 marks

7 A projectile is fired from a point $O$ on the slope of a hill which is inclined at an angle $\alpha$ to the horizontal. The projectile is fired up the hill with velocity $U$ at an angle $\theta$ above the hill and first strikes it at a point $A$. The projectile is modelled as a particle and the hill is modelled as a plane with $O A$ as a line of greatest slope.

1. Find, in terms of $U , g , \alpha$ and $\theta$, the time taken by the projectile to travel from $O$ to $A$.
2. Hence, or otherwise, show that the magnitude of the component of the velocity of the projectile perpendicular to the hill, when it strikes the hill at the point $A$, is the same as it was initially at $O$.
The projectile rebounds and strikes the hill again at a point $B$. The hill is smooth and the coefficient of restitution between the projectile and the hill is $e$.
\includegraphics[max width=\textwidth, alt={}, center]{fc5bfc4b-68bb-4a23-874b-87e9558dc990-06_428_1332_1023_338} Find the ratio of the time of flight from $O$ to $A$ to the time of flight from $A$ to $B$. Give your answer in its simplest form.

AQA M3 2006 June Q1

1 The time $T$ taken for a simple pendulum to make a single small oscillation is thought to depend only on its length $l$, its mass $m$ and the acceleration due to gravity $g$. By using dimensional analysis:

show that $T$ does not depend on $m$;
express $T$ in terms of $l , g$ and $k$, where $k$ is a dimensionless constant.

AQA M3 2006 June Q2

2 Three smooth spheres $A , B$ and $C$ of equal radii and masses $m , m$ and $2 m$ respectively lie at rest on a smooth horizontal table. The centres of the spheres lie in a straight line with $B$ between $A$ and $C$. The coefficient of restitution between any two spheres is $e$. The sphere $A$ is projected directly towards $B$ with speed $u$ and collides with $B$.

Find, in terms of $u$ and $e$, the speed of $B$ immediately after the impact between $A$ and $B$.
The sphere $B$ subsequently collides with $C$. The speed of $C$ immediately after this collision is $\frac { 3 } { 8 } u$. Find the value of $e$.

AQA M3 2006 June Q3

3 A ball of mass 0.45 kg is travelling horizontally with speed $15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when it strikes a fixed vertical bat directly and rebounds from it. The ball stays in contact with the bat for 0.1 seconds. At time $t$ seconds after first coming into contact with the bat, the force exerted on the ball by the bat is $1.4 \times 10 ^ { 5 } \left( t ^ { 2 } - 10 t ^ { 3 } \right)$ newtons, where $0 \leqslant t \leqslant 0.1$. In this simple model, ignore the weight of the ball and model the ball as a particle.

Show that the magnitude of the impulse exerted by the bat on the ball is 11.7 Ns , correct to three significant figures.
Find, to two significant figures, the speed of the ball immediately after the impact.
Give a reason why the speed of the ball immediately after the impact is different from the speed of the ball immediately before the impact.

AQA M3 2006 June Q4

4 The unit vectors $\mathbf { i }$ and $\mathbf { j }$ are directed due east and due north respectively.
Two cyclists, Aazar and Ben, are cycling on straight horizontal roads with constant velocities of $( 6 \mathbf { i } + 12 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$ and $( 12 \mathbf { i } - 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$ respectively. Initially, Aazar and Ben have position vectors $( 5 \mathbf { i } - \mathbf { j } ) \mathrm { km }$ and $( 18 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }$ respectively, relative to a fixed origin.

Find, as a vector in terms of $\mathbf { i }$ and $\mathbf { j }$, the velocity of Ben relative to Aazar.
The position vector of Ben relative to Aazar at time $t$ hours after they start is $\mathbf { r } \mathrm { km }$. Show that $$\mathbf { r } = ( 13 + 6 t ) \mathbf { i } + ( 6 - 20 t ) \mathbf { j }$$
Find the value of $t$ when Aazar and Ben are closest together.
Find the closest distance between Aazar and Ben.

AQA M3 2006 June Q5

Show that $x$ and $y$ satisfy the equation $$y = x \tan 30 ^ { \circ } - \frac { g x ^ { 2 } } { 800 \cos ^ { 2 } 30 ^ { \circ } }$$
On its downward flight the ball hits the horizontal crossbar of the goal at a point which is 2.5 m above the ground. Using the equation given in part (a), find the horizontal distance from $O$ to the goal.
(4 marks)
\includegraphics[max width=\textwidth, alt={}, center]{f8c04360-f54b-4d08-aee9-fe28612918d0-3_330_1411_1902_303}
State two modelling assumptions that you have made.

AQA M3 2006 June Q6

6 Two smooth billiard balls $A$ and $B$, of identical size and equal mass, move towards each other on a horizontal surface and collide. Just before the collision, $A$ has velocity $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a direction inclined at $30 ^ { \circ }$ to the line of centres of the balls, and $B$ has velocity $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a direction inclined at $60 ^ { \circ }$ to the line of centres, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{f8c04360-f54b-4d08-aee9-fe28612918d0-4_508_1420_532_294} The coefficient of restitution between the balls is $\frac { 1 } { 2 }$.

Find the speed of $B$ immediately after the collision.
Find the angle between the velocity of $B$ and the line of centres of the balls immediately after the collision.

AQA M3 2006 June Q7

1. Find, in terms of $U , g , \alpha$ and $\theta$, the time taken by the projectile to travel from $O$ to $A$.
2. Hence, or otherwise, show that the magnitude of the component of the velocity of the projectile perpendicular to the hill, when it strikes the hill at the point $A$, is the same as it was initially at $O$.
The projectile rebounds and strikes the hill again at a point $B$. The hill is smooth and the coefficient of restitution between the projectile and the hill is $e$.
\includegraphics[max width=\textwidth, alt={}, center]{f8c04360-f54b-4d08-aee9-fe28612918d0-5_428_1332_1023_338} Find the ratio of the time of flight from $O$ to $A$ to the time of flight from $A$ to $B$. Give your answer in its simplest form.

AQA M3 2007 June Q1

1 The magnitude of the gravitational force, $F$, between two planets of masses $m _ { 1 }$ and $m _ { 2 }$ with centres at a distance $x$ apart is given by $$F = \frac { G m _ { 1 } m _ { 2 } } { x ^ { 2 } }$$ where $G$ is a constant.

By using dimensional analysis, find the dimensions of $G$.
The lifetime, $t$, of a planet is thought to depend on its mass, $m$, its initial radius, $R$, the constant $G$ and a dimensionless constant, $k$, so that $$t = k m ^ { \alpha } R ^ { \beta } G ^ { \gamma }$$ where $\alpha , \beta$ and $\gamma$ are constants.
Find the values of $\alpha , \beta$ and $\gamma$.

AQA M3 2007 June Q2

2 The unit vectors $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$ are directed due east, due north and vertically upwards respectively. Two helicopters, $A$ and $B$, are flying with constant velocities of $( 20 \mathbf { i } - 10 \mathbf { j } + 20 \mathbf { k } ) \mathrm { ms } ^ { - 1 }$ and $( 30 \mathbf { i } + 10 \mathbf { j } + 10 \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ respectively. At noon, the position vectors of $A$ and $B$ relative to a fixed origin, $O$, are $( 8000 \mathbf { i } + 1500 \mathbf { j } + 3000 \mathbf { k } ) \mathrm { m }$ and $( 2000 \mathbf { i } + 500 \mathbf { j } + 1000 \mathbf { k } ) \mathrm { m }$ respectively.

Write down the velocity of $A$ relative to $B$.
Find the position vector of $A$ relative to $B$ at time $t$ seconds after noon.
Find the value of $t$ when $A$ and $B$ are closest together.

AQA M3 2007 June Q3

3 A particle $P$, of mass 2 kg , is initially at rest at a point $O$ on a smooth horizontal surface. The particle moves along a straight line, $O A$, under the action of a horizontal force. When the force has been acting for $t$ seconds, it has magnitude $( 4 t + 5 ) \mathrm { N }$.

Find the magnitude of the impulse exerted by the force on $P$ between the times $t = 0$ and $t = 3$.
Find the speed of $P$ when $t = 3$.
The speed of $P$ at $A$ is $37.5 \mathrm {~ms} ^ { - 1 }$. Find the time taken for the particle to reach $A$.

AQA M3 2007 June Q4

4 Two small smooth spheres, $A$ and $B$, of equal radii have masses 0.3 kg and 0.2 kg respectively. They are moving on a smooth horizontal surface directly towards each other with speeds $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively when they collide. The coefficient of restitution between $A$ and $B$ is 0.8 .

Find the speeds of $A$ and $B$ immediately after the collision.
Subsequently, $B$ collides with a fixed smooth vertical wall which is at right angles to the path of the sphere. The coefficient of restitution between $B$ and the wall is 0.7 . Show that $B$ will collide again with $A$.

AQA M3 2007 June Q5

5 A ball is projected with speed $u \mathrm {~ms} ^ { - 1 }$ at an angle of elevation $\alpha$ above the horizontal so as to hit a point $P$ on a wall. The ball travels in a vertical plane through the point of projection. During the motion, the horizontal and upward vertical displacements of the ball from the point of projection are $x$ metres and $y$ metres respectively.

Show that, during the flight, the equation of the trajectory of the ball is given by $$y = x \tan \alpha - \frac { g x ^ { 2 } } { 2 u ^ { 2 } } \left( 1 + \tan ^ { 2 } \alpha \right)$$
The ball is projected from a point 1 metre vertically below and $R$ metres horizontally from the point $P$.
1. By taking $g = 10 \mathrm {~ms} ^ { - 2 }$, show that $R$ satisfies the equation $$5 R ^ { 2 } \tan ^ { 2 } \alpha - u ^ { 2 } R \tan \alpha + 5 R ^ { 2 } + u ^ { 2 } = 0$$
2. Hence, given that $u$ and $R$ are constants, show that, for $\tan \alpha$ to have real values, $R$ must satisfy the inequality $$R ^ { 2 } \leqslant \frac { u ^ { 2 } \left( u ^ { 2 } - 20 \right) } { 100 }$$
3. Given that $R = 5$, determine the minimum possible speed of projection.

AQA M3 2007 June Q6

6 A smooth spherical ball, $A$, is moving with speed $u$ in a straight line on a smooth horizontal table when it hits an identical ball, $B$, which is at rest on the table. Just before the collision, the direction of motion of $A$ makes an angle of $30 ^ { \circ }$ with the line of the centres of the two balls, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{daea0765-041a-4569-a535-f90fe4708313-4_362_1632_621_242} The coefficient of restitution between $A$ and $B$ is $e$.

Given that $\cos 30 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }$, show that the speed of $B$ immediately after the collision is $$\frac { \sqrt { 3 } } { 4 } u ( 1 + e )$$
Find, in terms of $u$ and $e$, the components of the velocity of $A$, parallel and perpendicular to the line of centres, immediately after the collision.
Given that $e = \frac { 2 } { 3 }$, find the angle that the velocity of $A$ makes with the line of centres immediately after the collision. Give your answer to the nearest degree.
(3 marks)

AQA M3 2007 June Q7

7 A particle is projected from a point on a plane which is inclined at an angle $\alpha$ to the horizontal. The particle is projected up the plane with velocity $u$ at an angle $\theta$ above the plane. The motion of the particle is in a vertical plane containing a line of greatest slope of the inclined plane.
\includegraphics[max width=\textwidth, alt={}, center]{daea0765-041a-4569-a535-f90fe4708313-5_401_748_516_644}

Using the identity $\cos ( A + B ) = \cos A \cos B - \sin A \sin B$, show that the range up the plane is $$\frac { 2 u ^ { 2 } \sin \theta \cos ( \theta + \alpha ) } { g \cos ^ { 2 } \alpha }$$
Hence, using the identity $2 \sin A \cos B = \sin ( A + B ) + \sin ( A - B )$, show that, as $\theta$ varies, the range up the plane is a maximum when $\theta = \frac { \pi } { 4 } - \frac { \alpha } { 2 }$.
Given that the particle strikes the plane at right angles, show that $$2 \tan \theta = \cot \alpha$$

AQA M3 2008 June Q1

1 The speed, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, of a wave travelling along the surface of a sea is believed to depend on
the depth of the sea, $d \mathrm {~m}$,
the density of the water, $\rho \mathrm { kg } \mathrm { m } ^ { - 3 }$,
the acceleration due to gravity, $g$, and
a dimensionless constant, $k$
so that $$v = k d ^ { \alpha } \rho ^ { \beta } g ^ { \gamma }$$ where $\alpha , \beta$ and $\gamma$ are constants.
By using dimensional analysis, show that $\beta = 0$ and find the values of $\alpha$ and $\gamma$.

AQA M3 2008 June Q2

2 The unit vectors $\mathbf { i }$ and $\mathbf { j }$ are directed due east and due north respectively.
Two runners, Albina and Brian, are running on level parkland with constant velocities of $( 5 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and $( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ respectively. Initially, the position vectors of Albina and Brian are $( - 60 \mathbf { i } + 160 \mathbf { j } ) \mathrm { m }$ and $( 40 \mathbf { i } - 90 \mathbf { j } ) \mathrm { m }$ respectively, relative to a fixed origin in the parkland.

Write down the velocity of Brian relative to Albina.
Find the position vector of Brian relative to Albina $t$ seconds after they leave their initial positions.
Hence determine whether Albina and Brian will collide if they continue running with the same velocities.

AQA M3 2008 June Q3

3 A particle of mass 0.2 kg lies at rest on a smooth horizontal table. A horizontal force of magnitude $F$ newtons acts on the particle in a constant direction for 0.1 seconds. At time $t$ seconds, $$F = 5 \times 10 ^ { 3 } t ^ { 2 } , \quad 0 \leqslant t \leqslant 0.1$$ Find the value of $t$ when the speed of the particle is $2 \mathrm {~ms} ^ { - 1 }$.
(4 marks)

AQA M3 2008 June Q4

4 Two smooth spheres, $A$ and $B$, have equal radii and masses $m$ and $2 m$ respectively. The spheres are moving on a smooth horizontal plane. The sphere $A$ has velocity ( $4 \mathbf { i } + 3 \mathbf { j }$ ) when it collides with the sphere $B$ which has velocity $( - 2 \mathbf { i } + 2 \mathbf { j } )$. After the collision, the velocity of $B$ is $( \mathbf { i } + \mathbf { j } )$.

Find the velocity of $A$ immediately after the collision.
Find the angle between the velocities of $A$ and $B$ immediately after the collision.
Find the impulse exerted by $B$ on $A$.
State, as a vector, the direction of the line of centres of $A$ and $B$ when they collide.
(1 mark)

AQA M3 2008 June Q5

5 A boy throws a small ball from a height of 1.5 m above horizontal ground with initial velocity $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle $\alpha$ above the horizontal. The ball hits a small can placed on a vertical wall of height 2.5 m , which is at a horizontal distance of 5 m from the initial position of the ball, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{eed9842d-cd89-4eb7-b5ba-9380971be196-3_499_1180_1283_424}

Show that $\alpha$ satisfies the equation $$49 \tan ^ { 2 } \alpha - 200 \tan \alpha + 89 = 0$$
Find the two possible values of $\alpha$, giving your answers to the nearest $0.1 ^ { \circ }$.
1. To knock the can off the wall, the horizontal component of the velocity of the ball must be greater than $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Show that, for one of the possible values of $\alpha$ found in part (b), the can will be knocked off the wall, and for the other, it will not be knocked off the wall.
  (3 marks)
2. Given that the can is knocked off the wall, find the direction in which the ball is moving as it hits the can.

AQA M3 2008 June Q6

6 A small smooth ball of mass $m$, moving on a smooth horizontal surface, hits a smooth vertical wall and rebounds. The coefficient of restitution between the wall and the ball is $\frac { 3 } { 4 }$. Immediately before the collision, the ball has velocity $u$ and the angle between the ball's direction of motion and the wall is $\alpha$. The ball's direction of motion immediately after the collision is at right angles to its direction of motion before the collision, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{eed9842d-cd89-4eb7-b5ba-9380971be196-4_483_344_657_854}

Show that $\tan \alpha = \frac { 2 } { \sqrt { 3 } }$.
Find, in terms of $u$, the speed of the ball immediately after the collision.
The force exerted on the ball by the wall acts for 0.1 seconds. Given that $m = 0.2 \mathrm {~kg}$ and $u = 4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, find the average force exerted by the wall on the ball.

AQA M3 2008 June Q7

7 A projectile is fired with speed $u$ from a point $O$ on a plane which is inclined at an angle $\alpha$ to the horizontal. The projectile is fired at an angle $\theta$ to the inclined plane and moves in a vertical plane through a line of greatest slope of the inclined plane. The projectile lands at a point $P$, lower down the inclined plane, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{eed9842d-cd89-4eb7-b5ba-9380971be196-5_415_1098_495_463}

Find, in terms of $u , g , \theta$ and $\alpha$, the greatest perpendicular distance of the projectile from the plane.
1. Find, in terms of $u , g , \theta$ and $\alpha$, the time of flight from $O$ to $P$.
2. By using the identity $\cos A \cos B + \sin A \sin B = \cos ( A - B )$, show that the distance $O P$ is given by $\frac { 2 u ^ { 2 } \sin \theta \cos ( \theta - \alpha ) } { g \cos ^ { 2 } \alpha }$.
3. Hence, by using the identity $2 \sin A \cos B = \sin ( A + B ) + \sin ( A - B )$ or otherwise, show that, as $\theta$ varies, the maximum possible distance $O P$ is $\frac { u ^ { 2 } } { g ( 1 - \sin \alpha ) }$.
  (5 marks)

AQA M3 2010 June Q1

1 A tank containing a liquid has a small hole in the bottom through which the liquid escapes. The speed, $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$, at which the liquid escapes is given by $$u = C V \rho g$$ where $V \mathrm {~m} ^ { 3 }$ is the volume of the liquid in the tank, $\rho \mathrm { kg } \mathrm { m } ^ { - 3 }$ is the density of the liquid, $g$ is the acceleration due to gravity and $C$ is a constant. By using dimensional analysis, find the dimensions of $C$.
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\includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-03_2484_1709_223_153}

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