Questions M2 - A-Level Maths

Show that $k = 50$.
Find the maximum possible acceleration of the car when it is travelling at $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on a horizontal road.
The car climbs a hill, which is inclined at an angle of $10 ^ { \circ }$ to the horizontal, at a constant speed of $15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Calculate the power of the car's engine.

OCR M2 2008 January Q5

5 A particle $P$ of mass $2 m$ is moving on a smooth horizontal surface with speed $u$ when it collides directly with a particle $Q$ of mass $k m$ whose speed is $3 u$ in the opposite direction. As a result of the collision, the directions of motion of both particles are reversed and the speed of $P$ is halved.

Find, in terms of $u$ and $k$, the speed of $Q$ after the collision. Hence write down the range of possible values of $k$.
Calculate the magnitude of the impulse which $Q$ exerts on $P$.
Given that $k = \frac { 1 } { 2 }$, calculate the coefficient of restitution between $P$ and $Q$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{982647bd-8514-40cf-b4ee-674f51df32c5-3_472_1143_221_242} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} One end of a light inextensible string is attached to a point $P$. The other end is attached to a point $Q , 1.96 \mathrm {~m}$ vertically below $P$. A small smooth bead $B$, of mass 0.3 kg , is threaded on the string and moves in a horizontal circle with centre $Q$ and radius $1.96 \mathrm {~m} . B$ rotates about $Q$ with constant angular speed $\omega$ rad s $^ { - 1 }$ (see Fig. 1).

OCR M2 2008 January Q7

7 A missile is projected from a point $O$ on horizontal ground with speed $175 \mathrm {~ms} ^ { - 1 }$ at an angle of elevation $\theta$. The horizontal lower surface of a cloud is 650 m above the ground.

Find the value of $\theta$ for which the missile just reaches the cloud. It is given that $\theta = 55 ^ { \circ }$.
Find the length of time for which the missile is above the lower surface of the cloud.
Find the speed of the missile at the instant it enters the cloud.

OCR M2 2008 January Q8

A uniform semicircular lamina has radius 4 cm . Show that the distance from its centre to its centre of mass is 1.70 cm , correct to 3 significant figures.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{982647bd-8514-40cf-b4ee-674f51df32c5-4_429_947_405_640} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A model bridge is made from a uniform rectangular board, $A B C D$, with a semicircular section, $E F G$, removed. $O$ is the mid-point of $E G$. $A B = 8 \mathrm {~cm} , B C = 20 \mathrm {~cm} , A O = 12 \mathrm {~cm}$ and the radius of the semicircle is 4 cm (see Fig. 1).
(a) Show that the distance from $A B$ to the centre of mass of the model is 9.63 cm , correct to 3 significant figures.
(b) Calculate the distance from $A D$ to the centre of mass of the model.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{982647bd-8514-40cf-b4ee-674f51df32c5-4_572_945_1416_641} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The model bridge is smoothly pivoted at $A$ and is supported in equilibrium by a vertical wire attached to $D$. The weight of the model is 15 N and $A D$ makes an angle of $10 ^ { \circ }$ with the horizontal (see Fig. 2). Calculate the tension in the wire.

OCR M2 2009 January Q1

1 A stone is projected from a point on level ground with speed $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of elevation of $\theta ^ { \circ }$ above the horizontal. When the stone is at its greatest height it just passes over the top of a tree that is 17 m high. Calculate $\theta$.

OCR M2 2009 January Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{dd23f4a8-f7e7-4f80-bad6-7e9ec21565fc-2_465_643_495_749} A uniform right-angled triangular lamina $A B C$ with sides $A B = 12 \mathrm {~cm} , B C = 9 \mathrm {~cm}$ and $A C = 15 \mathrm {~cm}$ is freely suspended from a hinge at its vertex $A$. The lamina has mass 2 kg and is held in equilibrium with $A B$ horizontal by means of a string attached to $B$. The string is at an angle of $30 ^ { \circ }$ to the horizontal (see diagram). Calculate the tension in the string.

OCR M2 2009 January Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{dd23f4a8-f7e7-4f80-bad6-7e9ec21565fc-2_828_476_1338_836} A door is modelled as a lamina $A B C D E$ consisting of a uniform rectangular section $A B D E$ of weight 60 N and a uniform semicircular section $B C D$ of weight 10 N and radius $40 \mathrm {~cm} . A B$ is 200 cm and $A E$ is 80 cm . The door is freely hinged at $F$ and $G$, where $G$ is 30 cm below $B$ and $F$ is 30 cm above $A$ (see diagram).

Find the magnitudes and directions of the horizontal components of the forces on the door at each of $F$ and $G$.
Calculate the distance from $A E$ to the centre of mass of the door.

OCR M2 2009 January Q4

4 A car of mass 800 kg experiences a resistance of magnitude $k v ^ { 2 } \mathrm {~N}$, where $k$ is a constant and $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the car's speed. The car's engine is working at a constant rate of $P \mathrm {~W}$. At an instant when the car is travelling on a horizontal road with speed $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ its acceleration is $0.75 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. At an instant when the car is ascending a hill of constant slope $12 ^ { \circ }$ to the horizontal with speed $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ its acceleration is $0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.

Show that $k = 0.900$, correct to 3 decimal places, and find $P$. The power is increased to $1.5 P \mathrm {~W}$.
Calculate the maximum steady speed of the car on a horizontal road.

OCR M2 2009 January Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{dd23f4a8-f7e7-4f80-bad6-7e9ec21565fc-3_729_739_868_703} A particle $P$ of mass 0.2 kg is attached to one end of each of two light inextensible strings, one of length 0.4 m and one of length 0.3 m . The other end of the longer string is attached to a fixed point $A$, and the other end of the shorter string is attached to a fixed point $B$, which is vertically below $A$. The particle moves in a horizontal circle of radius 0.24 m at a constant angular speed of $8 \mathrm { rad } \mathrm { s } ^ { - 1 }$ (see diagram). Both strings are taut, the tension in $A P$ is $S \mathrm {~N}$ and the tension in $B P$ is $T \mathrm {~N}$.

By resolving vertically, show that $4 S = 3 T + 9.8$.
Find another equation connecting $S$ and $T$ and hence calculate the tensions, correct to 1 decimal place. \section*{[Questions 6 and 7 are printed overleaf.]}

OCR M2 2009 January Q6

6 A particle is projected from a point $O$ with speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of elevation $\theta$ above the horizontal and it moves freely under gravity. The horizontal and upward vertical displacements of the particle from $O$ at any subsequent time, $t$ seconds, are $x \mathrm {~m}$ and $y \mathrm {~m}$ respectively.

Express $x$ and $y$ in terms of $\theta$ and $t$, and hence show that $$y = x \tan \theta - \frac { 4.9 x ^ { 2 } } { v ^ { 2 } \cos ^ { 2 } \theta } .$$
\includegraphics[max width=\textwidth, alt={}]{dd23f4a8-f7e7-4f80-bad6-7e9ec21565fc-4_551_575_667_826}
The particle subsequently passes through the point $A$ with coordinates $( h , - h )$ as shown in the diagram. It is given that $v = 14$ and $\theta = 30 ^ { \circ }$.
Calculate $h$.
Calculate the direction of motion of the particle at $A$.
Calculate the speed of the particle at $A$.
\includegraphics[max width=\textwidth, alt={}, center]{dd23f4a8-f7e7-4f80-bad6-7e9ec21565fc-4_278_1061_1749_543} Two small spheres, $P$ and $Q$, are free to move on the inside of a smooth hollow cylinder, in such a way that they remain in contact with both the curved surface and the base of the cylinder. The mass of $P$ is 0.2 kg , the mass of $Q$ is 0.3 kg and the radius of the cylinder is $0.4 \mathrm {~m} . P$ and $Q$ are stationary at opposite ends of a diameter of the base of the cylinder (see diagram). The coefficient of restitution between $P$ and $Q$ is $0.5 . P$ is given an impulse of magnitude 0.8 Ns in a tangential direction.
Calculate the speeds of the particles after $P$ 's first impact with $Q$.
$Q$ subsequently catches up with $P$ and there is a second impact.
Calculate the speeds of the particles after this second impact.
Calculate the magnitude of the force exerted on $Q$ by the curved surface of the cylinder after the second impact.

OCR M2 2010 January Q1

1 Find the average power exerted by a climber of mass 75 kg when climbing a vertical distance of 40 m in 2 minutes.

OCR M2 2010 January Q2

2 A small sphere of mass 0.2 kg is dropped from rest at a height of 3 m above horizontal ground. It falls vertically, hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height of 1.8 m above the ground.

Calculate the magnitude of the impulse which the ground exerts on the sphere.
Calculate the coefficient of restitution between the sphere and the ground.

OCR M2 2010 January Q3

3 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8e1225a2-cb98-4b71-a4af-0150f093f852-2_528_688_845_731} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} A uniform conical shell has mass 0.2 kg , height 0.3 m and base diameter 0.8 m . A uniform hollow cylinder has mass 0.3 kg , length 0.7 m and diameter 0.8 m . The conical shell is attached to the cylinder, with the circumference of its base coinciding with one end of the cylinder (see Fig. 1).

Show that the distance of the centre of mass of the combined object from the vertex of the conical shell is 0.47 m . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8e1225a2-cb98-4b71-a4af-0150f093f852-2_497_572_1836_788} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The combined object is freely suspended from its vertex and is held with its axis horizontal. This is achieved by means of a wire attached to a point on the circumference of the base of the conical shell. The wire makes an angle of $80 ^ { \circ }$ with the slant edge of the conical shell (see Fig. 2).
Calculate the tension in the wire.

OCR M2 2010 January Q4

4 A car of mass 700 kg is moving along a horizontal road against a constant resistance to motion of 400 N . At an instant when the car is travelling at $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ its acceleration is $0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.

Find the driving force of the car at this instant.
Find the power at this instant. The maximum steady speed of the car on a horizontal road is $35 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find the maximum power of the car. The car now moves at maximum power against the same resistance up a slope of constant angle $\theta ^ { \circ }$ to the horizontal. The maximum steady speed up the slope is $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find $\theta$.

OCR M2 2010 January Q5

5 Two spheres of the same radius with masses 2 kg and 3 kg are moving directly towards each other on a smooth horizontal plane with speeds $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively. The spheres collide and the kinetic energy lost is 81 J . Calculate the speed and direction of motion of each sphere after the collision.

OCR M2 2010 January Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{8e1225a2-cb98-4b71-a4af-0150f093f852-3_698_1047_1297_550} A particle $P$ is projected with speed $V _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of elevation $\theta _ { 1 }$ from a point $O$ on horizontal ground. When $P$ is vertically above a point $A$ on the ground its height is 250 m and its velocity components are $40 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ horizontally and $30 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ vertically upwards (see diagram).

Show that $V _ { 1 } = 86.0$ and $\theta _ { 1 } = 62.3 ^ { \circ }$, correct to 3 significant figures. At the instant when $P$ is vertically above $A$, a second particle $Q$ is projected from $O$ with speed $V _ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of elevation $\theta _ { 2 } . P$ and $Q$ hit the ground at the same time and at the same place.
Calculate the total time of flight of $P$ and the total time of flight of $Q$.
Calculate the range of the particles and hence calculate $V _ { 2 }$ and $\theta _ { 2 }$.

OCR M2 2010 January Q7

7 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8e1225a2-cb98-4b71-a4af-0150f093f852-4_444_771_258_687} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} A particle $P$ of mass 0.2 kg is moving on the smooth inner surface of a fixed hollow hemisphere which has centre $O$ and radius $5 \mathrm {~m} . P$ moves with constant angular speed $\omega$ in a horizontal circle at a vertical distance of 3 m below the level of $O$ (see Fig.1).

Calculate the magnitude of the force exerted by the hemisphere on $P$.
Calculate $\omega$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8e1225a2-cb98-4b71-a4af-0150f093f852-4_592_773_1231_687} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A light inextensible string is now attached to $P$. The string passes through a small smooth hole at the lowest point of the hemisphere and a particle of mass 0.1 kg hangs in equilibrium at the end of the string. $P$ moves in the same horizontal circle as before (see Fig. 2).
Calculate the new angular speed of $P$.

OCR M2 2011 January Q1

1
\includegraphics[max width=\textwidth, alt={}, center]{941c0c81-a74f-49c0-acb7-1c23266fc2c8-02_378_471_260_836} A uniform square frame $A B C D$ has sides of length 0.6 m . The side $A D$ is removed from the frame, and the open frame $A B C D$ is attached at $A$ to a fixed point (see diagram).

Calculate the distance of the centre of mass of the open frame from $A$. The open frame rotates about $A$ in the plane $A B C D$ with angular speed $3 \mathrm { rad } \mathrm { s } ^ { - 1 }$.
Calculate the speed of the centre of mass of the open frame.

OCR M2 2011 January Q2

2 The resistance to the motion of a car is $k v ^ { \frac { 3 } { 2 } } \mathrm {~N}$, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the car's speed and $k$ is a constant. The power exerted by the car's engine is 15000 W , and the car has constant speed $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ along a horizontal road.

Show that $k = 4.8$. With the engine operating at a much lower power, the car descends a hill of inclination $\alpha$, where $\sin \alpha = \frac { 1 } { 15 }$. At an instant when the speed of the car is $16 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, its acceleration is $0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
Given that the mass of the car is 700 kg , calculate the power of the engine.
\includegraphics[max width=\textwidth, alt={}, center]{941c0c81-a74f-49c0-acb7-1c23266fc2c8-02_579_447_1658_849} A particle $P$ of mass 0.4 kg is attached to one end of each of two light inextensible strings which are both taut. The other end of the longer string is attached to a fixed point $A$, and the other end of the shorter string is attached to a fixed point $B$, which is vertically below $A$. The string $A P$ makes an angle of $30 ^ { \circ }$ with the vertical and is 0.5 m long. The string $B P$ makes an angle of $60 ^ { \circ }$ with the vertical. $P$ moves with constant angular speed in a horizontal circle with centre vertically below $B$ (see diagram). The tension in the string $A P$ is twice the tension in the string $B P$. Calculate

OCR M2 2011 January Q4

4 A block of mass 25 kg is dragged 30 m up a slope inclined at $5 ^ { \circ }$ to the horizontal by a rope inclined at $20 ^ { \circ }$ to the slope. The tension in the rope is 100 N and the resistance to the motion of the block is 70 N . The block is initially at rest. Calculate

the work done by the tension in the rope,
the change in the potential energy of the block,
the speed of the block after it has moved 30 m up the slope.

OCR M2 2011 January Q5

5 A uniform solid is made of a hemisphere with centre $O$ and radius 0.6 m , and a cylinder of radius 0.6 m and height 0.6 m . The plane face of the hemisphere and a plane face of the cylinder coincide. (The formula for the volume of a sphere is $\frac { 4 } { 3 } \pi r ^ { 3 }$.)

Show that the distance of the centre of mass of the solid from $O$ is 0.09 m .
\includegraphics[max width=\textwidth, alt={}, center]{941c0c81-a74f-49c0-acb7-1c23266fc2c8-03_636_1036_982_593} The solid is placed with the curved surface of the hemisphere on a rough horizontal surface and the axis inclined at $45 ^ { \circ }$ to the horizontal. The equilibrium of the solid is maintained by a horizontal force of 2 N applied to the highest point on the circumference of its plane face (see diagram). Calculate
(a) the mass of the solid,
(b) the set of possible values of the coefficient of friction between the surface and the solid.

OCR M2 2011 January Q6

6 A small ball $B$ is projected with speed $14 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of elevation $30 ^ { \circ }$ from a point $O$ on a horizontal plane, and moves freely under gravity.

Calculate the height of $B$ above the plane when moving horizontally.
$B$ has mass 0.4 kg . At the instant when $B$ is moving horizontally it receives an impulse of magnitude $I \mathrm { Ns }$ in its direction of motion which immediately increases the speed of $B$ to $15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Calculate $I$. For the instant when $B$ returns to the plane, calculate
the speed and direction of motion of $B$,
the time of flight, and the distance of $B$ from $O$.

OCR M2 2011 January Q8

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OCR M2 2011 January Q10

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\section*{OCR
RECOGNISING ACHIEVEMENT}

OCR M2 2012 January Q1

1 A particle $P$ is projected with speed $40 \mathrm {~ms} ^ { - 1 }$ at an angle of $35 ^ { \circ }$ above the horizontal from a point $O$. For the instant 3 s after projection, calculate the magnitude and direction of the velocity of $P$.

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