Questions M1 - A-Level Maths

Find the coefficient of friction between $B$ and the surface. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4b703cf9-b3d3-4210-b57b-89136595f8a5-04_307_751_1000_696} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} $B$ is now cut horizontally into two smaller blocks. The upper block has weight 9 N and the lower block has weight 3 N . The 5 N force now acts on the upper block and the 4 N force now acts on the lower block (see Fig. 2). The coefficient of friction between the two blocks is $\mu$.
Given that the upper block is in limiting equilibrium, find $\mu$.
Given instead that $\mu = 0.1$, find the accelerations of the two blocks.

OCR M1 2010 June Q8

Moderate -0.8

8 \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}

6 (ii)

\href{http://physicsandmathstutor.com}{physicsandmathstutor.com}

OCR M1 2010 June Q10

Moderate -0.8

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\section*{PLEASE DO NOT WRITE ON THIS PAGE} RECOGNISING ACHIEVEMENT

OCR M1 Specimen Q1

4 marks Easy -1.2

1
\includegraphics[max width=\textwidth, alt={}, center]{463347e9-b850-4f4a-b2d2-423cf142e30f-2_99_812_310_635} An engine pulls a truck of mass 6000 kg along a straight horizontal track, exerting a constant horizontal force of magnitude $E$ newtons on the truck (see diagram). The resistance to motion of the truck has magnitude 400 N , and the acceleration of the truck is $0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. Find the value of $E$.

OCR M1 Specimen Q2

7 marks Moderate -0.8

2 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{463347e9-b850-4f4a-b2d2-423cf142e30f-2_166_518_824_351} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{463347e9-b850-4f4a-b2d2-423cf142e30f-2_168_755_822_1043} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} Forces of magnitudes 8 N and 5 N act on a particle. The angle between the directions of the two forces is $30 ^ { \circ }$, as shown in Fig. 1. The resultant of the two forces has magnitude $R \mathrm {~N}$ and acts at an angle $\theta ^ { \circ }$ to the force of magnitude 8 N , as shown in Fig. 2. Find $R$ and $\theta$.

OCR M1 Specimen Q3

10 marks Moderate -0.8

3 A particle is projected vertically upwards, from the ground, with a speed of $28 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Ignoring air resistance, find

the maximum height reached by the particle,
the speed of the particle when it is 30 m above the ground,
the time taken for the particle to fall from its highest point to a height of 30 m ,
the length of time for which the particle is more than 30 m above the ground. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{463347e9-b850-4f4a-b2d2-423cf142e30f-3_569_1132_258_516} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A woman runs from $A$ to $B$, then from $B$ to $A$ and then from $A$ to $B$ again, on a straight track, taking 90 s . The woman runs at a constant speed throughout. Fig. 1 shows the $( t , v )$ graph for the woman.
Find the total distance run by the woman.
Find the distance of the woman from $A$ when $t = 50$ and when $t = 80$, \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{463347e9-b850-4f4a-b2d2-423cf142e30f-3_424_1135_1233_513} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} At time $t = 0$, a child also starts to move, from $A$, along $A B$. The child walks at a constant speed for the first 50 s and then at an increasing speed for the next 40 s . Fig. 2 shows the ( $t , v$ ) graph for the child; it consists of two straight line segments.
At time $t = 50$, the woman and the child pass each other, moving in opposite directions. Find the speed of the child during the first 50 s .
At time $t = 80$, the woman overtakes the child. Find the speed of the child at this instant.

OCR M1 Specimen Q5

13 marks Standard +0.3

5 A particle $P$ moves in a straight line so that, at time $t$ seconds after leaving a fixed point $O$, its acceleration is $- \frac { 1 } { 10 } t \mathrm {~m} \mathrm {~s} ^ { - 2 }$. At time $t = 0$, the velocity of $P$ is $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Find, by integration, an expression in terms of $t$ and $V$ for the velocity of $P$.
Find the value of $V$, given that $P$ is instantaneously at rest when $t = 10$.
Find the displacement of $P$ from $O$ when $t = 10$.
Find the speed with which the particle returns to $O$.

OCR M1 Specimen Q6

13 marks Standard +0.8

6
\includegraphics[max width=\textwidth, alt={}, center]{463347e9-b850-4f4a-b2d2-423cf142e30f-4_168_1032_292_552} Three uniform spheres $A , B$ and $C$ have masses $0.3 \mathrm {~kg} , 0.4 \mathrm {~kg}$ and $m \mathrm {~kg}$ respectively. The spheres lie in a smooth horizontal groove with $B$ between $A$ and $C$. Sphere $B$ is at rest and spheres $A$ and $C$ are each moving with speed $3.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ towards $B$ (see diagram). Air resistance may be ignored.

$A$ collides with $B$. After this collision $A$ continues to move in the same direction as before, but with speed $0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find the speed with which $B$ starts to move.
$B$ and $C$ then collide, after which they both move towards $A$, with speeds of $3.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively. Find the value of $m$.
The next collision is between $A$ and $B$. Explain briefly how you can tell that, after this collision, $A$ and $B$ cannot both be moving towards $C$.
When the spheres have finished colliding, which direction is $A$ moving in? What can you say about its speed? Justify your answers.

OCR M1 Specimen Q7

13 marks Standard +0.3

7 A sledge of mass 25 kg is on a plane inclined at $30 ^ { \circ }$ to the horizontal. The coefficient of friction between the sledge and the plane is 0.2 .

\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{463347e9-b850-4f4a-b2d2-423cf142e30f-4_289_488_1493_849} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} The sledge is pulled up the plane, with constant acceleration, by means of a light cable which is parallel to a line of greatest slope (see Fig. 1). The sledge starts from rest and acquires a speed of $0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ after being pulled for 10 s . Ignoring air resistance, find the tension in the cable.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{463347e9-b850-4f4a-b2d2-423cf142e30f-4_291_490_2149_849} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} On a subsequent occasion the cable is not in use and two people of total mass 150 kg are seated in the sledge. The sledge is held at rest by a horizontal force of magnitude $P$ newtons, as shown in Fig. 2. Find the least value of $P$ which will prevent the sledge from sliding down the plane.

OCR MEI M1 2005 January Q1

7 marks Moderate -0.8

1 The position vector, $\mathbf { r }$, of a particle of mass 4 kg at time $t$ is given by $$\mathbf { r } = t ^ { 2 } \mathbf { i } + \left( 5 t - 2 t ^ { 2 } \right) \mathbf { j } ,$$ where $\mathbf { i }$ and $\mathbf { j }$ are the standard unit vectors, lengths are in metres and time is in seconds.

Find an expression for the acceleration of the particle. The particle is subject to a force $\mathbf { F }$ and a force $12 \mathbf { j } \mathbf { N }$.
Find $\mathbf { F }$.

OCR MEI M1 2005 January Q2

8 marks Standard +0.3

2 Particles of mass 2 kg and 4 kg are attached to the ends $X$ and $Y$ of a light, inextensible string. The string passes round fixed, smooth pulleys at $\mathrm { P } , \mathrm { Q }$ and R , as shown in Fig. 2. The system is released from rest with the string taut. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c84a748a-a6f4-48c5-b864-fe543569bdf5-2_478_397_1211_872} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure}

State what information in the question tells you that
(A) the tension is the same throughout the string,
(B) the magnitudes of the accelerations of the particles at X and Y are the same. The tension in the string is $T \mathrm {~N}$ and the magnitude of the acceleration of the particles is $a \mathrm {~ms} ^ { - 2 }$.
Draw a diagram showing the forces acting at X and a diagram showing the forces acting at Y .
Write down equations of motion for the particles at X and at Y . Hence calculate the values of $T$ and $a$.

OCR MEI M1 2005 January Q3

6 marks Moderate -0.8

3 A particle is in equilibrium when acted on by the forces $\left( \begin{array} { r } x \\ - 7 \\ z \end{array} \right) , \left( \begin{array} { r } 4 \\ y \\ - 5 \end{array} \right)$ and $\left( \begin{array} { r } 5 \\ 4 \\ - 7 \end{array} \right)$, where the units are newtons.

Find the values of $x , y$ and $z$.
Calculate the magnitude of $\left( \begin{array} { r } 5 \\ 4 \\ - 7 \end{array} \right)$.

OCR MEI M1 2005 January Q4

8 marks Standard +0.3

4 A particle is projected vertically upwards from a point O at $21 \mathrm {~ms} ^ { - 1 }$.

Calculate the greatest height reached by the particle. When this particle is at its highest point, a second particle is projected vertically upwards from $O$ at $15 \mathrm {~ms} ^ { - 1 }$.
Show that the particles collide 1.5 seconds later and determine the height above O at which the collision takes place.

OCR MEI M1 2005 January Q5

7 marks Moderate -0.8

5 A small box B of weight 400 N is held in equilibrium by two light strings AB and BC . The string BC is fixed at C . The end A of string AB is fixed so that AB is at an angle $\alpha$ to the vertical where $\alpha < 60 ^ { \circ }$. String BC is at $60 ^ { \circ }$ to the vertical. This information is shown in Fig. 5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c84a748a-a6f4-48c5-b864-fe543569bdf5-3_424_472_1599_774} \captionsetup{labelformat=empty} \caption{Fig. 5}

\end{figure}

Draw a labelled diagram showing all the forces acting on the box.
In one situation string AB is fixed so that $\alpha = 30 ^ { \circ }$. By drawing a triangle of forces, or otherwise, calculate the tension in the string BC and the tension in the string AB .
Show carefully, but briefly, that the box cannot be in equilibrium if $\alpha = 60 ^ { \circ }$ and BC remains at $60 ^ { \circ }$ to the vertical.

OCR MEI M1 2005 January Q7

17 marks Standard +0.3

7 The trajectory ABCD of a small stone moving with negligible air resistance is shown in Fig. 7. AD is horizontal and BC is parallel to AD . The stone is projected from A with speed $40 \mathrm {~ms} ^ { - 1 }$ at $50 ^ { \circ }$ to the horizontal. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c84a748a-a6f4-48c5-b864-fe543569bdf5-4_341_1107_484_498} \captionsetup{labelformat=empty} \caption{Fig. 7}

\end{figure}

Write down an expression for the horizontal displacement from A of the stone $t$ seconds after projection. Write down also an expression for the vertical displacement at time $t$.
Show that the stone takes 6.253 seconds (to three decimal places) to travel from A to D . Calculate the range of the stone. You are given that $X = 30$.
Calculate the time it takes the stone to reach B . Hence determine the time for it to travel from A to C.
Calculate the direction of the motion of the stone at $\mathbf { C }$. Section B (36 marks)

OCR MEI M1 2006 January Q1

6 marks Easy -1.3

1 A particle travels in a straight line during the time interval $0 \leqslant t \leqslant 12$, where $t$ is the time in seconds. Fig. 1 is the velocity-time graph for the motion. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{19d42df9-e752-4d33-85e1-4ec59b32135a-2_455_874_484_593} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure}

Calculate the acceleration of the particle in the interval $0 < t < 6$.
Calculate the distance travelled by the particle from $t = 0$ to $t = 4$.
When $t = 0$ the particle is at A . Calculate how close the particle gets to A during the interval $4 \leqslant t \leqslant 12$.

OCR MEI M1 2006 January Q2

5 marks Moderate -0.8

2 Fig. 2 shows a light string with an object of mass 4 kg attached at end A . The string passes over a smooth pulley and its other end B is attached to two light strings BC and BD of the same length. The strings BC and BD are attached to horizontal ground and are each inclined at $20 ^ { \circ }$ to the vertical. The system is in equilibrium. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{19d42df9-e752-4d33-85e1-4ec59b32135a-2_588_451_1749_806} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure}

What information in the question tells you that the tension is the same throughout the string AB ?
What is the tension in the string AB ?
Calculate the tension in the strings BC and BD .

OCR MEI M1 2006 January Q3

7 marks Moderate -0.8

3 A force $\mathbf { F }$ is given by $\mathbf { F } = ( 3.5 \mathbf { i } + 12 \mathbf { j } ) \mathrm { N }$, where $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors east and north respectively.

Calculate the magnitude of $\mathbf { F }$ and also its direction as a bearing.
$\mathbf { G }$ is the force $( 7 \mathbf { i } + 24 \mathbf { j } )$ N. Show that $\mathbf { G }$ and $\mathbf { F }$ are in the same direction and compare their magnitudes.
Force $\mathbf { F } _ { 1 }$ is $( 9 \mathbf { i } - 18 \mathbf { j } ) \mathrm { N }$ and force $\mathbf { F } _ { 2 }$ is $( 12 \mathbf { i } + q \mathbf { j } ) \mathrm { N }$. Find $q$ so that the sum $\mathbf { F } _ { 1 } + \mathbf { F } _ { 2 }$ is in the direction of $\mathbf { F }$.

OCR MEI M1 2006 January Q4

5 marks Moderate -0.8

4 A car and its trailer travel along a straight, horizontal road. The coupling between them is light and horizontal. The car has mass 900 kg and resistance to motion 100 N , the trailer has mass 700 kg and resistance to motion 300 N , as shown in Fig. 4. The car and trailer have an acceleration of $1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{19d42df9-e752-4d33-85e1-4ec59b32135a-3_400_753_1037_657} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure}

Calculate the driving force of the car.
Calculate the force in the coupling.

OCR MEI M1 2006 January Q5

6 marks Moderate -0.3

5 The acceleration of a particle of mass 4 kg is given by $\mathbf { a } = ( 9 \mathbf { i } - 4 t \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }$, where $\mathbf { i }$ and $\mathbf { j }$ are unit vectors and $t$ is the time in seconds.

Find the acceleration of the particle when $t = 0$ and also when $t = 3$.
Calculate the force acting on the particle when $t = 3$. The particle has velocity $( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ when $t = 1$.
Find an expression for the velocity of the particle at time $t$.

OCR MEI M1 2006 January Q6

7 marks Moderate -0.3

6 A car is driven with constant acceleration, $a \mathrm {~m} \mathrm {~s} ^ { - 2 }$, along a straight road. Its speed when it passes a road sign is $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The car travels 14 m in the 2 seconds after passing the sign; 5 seconds after passing the sign it has a speed of $19 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Write down two equations connecting $a$ and $u$. Hence find the values of $a$ and $u$.
What distance does the car travel in the 5 seconds after passing the road sign? Section B (36 marks)

OCR MEI M1 2006 January Q7

16 marks Moderate -0.3

7 Clive and Ken are trying to move a box of mass 50 kg on a rough, horizontal floor. As shown in Fig. 7, Clive always pushes horizontally and Ken always pulls at an angle of $30 ^ { \circ }$ to the horizontal. Each of them applies forces to the box in the same vertical plane as described below. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{19d42df9-e752-4d33-85e1-4ec59b32135a-4_360_745_995_660} \captionsetup{labelformat=empty} \caption{Fig. 7}

\end{figure} Initially, the box is in equilibrium with Clive pushing with a force of 60 N and Ken not pulling at all.

What is the resistance to motion of the box? Ken now adds a pull of 70 N to Clive's push of 60 N . The box remains in equilibrium.
What now is the resistance to motion of the box?
Calculate the normal reaction of the floor on the box. The frictional resistance to sliding of the box is 125 N .
Clive now pushes with a force of 160 N but Ken does not pull at all.
Calculate the acceleration of the box. Clive stops pushing when the box has a speed of $1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
How far does the box then slide before coming to rest? Ken and Clive now try again. Ken pulls with a force of $Q \mathrm {~N}$ and Clive pushes with a force of 160 N . The frictional resistance to sliding of the box is now 115 N and the acceleration of the box is $3 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
Calculate the value of $Q$.

OCR MEI M1 2006 January Q8

20 marks Standard +0.3

8 A girl throws a small stone with initial speed $14 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $60 ^ { \circ }$ to the horizontal from a point 1 m above the ground. She throws the stone directly towards a vertical wall of height 6 m standing on horizontal ground. The point O is on the ground directly below the point of projection, as shown in Fig. 8. Air resistance is negligible. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{19d42df9-e752-4d33-85e1-4ec59b32135a-5_658_757_482_648} \captionsetup{labelformat=empty} \caption{Fig. 8}

\end{figure}

Write down an expression in terms of $t$ for the horizontal displacement of the stone from O , $t$ seconds after projection. Find also an expression for the height of the stone above O at this time. The stone is at the top of its trajectory when it passes over the wall.
(A) Find the time it takes for the stone to reach its highest point.
(B) Calculate the distance of O from the base of the wall.
(C) Show that the stone passes over the wall with 2.5 m clearance.
Find the cartesian equation of the trajectory of the stone referred to the horizontal and vertical axes, $\mathrm { O } x$ and $\mathrm { O } y$. There is no need to simplify your answer. The girl now moves away a further distance $d \mathrm {~m}$ from the wall. She throws a stone as before and it just passes over the wall.
Calculate $d$.