Questions M3 - A-Level Maths

Edexcel M3 Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e0668f31-4b72-4dfd-9cf7-470acef0bfdb-2_469_465_776_680} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} A particle $P$ of mass 0.5 kg is at rest at the highest point $A$ of a smooth sphere, centre $O$, of radius 1.25 m which is fixed to a horizontal surface. When $P$ is slightly disturbed it slides along the surface of the sphere. Whilst $P$ is in contact with the sphere it has speed $v \mathrm {~ms} ^ { - 1 }$ when $\angle A O P = \theta$ as shown in Figure 1.

Show that $v ^ { 2 } = 24.5 ( 1 - \cos \theta )$.
Find the value of $\cos \theta$ when $P$ leaves the surface of the sphere.

Edexcel M3 Q3

3. A car starts from rest at the point $O$ and moves along a straight line. The car accelerates to a maximum velocity, $V \mathrm {~ms} ^ { - 1 }$, before decelerating and coming to rest again at the point $A$. The acceleration of the car during this journey, $a \mathrm {~ms} ^ { - 2 }$, is modelled by the formula $$a = \frac { 500 - k x } { 150 }$$ where $x$ is the distance in metres of the car from $O$.
Using this model and given that the car is travelling at $16 \mathrm {~ms} ^ { - 1 }$ when it is 40 m from $O$,

find $k$,
show that $V = 41$, correct to 2 significant figures,
find the distance $O A$.

Edexcel M3 Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e0668f31-4b72-4dfd-9cf7-470acef0bfdb-3_316_536_1087_639} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} A particle $P$ of mass 2 kg is attached to one end of a light elastic string of natural length 1.5 m and modulus of elasticity $\lambda$. The other end of the string is fixed to a point $A$ on a rough plane inclined at an angle of $30 ^ { \circ }$ to the horizontal as shown in Figure 2. The coefficient of friction between $P$ and the plane is $\frac { 1 } { 6 } \sqrt { 3 }$.
$P$ is held at rest at $A$ and then released. It first comes to instantaneous rest at the point $B , 2.2 \mathrm {~m}$ from $A$. For the motion of $P$ from $A$ to $B$,

show that the work done against friction is 10.78 J ,
find the change in the gravitational potential energy of $P$. By using the work-energy principle, or otherwise,
find $\lambda$.

Edexcel M3 Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e0668f31-4b72-4dfd-9cf7-470acef0bfdb-4_693_554_196_717} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} A flask is modelled as a uniform solid formed by removing a cylinder of radius $r$ and height $h$ from a cylinder of radius $\frac { 4 } { 3 } r$ and height $\frac { 3 } { 2 } h$ with the same axis of symmetry and a common plane as shown in Figure 3.

Show that the centre of mass of the flask is a distance of $\frac { 9 } { 10 } h$ from the open end of the flask. The flask is made from a material of density $\rho$ and is filled to the level of the open plane face with a liquid of density $k \rho$. Given that the centre of mass of the flask and liquid together is a distance of $\frac { 15 } { 22 } h$ from the open end of the flask,
find the value of $k$.
Explain why it may be advantageous to make the base of the flask from a more dense material.
(2 marks)

Edexcel M3 Q6

6. A particle $P$ of mass 2.5 kg is moving with simple harmonic motion in a straight line between two points $A$ and $B$ on a smooth horizontal table. When $P$ is 3 m from $O$, the centre of the oscillations, its speed is $6 \mathrm {~ms} ^ { - 1 }$. When $P$ is 2.25 m from $O$, its speed is $8 \mathrm {~ms} ^ { - 1 }$.

Show that $A B = 7.5 \mathrm {~m}$.
Find the period of the motion.
Find the kinetic energy of $P$ when it is 2.7 m from $A$.
Show that the time taken by $P$ to travel directly from $A$ to the midpoint of $O B$ is $\frac { \pi } { 4 }$ seconds.

Edexcel M3 2003 January Q3

Show that the distance $d$ of the centre of mass of the toy from its lowest point $O$ is given by $$d = \frac { h ^ { 2 } + 2 h r + 5 r ^ { 2 } } { 2 ( h + 4 r ) } .$$ When the toy is placed with any point of the curved surface of the hemisphere resting on the plane it will remain in equilibrium.
Find $h$ in terms of $r$.
(3)

Edexcel M3 Q2

Show that $\lambda = 4 m g \sin \alpha$. The particle is now moved and held at rest at $A$ with the string slack. It is then gently released so that it moves down the plane along a line of greatest slope.
Find the greatest distance from $A$ that the particle reaches down the plane.

OCR M3 2009 January Q4

Show that $v ^ { 2 } = 9 + 9.8 \sin \theta$.
Find, in terms of $\theta$, the radial and tangential components of the acceleration of $P$.
Show that the tension in the string is $( 3.6 + 5.88 \sin \theta ) \mathrm { N }$ and hence find the value of $\theta$ at the instant when the string becomes slack, giving your answer correct to 1 decimal place.

OCR M3 2010 January Q6

By considering the total energy of the system, obtain an expression for $v ^ { 2 }$ in terms of $\theta$.
Show that the magnitude of the force exerted on $P$ by the cylinder is $( 7.12 \sin \theta - 4.64 \theta ) \mathrm { N }$.
Given that $P$ leaves the surface of the cylinder when $\theta = \alpha$, show that $1.53 < \alpha < 1.54$.

OCR M3 2007 June Q6

Show that, when $P$ is in equilibrium, $O P = 7.25 \mathrm {~m}$.
Verify that $P$ and $Q$ together just reach the safety net.
At the lowest point of their motion $P$ releases $Q$. Prove that $P$ subsequently just reaches $O$.
State two additional modelling assumptions made when answering this question.

OCR M3 2010 June Q3

Given that the coefficient of restitution between the sphere and the wall is $\frac { 1 } { 2 }$, state the values of $u$ and $v$. Shortly after hitting the wall the sphere $A$ comes into contact with another uniform smooth sphere $B$, which has the same mass and radius as $A$. The sphere $B$ is stationary and at the instant of contact the line of centres of the spheres is parallel to the wall (see Fig. 2). The contact between the spheres is perfectly elastic. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a8c1e5b3-4d8b-4795-9e9f-4c0db374112e-3_524_371_1503_888} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
Find, for each sphere, its speed and its direction of motion immediately after the contact.
$4 ~ O$ is a fixed point on a horizontal plane. A particle $P$ of mass 0.25 kg is released from rest at $O$ and moves in a straight line on the plane. At time $t \mathrm {~s}$ after release the only horizontal force acting on $P$ has magnitude $$\frac { 1 } { 2400 } \left( 144 - t ^ { 2 } \right) \mathrm { N } \quad \text { for } 0 \leqslant t \leqslant 12$$ and $$\frac { 1 } { 2400 } \left( t ^ { 2 } - 144 \right) \mathrm { N } \text { for } t \geqslant 12 .$$ The force acts in the direction of $P$ 's motion. $P$ 's velocity at time $t \mathrm {~s}$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find an expression for $v$ in terms of $t$, valid for $t \geqslant 12$, and hence show that $v$ is three times greater when $t = 24$ than it is when $t = 12$.
Sketch the $( t , v )$ graph for $0 \leqslant t \leqslant 24$.

OCR M3 2016 June Q3

Find the speed of $A$ after the collision. Find also the component of the velocity of $B$ along the line of centres after the collision.
$B$ subsequently hits the wall.
Explain why $A$ and $B$ will have a second collision if the coefficient of restitution between $B$ and the wall is sufficiently large. Find the set of values of the coefficient of restitution for which this second collision will occur. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{4} \includegraphics[alt={},max width=\textwidth]{c0f31235-80aa-4838-844f-b706de55e7cd-3_193_1451_705_306}
\end{figure} $A$ and $C$ are two fixed points, 1.5 m apart, on a smooth horizontal plane. A light elastic string of natural length 0.4 m and modulus of elasticity 20 N has one end fixed to point $A$ and the other end fixed to a particle $B$. Another light elastic string of natural length 0.6 m and modulus of elasticity 15 N has one end fixed to point $C$ and the other end fixed to the particle $B$. The particle is released from rest when $A B C$ forms a straight line and $A B = 0.4 \mathrm {~m}$ (see diagram). Find the greatest kinetic energy of particle $B$ in the subsequent motion.
\includegraphics[max width=\textwidth, alt={}, center]{c0f31235-80aa-4838-844f-b706de55e7cd-3_586_533_1409_758} One end of a light inextensible string of length $a$ is attached to a fixed point $O$. A particle $P$ of mass $m$ is attached to the other end of the string and hangs at rest. $P$ is then projected horizontally from this position with speed $2 \sqrt { a g }$. When the string makes an angle $\theta$ with the upward vertical $P$ has speed $v$ (see diagram). The tension in the string is $T$.
Find an expression for $T$ in terms of $m , g$ and $\theta$, and hence find the height of $P$ above its initial level when the string becomes slack.
$P$ is now projected horizontally from the same initial position with speed $U$.
Find the set of values of $U$ for which the string does not remain taut in the subsequent motion.
\includegraphics[max width=\textwidth, alt={}, center]{c0f31235-80aa-4838-844f-b706de55e7cd-4_566_1013_255_525} Two uniform rods $A B$ and $A C$ are freely jointed at $A$. Rod $A B$ is of length $2 l$ and weight $W$; $\operatorname { rod } A C$ is of length $4 l$ and weight $2 W$. The rods rest in equilibrium in a vertical plane on two rough horizontal steps, so that $A B$ makes an angle of $\theta$ with the horizontal, where $\sin \theta = \frac { 4 } { 5 }$, and $A C$ makes an angle of $\varphi$ with the horizontal, where $\sin \varphi = \frac { 3 } { 5 }$ (see diagram). The force of the step acting on $A B$ at $B$ has vertical component $R$ and horizontal component $F$.
By taking moments about $A$ for the rod $A B$, find an equation relating $W , R$ and $F$.
Show that $R = \frac { 73 } { 50 } W$, and find the vertical component of the force acting on $A C$ at $C$.
The coefficient of friction at $B$ is equal to that at $C$. Given that one of the rods is on the point of slipping, explain which rod this must be, and find the coefficient of friction.

OCR M3 2006 January Q10

10 JANUARY 2006 Afternoon
1 hour 30 minutes
Additional materials:
8 page answer booklet
Graph paper
List of Formulae (MF1) TIME
1 hour 30 minutes

Write your name, centre number and candidate number in the spaces provided on the answer booklet.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures unless a different degree of accuracy is specified in the question or is clearly appropriate.
The acceleration due to gravity is denoted by $\mathrm { g } \mathrm { m } \mathrm { s } ^ { - 2 }$. Unless otherwise instructed, when a numerical value is needed, use $g = 9.8$.
You are permitted to use a graphical calculator in this paper.

The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 72.
Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper.
You are reminded of the need for clear presentation in your answers.

AQA M3 2009 June Q1

1 A ball of mass $m$ is travelling vertically downwards with speed $u$ when it hits a horizontal floor. The ball bounces vertically upwards to a height $h$. It is thought that $h$ depends on $m , u$, the acceleration due to gravity $g$, and a dimensionless constant $k$, such that $$h = k m ^ { \alpha } u ^ { \beta } g ^ { \gamma }$$ where $\alpha , \beta$ and $\gamma$ are constants.
By using dimensional analysis, find the values of $\alpha , \beta$ and $\gamma$.

AQA M3 2009 June Q2

2 A particle is projected from a point $O$ on a horizontal plane and has initial velocity components of $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $10 \mathrm {~ms} ^ { - 1 }$ parallel to and perpendicular to the plane respectively. At time $t$ seconds after projection, the horizontal and upward vertical distances of the particle from the point $O$ are $x$ metres and $y$ metres respectively.

Show that $x$ and $y$ satisfy the equation $$y = - \frac { g } { 8 } x ^ { 2 } + 5 x$$
By using the equation in part (a), find the horizontal distance travelled by the particle whilst it is more than 1 metre above the plane.
Hence find the time for which the particle is more than 1 metre above the plane.

AQA M3 2009 June Q3

3 A fishing boat is travelling between two ports, $A$ and $B$, on the shore of a lake. The bearing of $B$ from $A$ is $130 ^ { \circ }$. The fishing boat leaves $A$ and travels directly towards $B$ with speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. A patrol boat on the lake is travelling with speed $4 \mathrm {~ms} ^ { - 1 }$ on a bearing of $040 ^ { \circ }$.
\includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-3_713_1319_443_406}

Find the velocity of the fishing boat relative to the patrol boat, giving your answer as a speed together with a bearing.
When the patrol boat is 1500 m due west of the fishing boat, it changes direction in order to intercept the fishing boat in the shortest possible time.
1. Find the bearing on which the patrol boat should travel in order to intercept the fishing boat.
2. Given that the patrol boat intercepts the fishing boat before it reaches $B$, find the time, in seconds, that it takes the patrol boat to intercept the fishing boat after changing direction.
3. State a modelling assumption necessary for answering this question, other than the boats being particles.

AQA M3 2009 June Q4

4 A particle of mass 0.5 kg is initially at rest. The particle then moves in a straight line under the action of a single force. This force acts in a constant direction and has magnitude $\left( t ^ { 3 } + t \right) \mathrm { N }$, where $t$ is the time, in seconds, for which the force has been acting.

Find the magnitude of the impulse exerted by the force on the particle between the times $t = 0$ and $t = 4$.
Hence find the speed of the particle when $t = 4$.
Find the time taken for the particle to reach a speed of $12 \mathrm {~ms} ^ { - 1 }$.

AQA M3 2009 June Q5

5 Two smooth spheres, $A$ and $B$, of equal radii and different masses are moving on a smooth horizontal surface when they collide. Just before the collision, $A$ is moving with speed $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $30 ^ { \circ }$ to the line of centres of the spheres, and $B$ is moving with speed $3 \mathrm {~ms} ^ { - 1 }$ perpendicular to the line of centres, as shown in the diagram below. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{719b82f7-2ab5-48db-9b2a-98284096a78a-4_314_1100_593_392} \captionsetup{labelformat=empty} \caption{Before collision}

\end{figure} Immediately after the collision, $A$ and $B$ move with speeds $u$ and $v$ in directions which make angles of $90 ^ { \circ }$ and $40 ^ { \circ }$ respectively with the line of centres, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-4_392_1102_1155_392}

Show that $v = 4.67 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, correct to three significant figures.
Find the coefficient of restitution between the spheres.
Given that the mass of $A$ is 0.5 kg , show that the magnitude of the impulse exerted on $A$ during the collision is 2.17 Ns , correct to three significant figures.
Find the mass of $B$.

AQA M3 2009 June Q6

6 A smooth sphere $A$ of mass $m$ is moving with speed $5 u$ in a straight line on a smooth horizontal table. The sphere $A$ collides directly with a smooth sphere $B$ of mass $7 m$, having the same radius as $A$ and moving with speed $u$ in the same direction as $A$. The coefficient of restitution between $A$ and $B$ is $e$.
\includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-5_287_880_529_571}

Show that the speed of $B$ after the collision is $\frac { u } { 2 } ( e + 3 )$.
Given that the direction of motion of $A$ is reversed by the collision, show that $e > \frac { 3 } { 7 }$.
Subsequently, $B$ hits a wall fixed at right angles to the direction of motion of $A$ and $B$. The coefficient of restitution between $B$ and the wall is $\frac { 1 } { 2 }$. Given that after $B$ rebounds from the wall both spheres move in the same direction and collide again, show also that $e < \frac { 9 } { 13 }$.
(4 marks)

AQA M3 2009 June Q7

7 A particle is projected from a point $O$ on a smooth plane which is inclined at $30 ^ { \circ }$ to the horizontal. The particle is projected down the plane with velocity $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $40 ^ { \circ }$ above the plane and first strikes it at a point $A$. 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]{719b82f7-2ab5-48db-9b2a-98284096a78a-6_449_963_488_529}

Show that the time taken by the particle to travel from $O$ to $A$ is $$\frac { 20 \sin 40 ^ { \circ } } { g \cos 30 ^ { \circ } }$$
Find the components of the velocity of the particle parallel to and perpendicular to the slope as it hits the slope at $A$.
The coefficient of restitution between the slope and the particle is 0.5 . Find the speed of the particle as it rebounds from the slope.

Questions M3 (745 questions)

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