Questions M3 - A-Level Maths

the direction of the impulse,
the speed of the ball immediately after it is struck.

OCR M3 2006 June Q2

2 A duck of mass 2 kg is travelling with horizontal speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when it lands on a lake. The duck is brought to rest by the action of resistive forces, acting in the direction opposite to the duck's motion and having total magnitude $\left( 2 v + 3 v ^ { 2 } \right) \mathrm { N }$, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the speed of the duck. Show that the duck comes to rest after travelling approximately 1.30 m from the point of its initial contact with the surface of the lake.

OCR M3 2006 June Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{09d3e8ca-0062-4f62-8453-7acaff591db5-2_661_711_918_717} Two uniform rods $A B$ and $A C$, of equal lengths, and of weights 200 N and 360 N respectively, are freely jointed at $A$. The mid-points of the rods are joined by a taut light inextensible string. The rods are in equilibrium in a vertical plane with $B$ and $C$ in contact with a smooth horizontal surface. The point $A$ is 2.1 m above the surface and $B C = 1.4 \mathrm {~m}$ (see diagram).

Show that the force exerted on $A B$ at $B$ has magnitude 240 N and find the tension in the string.
Find the horizontal and vertical components of the force exerted on $A B$ at $A$ and state their directions.

OCR M3 2006 June Q4

4 A particle is connected to a fixed point by a light inextensible string of length 2.45 m to make a simple pendulum. The particle is released from rest with the string taut and inclined at 0.1 radians to the downward vertical.

Show that the motion of the particle is approximately simple harmonic with period 3.14 s , correct to 3 significant figures. Calculate, in either order,
the angular speed of the pendulum when it has moved 0.04 radians from the initial position,
the time taken by the pendulum to move 0.04 radians from the initial position.

OCR M3 2006 June Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{09d3e8ca-0062-4f62-8453-7acaff591db5-3_362_841_264_651} Two uniform smooth spheres $A$ and $B$, of equal radius, have masses 2 kg and 3 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision $A$ is moving with speed $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at $60 ^ { \circ }$ to the line of centres, and $B$ is moving with speed $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ along the line of centres (see diagram). The coefficient of restitution between the spheres is 0.5 . Find the speed and direction of motion of each sphere after the collision.

OCR M3 2006 June Q6

6 A bungee jumper of mass 70 kg is joined to a fixed point $O$ by a light elastic rope of natural length 30 m and modulus of elasticity 1470 N . The jumper starts from rest at $O$ and falls vertically. The jumper is modelled as a particle and air resistance is ignored.

Find the distance fallen by the jumper when maximum speed is reached.
Show that this maximum speed is $26.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, correct to 3 significant figures.
Find the extension of the rope when the jumper is at the lowest position. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{09d3e8ca-0062-4f62-8453-7acaff591db5-4_543_616_310_301} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{09d3e8ca-0062-4f62-8453-7acaff591db5-4_668_709_267_1135} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A smooth horizontal cylinder of radius 0.6 m is fixed with its axis horizontal and passing through a fixed point $O$. A light inextensible string of length $0.6 \pi \mathrm {~m}$ has particles $P$ and $Q$, of masses 0.3 kg and 0.4 kg respectively, attached at its ends. The string passes over the cylinder and is held at rest with $P , O$ and $Q$ in a straight horizontal line (see Fig. 1). The string is released and $Q$ begins to descend. When the line $O P$ makes an angle $\theta$ radians, $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$, with the horizontal, the particles have speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ (see Fig. 2).
By considering the total energy of the system, or otherwise, show that $$v ^ { 2 } = 6.72 \theta - 5.04 \sin \theta .$$
Show that the magnitude of the contact force between $P$ and the cylinder is $$( 5.46 \sin \theta - 3.36 \theta ) \text { newtons. }$$ Hence find the value of $\theta$ for which the magnitude of the contact force is greatest.
Find the transverse component of the acceleration of $P$ in terms of $\theta$.

OCR M3 2007 June Q1

1 A particle $P$ is moving with simple harmonic motion in a straight line. The period is 6.1 s and the amplitude is 3 m . Calculate, in either order,

the maximum speed of $P$,
the distance of $P$ from the centre of motion when $P$ has speed $2.5 \mathrm {~ms} ^ { - 1 }$.

OCR M3 2007 June Q2

2 A tennis ball of mass 0.057 kg has speed $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The ball receives an impulse of magnitude 0.6 N s which reduces the speed of the ball to $7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Using an impulse-momentum triangle, or otherwise, find the angle the impulse makes with the original direction of motion of the ball.

OCR M3 2007 June Q3

3 A particle $P$ of mass 0.2 kg is projected horizontally with speed $u \mathrm {~ms} ^ { - 1 }$ from a fixed point $O$ on a smooth horizontal surface. $P$ moves in a straight line and, at time $t \mathrm {~s}$ after projection, $P$ has speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and is $x \mathrm {~m}$ from $O$. The only force acting on $P$ has magnitude $0.4 v ^ { 2 } \mathrm {~N}$ and is directed towards $O$.

Show that $\frac { 1 } { v } \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 2$.
Hence show that $v = u \mathrm { e } ^ { - 2 x }$.
Find $u$, given that $x = 2$ when $t = 4$.

OCR M3 2007 June Q4

10 marks

4
\includegraphics[max width=\textwidth, alt={}, center]{a04e6d4e-2437-4761-87ee-43e6771fbbd9-2_332_995_1375_575} Two uniform smooth spheres $A$ and $B$, of equal radius, have masses 4 kg and 3 kg respectively. They are moving on a horizontal surface, and they collide. Immediately before the collision, $A$ is moving with speed $15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle $\alpha$ to the line of centres, where $\sin \alpha = 0.8$, and $B$ is moving along the line of centres with speed $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ (see diagram). The coefficient of restitution between the spheres is 0.5 . Find the speed and direction of motion of each sphere after the collision.
[0pt] [10]

OCR M3 2007 June Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{a04e6d4e-2437-4761-87ee-43e6771fbbd9-3_549_447_253_849} Two uniform rods $A B$ and $B C$, each of length 1.4 m and weight 80 N , are freely jointed to each other at $B$, and $A B$ is freely jointed to a fixed point at $A$. They are held in equilibrium with $A B$ at an angle $\alpha$ to the horizontal, and $B C$ at an angle of $60 ^ { \circ }$ to the horizontal, by a light string, perpendicular to $B C$, attached to $C$ (see diagram).

By taking moments about $B$ for $B C$, calculate the tension in the string. Hence find the horizontal and vertical components of the force acting on $B C$ at $B$.
Find $\alpha$.
\includegraphics[max width=\textwidth, alt={}, center]{a04e6d4e-2437-4761-87ee-43e6771fbbd9-3_691_665_1370_740} A circus performer $P$ of mass 80 kg is suspended from a fixed point $O$ by an elastic rope of natural length 5.25 m and modulus of elasticity $2058 \mathrm {~N} . P$ is in equilibrium at a point 5 m above a safety net. A second performer $Q$, also of mass 80 kg , falls freely under gravity from a point above $P$. $P$ catches $Q$ and together they begin to descend vertically with initial speed $3.5 \mathrm {~ms} ^ { - 1 }$ (see diagram). The performers are modelled as particles.

OCR M3 2007 June Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{a04e6d4e-2437-4761-87ee-43e6771fbbd9-4_588_629_274_758} A particle $P$ of mass 0.8 kg is attached to a fixed point $O$ by a light inextensible string of length 0.4 m . A particle $Q$ is suspended from $O$ by an identical string. With the string $O P$ taut and inclined at $\frac { 1 } { 3 } \pi$ radians to the vertical, $P$ is projected with speed $0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a direction perpendicular to the string so as to strike $Q$ directly (see diagram). The coefficient of restitution between $P$ and $Q$ is $\frac { 1 } { 7 }$.

Calculate the tension in the string immediately after $P$ is set in motion.
Immediately after $P$ and $Q$ collide they have equal speeds and are moving in opposite directions. Show that $Q$ starts to move with speed $0.15 \mathrm {~ms} ^ { - 1 }$.
Prove that before the second collision between $P$ and $Q , Q$ is moving with approximate simple harmonic motion.
Hence find the time interval between the first and second collisions of $P$ and $Q$. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

OCR M3 2008 June Q1

1 A particle $P$ of mass $m \mathrm {~kg}$ is attached to one end of a light elastic string of natural length 1.8 m and modulus of elasticity 1.35 mg N . The other end of the string is attached to a fixed point $O$ on a smooth horizontal surface. $P$ is held at rest at a point on the surface 3 m from $O$. The particle is then released. Find

the initial acceleration of $P$,
the speed of $P$ at the instant the string becomes slack.

OCR M3 2008 June Q2

2 A particle $P$ of mass 0.2 kg is moving with speed $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when it hits a horizontal smooth surface. The direction of motion of $P$ immediately before impact makes an angle of $27 ^ { \circ }$ with the surface. Given that the coefficient of restitution between the particle and the surface is 0.6 , find

the vertical component of the velocity of $P$ immediately after impact,
the magnitude of the impulse exerted on $P$.

OCR M3 2008 June Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{85402f4a-8d55-47d8-ba48-5b837609b0f4-2_387_561_1055_794} Two uniform smooth spheres $A$ and $B$, of equal radius, have masses 0.8 kg and 2.0 kg respectively. The spheres are on a horizontal surface. $A$ is moving with speed $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at $60 ^ { \circ }$ to the line of centres when it collides with $B$, which is stationary (see diagram). The coefficient of restitution between the spheres is 0.75 . Find the speed and direction of motion of $A$ immediately after the collision.

OCR M3 2008 June Q4

4 A particle $P$ of mass $m \mathrm {~kg}$ is held at rest at a point $O$ on a fixed plane inclined at an angle $\sin ^ { - 1 } \left( \frac { 4 } { 7 } \right)$ to the horizontal. $P$ is released and moves down the plane. The total resistance acting on $P$ is $0.2 m v \mathrm {~N}$, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the velocity of $P$ at time $t \mathrm {~s}$ after leaving $O$.

Show that $5 \frac { \mathrm {~d} v } { \mathrm {~d} t } = 28 - v$ and hence find an expression for $v$ in terms of $t$.
Find the acceleration of $P$ when $t = 10$.

OCR M3 2008 June Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{85402f4a-8d55-47d8-ba48-5b837609b0f4-3_581_903_267_621} Two uniform rods $X A$ and $X B$ are freely jointed at $X$. The lengths of the rods are 1.5 m and 1.3 m respectively, and their weights are 150 N and 130 N respectively. The rods are in equilibrium in a vertical plane with $A$ and $B$ in contact with a rough horizontal surface. $A$ and $B$ are at distances horizontally from $X$ of 0.9 m and 0.5 m respectively, and $X$ is 1.2 m above the surface (see diagram).

The normal components of the contact forces acting on the rods at $A$ and $B$ are $R _ { A } \mathrm {~N}$ and $R _ { B } \mathrm {~N}$ respectively. Show that $R _ { A } = 125$ and find $R _ { B }$.
Find the frictional components of the contact forces acting on the rods at $A$ and $B$.
Find the horizontal and vertical components of the force exerted on $X A$ at $X$, stating their directions.

OCR M3 2008 June Q6

6 A particle $P$ of mass 0.1 kg moves in a straight line on a smooth horizontal surface. A force of $( 0.36 - 0.144 x ) \mathrm { N }$ acts on $P$ in the direction from $O$ to $P$, where $x \mathrm {~m}$ is the displacement of $P$ from a point $O$ on the surface at time $t \mathrm {~s}$.

By using the substitution $x = y + 2.5$, or otherwise, show that $P$ moves with simple harmonic motion of period 5.24 s , correct to 3 significant figures. The maximum value of $x$ during the motion is 3 .
Write down the amplitude of $P$ 's motion and find the two possible values of $x$ for which $P$ 's speed is $0.48 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
On each of the first two occasions when $P$ has speed $0.48 \mathrm {~m} \mathrm {~s} ^ { - 1 } , P$ is moving towards $O$. Find the time interval between
(a) these first two occasions,
(b) the second and third occasions when $P$ has speed $0.48 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

OCR M3 2008 June Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{85402f4a-8d55-47d8-ba48-5b837609b0f4-4_517_677_267_733} A particle $P$ of mass $m \mathrm {~kg}$ is slightly disturbed from rest at the highest point on the surface of a smooth fixed sphere of radius $a$ m and centre $O$. The particle starts to move downwards on the surface. While $P$ remains on the surface $O P$ makes an angle of $\theta$ radians with the upward vertical and has angular speed $\omega$ rad s $^ { - 1 }$ (see diagram). The sphere exerts a force of magnitude $R \mathrm {~N}$ on $P$.

Show that $a \omega ^ { 2 } = 2 g ( 1 - \cos \theta )$.
Find an expression for $R$ in terms of $m , g$ and $\theta$. At the instant that $P$ loses contact with the surface of the sphere, find
the transverse component of the acceleration of $P$,
the rate of change of $R$ with respect to time $t$, in terms of $m , g$ and $a$.

OCR M3 2009 June Q1

1 A smooth sphere of mass 0.3 kg bounces on a fixed horizontal surface. Immediately before the sphere bounces the components of its velocity horizontally and vertically downwards are $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively. The speed of the sphere immediately after it bounces is $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Show that the vertical component of the velocity of the sphere immediately after impact is $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, and hence find the coefficient of restitution between the surface and the sphere.
State the direction of the impulse on the sphere and find its magnitude.

OCR M3 2009 June Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{7a67db39-4934-4808-a56b-c6841950d324-2_460_725_731_708} Two uniform rods, $A B$ and $B C$, are freely jointed to each other at $B$, and $C$ is freely jointed to a fixed point. The rods are in equilibrium in a vertical plane with $A$ resting on a rough horizontal surface. This surface is 1.5 m below the level of $B$ and the horizontal distance between $A$ and $B$ is 3 m (see diagram). The weight of $A B$ is 80 N and the frictional force acting on $A B$ at $A$ is 14 N .

Write down the horizontal component of the force acting on $A B$ at $B$ and show that the vertical component of this force is 33 N upwards.
Given that the force acting on $B C$ at $C$ has magnitude 50 N , find the weight of $B C$.
\includegraphics[max width=\textwidth, alt={}, center]{7a67db39-4934-4808-a56b-c6841950d324-2_421_949_1793_598} Two uniform smooth spheres $A$ and $B$, of equal radius, have masses 4 kg and 2 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision both spheres have speed $3 \mathrm {~ms} ^ { - 1 }$. The spheres are moving in opposite directions, each at $60 ^ { \circ }$ to the line of centres (see diagram). After the collision $A$ moves in a direction perpendicular to the line of centres.
Show that the speed of $B$ is unchanged as a result of the collision, and find the angle that the new direction of motion of $B$ makes with the line of centres.
Find the coefficient of restitution between the spheres.

OCR M3 2009 June Q4

4 A motor-cycle, whose mass including the rider is 120 kg , is decelerating on a horizontal straight road. The motor-cycle passes a point $A$ with speed $40 \mathrm {~ms} ^ { - 1 }$ and when it has travelled a distance of $x \mathrm {~m}$ beyond $A$ its speed is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The engine develops a constant power of 8 kW and resistances are modelled by a force of $0.25 v ^ { 2 } \mathrm {~N}$ opposing the motion.

Show that $\frac { 480 v ^ { 2 } } { v ^ { 3 } - 32000 } \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 1$.
Find the speed of the motor-cycle when it has travelled 500 m beyond $A$.

OCR M3 2009 June Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{7a67db39-4934-4808-a56b-c6841950d324-3_591_668_776_737} Each of two identical strings has natural length 1.5 m and modulus of elasticity 18 N . One end of one of the strings is attached to $A$ and one end of the other string is attached to $B$, where $A$ and $B$ are fixed points which are 3 m apart and at the same horizontal level. $M$ is the mid-point of $A B$. A particle $P$ of mass $m \mathrm {~kg}$ is attached to the other end of each of the strings. $P$ is held at rest at the point 0.8 m vertically above $M$, and then released. The lowest point reached by $P$ in the subsequent motion is 2 m below $M$ (see diagram).

Find the maximum tension in each of the strings during $P$ 's motion.
By considering energy,
(a) show that the value of $m$ is 0.42 , correct to 2 significant figures,
(b) find the speed of $P$ at $M$.

OCR M3 2009 June Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{7a67db39-4934-4808-a56b-c6841950d324-4_368_131_274_1005} A particle $P$ of mass $m \mathrm {~kg}$ is attached to one end of a light inextensible string of length $L \mathrm {~m}$. The other end of the string is attached to a fixed point $O$. The particle is held at rest with the string taut and then released. $P$ starts to move and in the subsequent motion the angular displacement of $O P$, at time $t \mathrm {~s}$, is $\theta$ radians from the downward vertical (see diagram). The initial value of $\theta$ is 0.05 .

Show that $\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - \frac { g } { L } \sin \theta$.
Hence show that the motion of $P$ is approximately simple harmonic.
Given that the period of the approximate simple harmonic motion is $\frac { 4 } { 7 } \pi \mathrm {~s}$, find the value of $L$.
Find the value of $\theta$ when $t = 0.7 \mathrm {~s}$, and the value of $t$ when $\theta$ next takes this value.
Find the speed of $P$ when $t = 0.7 \mathrm {~s}$.
\includegraphics[max width=\textwidth, alt={}, center]{7a67db39-4934-4808-a56b-c6841950d324-4_422_501_1500_822} A hollow cylinder has internal radius $a$. The cylinder is fixed with its axis horizontal. A particle $P$ of mass $m$ is at rest in contact with the smooth inner surface of the cylinder. $P$ is given a horizontal velocity $u$, in a vertical plane perpendicular to the axis of the cylinder, and begins to move in a vertical circle. While $P$ remains in contact with the surface, $O P$ makes an angle $\theta$ with the downward vertical, where $O$ is the centre of the circle. The speed of $P$ is $v$ and the magnitude of the force exerted on $P$ by the surface is $R$ (see diagram).
Find $v ^ { 2 }$ in terms of $u , a , g$ and $\theta$ and show that $R = \frac { m u ^ { 2 } } { a } + m g ( 3 \cos \theta - 2 )$.
Given that $P$ just reaches the highest point of the circle, find $u ^ { 2 }$ in terms of $a$ and $g$, and show that in this case the least value of $v ^ { 2 }$ is $a g$.
Given instead that $P$ oscillates between $\theta = \pm \frac { 1 } { 6 } \pi$ radians, find $u ^ { 2 }$ in terms of $a$ and $g$.

OCR M3 2010 June Q1

1 A small ball of mass 0.8 kg is moving with speed $10.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when it receives an impulse of magnitude 4 Ns . The speed of the ball immediately afterwards is $8.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The angle between the directions of motion before and after the impulse acts is $\alpha$. Using an impulse-momentum triangle, or otherwise, find $\alpha$.

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