Questions M1 - A-Level Maths

Find the value of the normal reaction of the floor on the box. The box remains at rest on the floor when a force of 10 N is applied to it at an angle of $40 ^ { \circ }$ to the upward vertical, as shown in Fig. 3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{52d6c914-b204-4587-a82e-fbab6693fcf8-2_293_472_2131_794} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
Draw a diagram showing all the forces acting on the box.
Calculate the new value of the normal reaction of the floor on the box and also the frictional force.

OCR MEI M1 2007 January Q4

7 marks Moderate -0.8

4 Fig. 4 shows forces of magnitudes 20 N and 16 N inclined at $60 ^ { \circ }$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{52d6c914-b204-4587-a82e-fbab6693fcf8-3_191_346_328_858} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure}

Calculate the component of the resultant of these two forces in the direction of the 20 N force.
Calculate the magnitude of the resultant of these two forces. These are the only forces acting on a particle of mass 2 kg .
Find the magnitude of the acceleration of the particle and the angle the acceleration makes with the 20 N force.

OCR MEI M1 2007 January Q5

6 marks Moderate -0.3

5 A block of mass 4 kg slides on a horizontal plane against a constant resistance of 14.8 N . A light, inextensible string is attached to the block and, after passing over a smooth pulley, is attached to a freely hanging sphere of mass 2 kg . The part of the string between the block and the pulley is horizontal. This situation is shown in Fig. 5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{52d6c914-b204-4587-a82e-fbab6693fcf8-3_250_671_1466_696} \captionsetup{labelformat=empty} \caption{Fig. 5}

\end{figure} The tension in the string is $T \mathrm {~N}$ and the acceleration of the block and of the sphere is $a \mathrm {~ms} ^ { - 2 }$.

Write down the equation of motion of the block and also the equation of motion of the sphere, each in terms of $T$ and $a$.
Find the values of $T$ and $a$.

OCR MEI M1 2007 January Q6

7 marks Moderate -0.8

6 The velocity of a model boat, $\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }$, is given by $$\mathbf { v } = \binom { - 5 } { 10 } + t \binom { 6 } { - 8 }$$ where $t$ is the time in seconds and the vectors $\binom { 1 } { 0 }$ and $\binom { 0 } { 1 }$ are east and north respectively.

Show that when $t = 2.5$ the boat is travelling south-east (i.e. on a bearing of $135 ^ { \circ }$ ). Calculate its speed at this time. The boat is at a point O when $t = 0$.
Calculate the bearing of the boat from O when $t = 2.5$.

OCR MEI M1 2007 January Q7

18 marks Standard +0.3

7 A horizontal force of 24 N acts on a block of mass 12 kg on a horizontal plane. The block is initially at rest. This situation is first modelled assuming the plane is smooth.

Write down the acceleration of the block according to this model. The situation is now modelled assuming a constant resistance to motion of 15 N .
Calculate the acceleration of the block according to this new model. How much less distance does the new model predict that the block will travel in the first 4 seconds? The 24 N force is removed and the block slides down a slope at $5 ^ { \circ }$ to the horizontal. The speed of the block at the top of the slope is $1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, as shown in Fig. 7. The answers to parts (iii) and (iv) should be found using the assumption that the resistance to the motion of the block is still a constant 15 N . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{52d6c914-b204-4587-a82e-fbab6693fcf8-5_255_901_1128_575} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
Calculate the acceleration of the block in the direction of its motion.
For how much time does the block slide down the slope before coming to rest and how far does it slide in that time? Measurements show that the block actually comes to rest in 3.5 seconds.
Assuming that the error in the prediction is due only to the value of the resistance, calculate the true value of the resistance.

OCR MEI M1 2008 January Q1

6 marks Easy -1.3

1 A cyclist starts from rest and takes 10 seconds to accelerate at a constant rate up to a speed of $15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. After travelling at this speed for 20 seconds, the cyclist then decelerates to rest at a constant rate over the next 5 seconds.

Sketch a velocity-time graph for the motion.
Calculate the distance travelled by the cyclist.

OCR MEI M1 2008 January Q2

7 marks Moderate -0.8

2 The force acting on a particle of mass 1.5 kg is given by the vector $\binom { 6 } { 9 } \mathrm {~N}$.

Give the acceleration of the particle as a vector.
Calculate the angle that the acceleration makes with the direction $\binom { 1 } { 0 }$.
At a certain point of its motion, the particle has a velocity of $\binom { - 2 } { 3 } \mathrm {~ms} ^ { - 1 }$. Calculate the displacement of the particle over the next two seconds.

OCR MEI M1 2008 January Q3

8 marks Moderate -0.8

3 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5211a643-307a-4886-a2e2-c11b28e05216-2_344_716_1324_717} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} Fig. 3 shows a block of mass 15 kg on a rough, horizontal plane. A light string is fixed to the block at A, passes over a smooth, fixed pulley B and is attached at C to a sphere. The section of the string between the block and the pulley is inclined at $40 ^ { \circ }$ to the horizontal and the section between the pulley and the sphere is vertical. The system is in equilibrium and the tension in the string is 58.8 N .

The sphere has a mass of $m \mathrm {~kg}$. Calculate the value of $m$.
Calculate the frictional force acting on the block.
Calculate the normal reaction of the plane on the block.

OCR MEI M1 2008 January Q4

7 marks Moderate -0.8

4 Force $\mathbf { F }$ is $\left( \begin{array} { l } 4 \\ 1 \\ 2 \end{array} \right) \mathrm { N }$ and force $\mathbf { G }$ is $\left( \begin{array} { r } - 6 \\ 2 \\ 4 \end{array} \right) \mathrm { N }$.

Find the resultant of $\mathbf { F }$ and $\mathbf { G }$ and calculate its magnitude.
Forces $\mathbf { F } , 2 \mathbf { G }$ and $\mathbf { H }$ act on a particle which is in equilibrium. Find $\mathbf { H }$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5211a643-307a-4886-a2e2-c11b28e05216-3_99_841_676_651} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} A toy car is moving along the straight line $\mathrm { O } x$, where O is the origin. The time $t$ is in seconds. At time $t = 0$ the car is at $\mathrm { A } , 3 \mathrm {~m}$ from O as shown in Fig. 5. The velocity of the car, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, is given by $$v = 2 + 12 t - 3 t ^ { 2 }$$ Calculate the distance of the car from O when its acceleration is zero.

OCR MEI M1 2008 January Q6

17 marks Moderate -0.3

6 A helicopter rescue activity at sea is modelled as follows. The helicopter is stationary and a man is suspended from it by means of a vertical, light, inextensible wire that may be raised or lowered, as shown in Fig. 6.1.

When the man is descending with an acceleration $1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ downwards, how much time does it take for his speed to increase from $0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ downwards to $3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ downwards? \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5211a643-307a-4886-a2e2-c11b28e05216-4_373_460_365_1242} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
\end{figure} How far does he descend in this time? The man has a mass of 80 kg . All resistances to motion may be neglected.
Calculate the tension in the wire when the man is being lowered
(A) with an acceleration of $1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ downwards,
(B) with an acceleration of $1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ upwards. Subsequently, the man is raised and this situation is modelled with a constant resistance of 116 N to his upward motion.
For safety reasons, the tension in the wire should not exceed 2500 N . What is the maximum acceleration allowed when the man is being raised? At another stage of the rescue, the man has equipment of mass 10 kg at the bottom of a vertical rope which is hanging from his waist, as shown in Fig. 6.2. The man and his equipment are being raised; the rope is light and inextensible and the tension in it is 80 N .
Assuming that the resistance to the upward motion of the man is still 116 N and that there is negligible resistance to the motion of the equipment, calculate the \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5211a643-307a-4886-a2e2-c11b28e05216-4_442_460_1589_1242} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
\end{figure} tension in the wire.

OCR MEI M1 2008 January Q7

19 marks Moderate -0.3

7 A small firework is fired from a point O at ground level over horizontal ground. The highest point reached by the firework is a horizontal distance of 60 m from O and a vertical distance of 40 m from O , as shown in Fig. 7. Air resistance is negligible. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5211a643-307a-4886-a2e2-c11b28e05216-5_600_1029_447_557} \captionsetup{labelformat=empty} \caption{Fig. 7}

\end{figure} The initial horizontal component of the velocity of the firework is $21 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Calculate the time for the firework to reach its highest point and show that the initial vertical component of its velocity is $28 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Show that the firework is $\left( 28 t - 4.9 t ^ { 2 } \right) \mathrm { m }$ above the ground $t$ seconds after its projection. When the firework is at its highest point it explodes into several parts. Two of the parts initially continue to travel horizontally in the original direction, one with the original horizontal speed of $21 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the other with a quarter of this speed.
State why the two parts are always at the same height as one another above the ground and hence find an expression in terms of $t$ for the distance between the parts $t$ seconds after the explosion.
Find the distance between these parts of the firework
(A) when they reach the ground,
(B) when they are 10 m above the ground.
Show that the cartesian equation of the trajectory of the firework before it explodes is $y = \frac { 1 } { 90 } \left( 120 x - x ^ { 2 } \right)$, referred to the coordinate axes shown in Fig. 7.

OCR MEI M1 2010 January Q1

5 marks Easy -1.2

1 A ring is moving up and down a vertical pole. The displacement, $s \mathrm {~m}$, of the ring above a mark on the pole is modelled by the displacement-time graph shown in Fig. 1. The three sections of the graph are straight lines. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{eafaf02f-bcd4-4368-a282-61ef1ad074da-2_766_1065_500_539} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure}

Calculate the velocity of the ring in the interval $0 < t < 2$ and in the interval $2 < t < 3.5$.
Sketch a velocity-time graph for the motion of the ring during the 4 seconds.
State the direction of motion of the ring when
(A) $t = 1$,
(B) $t = 2.75$,
(C) $t = 3.25$.

OCR MEI M1 2010 January Q2

7 marks Moderate -0.8

2 A particle of mass 5 kg has constant acceleration. Initially, the particle is at $\binom { - 1 } { 2 } \mathrm {~m}$ with velocity $\binom { 2 } { - 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$; after 4 seconds the particle has velocity $\binom { 12 } { 9 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Calculate the acceleration of the particle.
Calculate the position of the particle at the end of the 4 seconds.
Calculate the force acting on the particle.

OCR MEI M1 2010 January Q3

8 marks Moderate -0.8

3 In this question, $\mathbf { i }$ is a horizontal unit vector and $\mathbf { j }$ is a unit vector pointing vertically upwards.
A force $\mathbf { F }$ is $- \mathbf { i } + 5 \mathbf { j }$.

Calculate the magnitude of $\mathbf { F }$. Calculate also the angle between $\mathbf { F }$ and the upward vertical. Force $\mathbf { G }$ is $2 a \mathbf { i } + a \mathbf { j }$ and force $\mathbf { H }$ is $- 2 \mathbf { i } + 3 b \mathbf { j }$, where $a$ and $b$ are constants. The force $\mathbf { H }$ is the resultant of forces $4 \mathbf { F }$ and $\mathbf { G }$.
Find $\mathbf { G }$ and $\mathbf { H }$.

OCR MEI M1 2010 January Q4

8 marks Moderate -0.8

4 A box of mass 2.5 kg is on a smooth horizontal table, as shown in Fig. 4. A light string AB is attached to the table at A and the box at B . AB is at an angle of $50 ^ { \circ }$ to the vertical. Another light string is attached to the box at C ; this string is inclined at $15 ^ { \circ }$ above the horizontal and the tension in it is 20 N . The box is in equilibrium. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{eafaf02f-bcd4-4368-a282-61ef1ad074da-3_403_1063_1085_539} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure}

Calculate the horizontal component of the force exerted on the box by the string at C .
Calculate the tension in the string AB .
Calculate the normal reaction of the table on the box. The string at C is replaced by one inclined at $15 ^ { \circ }$ below the horizontal with the same tension of 20 N .
Explain why this has no effect on the tension in string AB .

OCR MEI M1 2011 January Q1

8 marks Moderate -0.8

1 An object C is moving along a vertical straight line. Fig. 1 shows the velocity-time graph for part of its motion. Initially C is moving upwards at $14 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and after 10 s it is moving downwards at $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e36ef805-beff-4125-b332-439ccb0d91c4-2_878_933_479_607} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} C then moves as follows.

In the interval $10 \leqslant t \leqslant 15$, the velocity of C is constant at $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ downwards.
In the interval $15 \leqslant t \leqslant 20$, the velocity of C increases uniformly so that C has zero velocity at $t = 20$.
1. Complete the velocity-time graph for the motion of C in the time interval $0 \leqslant t \leqslant 20$.
2. Calculate the acceleration of C in the time interval $0 < t < 10$.
3. Calculate the displacement of C from $t = 0$ to $t = 20$.

OCR MEI M1 2011 January Q2

6 marks Moderate -0.8

2 Fig. 2 shows two forces acting at A. The figure also shows the perpendicular unit vectors $\mathbf { i }$ and $\mathbf { j }$ which are respectively horizontal and vertically upwards. The resultant of the two forces is $\mathbf { F } \mathbf { N }$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e36ef805-beff-4125-b332-439ccb0d91c4-3_264_922_479_609} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure}

Find $\mathbf { F }$ in terms of $\mathbf { i }$ and $\mathbf { j }$, giving your answer correct to three significant figures.
Calculate the magnitude of $\mathbf { F }$ and the angle that $\mathbf { F }$ makes with the upward vertical.

OCR MEI M1 2011 January Q3

7 marks Moderate -0.3

3 Two cars, P and Q, are being crashed as part of a film 'stunt'.
At the start

P is travelling directly towards Q with a speed of $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$,
Q is instantaneously at rest and has an acceleration of $4 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ directly towards P .
$P$ continues with the same velocity and $Q$ continues with the same acceleration. The cars collide $T$ seconds after the start.
1. Find expressions in terms of $T$ for how far each of the cars has travelled since the start.

At the start, $P$ is 90 m from $Q$.

Show that $T ^ { 2 } + 4 T - 45 = 0$ and hence find $T$.

OCR MEI M1 2011 January Q4

8 marks Standard +0.3

4 At time $t$ seconds, a particle has position with respect to an origin O given by the vector $$\mathbf { r } = \binom { 8 t } { 10 t ^ { 2 } - 2 t ^ { 3 } } ,$$ where $\binom { 1 } { 0 }$ and $\binom { 0 } { 1 }$ are perpendicular unit vectors east and north respectively and distances are in metres.

When $t = 1$, the particle is at P . Find the bearing of P from O .
Find the velocity of the particle at time $t$ and show that it is never zero.
Determine the time(s), if any, when the acceleration of the particle is zero.

OCR MEI M1 2011 January Q5

7 marks Moderate -0.3

5 Fig. 5 shows two boxes, A of mass 12 kg and B of mass 6 kg , sliding in a straight line on a rough horizontal plane. The boxes are connected by a light rigid rod which is parallel to the line of motion. The only forces acting on the boxes in the line of motion are those due to the rod and a constant force of $F \mathrm {~N}$ on each box. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e36ef805-beff-4125-b332-439ccb0d91c4-4_246_1006_479_568} \captionsetup{labelformat=empty} \caption{Fig. 5}

\end{figure} The boxes have an initial speed of $1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and come to rest after sliding a distance of 0.375 m .

Calculate the deceleration of the boxes and the value of $F$.
Calculate the magnitude of the force in the rod and state, with a reason, whether it is a tension or a thrust (compression).

OCR MEI M1 2012 January Q1

5 marks Moderate -0.8

1 Fig. 1 shows two blocks of masses 3 kg and 5 kg connected by a light string which passes over a smooth, fixed pulley. Initially the blocks are held at rest but then they are released. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0330185f-d79d-4a78-9fa2-29ec345c2856-2_490_303_520_881} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} Find the acceleration of the blocks when they start to move, and the tension in the string.

OCR MEI M1 2012 January Q2

7 marks Moderate -0.3

2 Fig. 2 shows a small object, P , of weight 20 N , suspended by two light strings. The strings are tied to points A and B on a sloping ceiling which is at an angle of $60 ^ { \circ }$ to the upward vertical. The string AP is at $60 ^ { \circ }$ to the downward vertical and the string BP makes an angle of $30 ^ { \circ }$ with the ceiling. The object is in equilibrium. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0330185f-d79d-4a78-9fa2-29ec345c2856-2_430_670_1546_699} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure}

Show that $\angle \mathrm { APB } = 90 ^ { \circ }$.
Draw a labelled triangle of forces to represent the three forces acting on P .
Hence, or otherwise, find the tensions in the two strings.

OCR MEI M1 2012 January Q3

8 marks Moderate -0.3

3 Two girls, Marie and Nina, are members of an Olympic hockey team. They are doing fitness training. Marie runs along a straight line at a constant speed of $6 \mathrm {~ms} ^ { - 1 }$. Nina is stationary at a point O on the line until Marie passes her. Nina immediately runs after Marie until she catches up with her. The time, $t \mathrm {~s}$, is measured from the moment when Nina starts running. So when $t = 0$, both girls are at O .
Nina's acceleration, $a \mathrm {~ms} ^ { - 2 }$, is given by $$\begin{array} { l l } a = 4 - t & \text { for } 0 \leqslant t \leqslant 4 , \\ a = 0 & \text { for } t > 4 . \end{array}$$

Show that Nina's speed, $v \mathrm {~ms} ^ { - 1 }$, is given by $$\begin{array} { l l } v = 4 t - \frac { 1 } { 2 } t ^ { 2 } & \text { for } 0 \leqslant t \leqslant 4 , \\ v = 8 & \text { for } t > 4 . \end{array}$$
Find an expression for the distance Nina has run at time $t$, for $0 \leqslant t \leqslant 4$. Find how far Nina has run when $t = 4$ and when $t = 5 \frac { 1 } { 3 }$.
Show that Nina catches up with Marie when $t = 5 \frac { 1 } { 3 }$.

OCR MEI M1 2012 January Q4

8 marks Moderate -0.8

4 A projectile P travels in a vertical plane over level ground. Its position vector $\mathbf { r }$ at time $t$ seconds after projection is modelled by $$\mathbf { r } = \binom { x } { y } = \binom { 0 } { 5 } + \binom { 30 } { 40 } t - \binom { 0 } { 5 } t ^ { 2 } ,$$ where distances are in metres and the origin is a point on the level ground.

Write down
(A) the height from which P is projected,
(B) the value of $g$ in this model.
Find the displacement of P from $t = 3$ to $t = 5$.
Show that the equation of the trajectory is $$y = 5 + \frac { 4 } { 3 } x - \frac { x ^ { 2 } } { 180 } .$$

OCR MEI M1 2012 January Q5

8 marks Moderate -0.8

5 The vectors $\mathbf { p }$ and $\mathbf { q }$ are given by $$\mathbf { p } = 8 \mathbf { i } + \mathbf { j } \text { and } \mathbf { q } = 4 \mathbf { i } - 7 \mathbf { j } .$$

Show that $\mathbf { p }$ and $\mathbf { q }$ are equal in magnitude.
Show that $\mathbf { p } + \mathbf { q }$ is parallel to $2 \mathbf { i } - \mathbf { j }$.
Draw $\mathbf { p } + \mathbf { q }$ and $\mathbf { p } - \mathbf { q }$ on the grid. Write down the angle between these two vectors.

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