Questions - OCR M1 - A-Level Maths

OCR M1 2016 June Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{c6bac5bf-960e-4c3d-b9fa-c52de66ba719-2_144_1317_1655_372} Four particles $A , B , C$ and $D$ are on the same straight line on a smooth horizontal table. $A$ has speed $6 \mathrm {~ms} ^ { - 1 }$ and is moving towards $B$. The speed of $B$ is $2 \mathrm {~ms} ^ { - 1 }$ and $B$ is moving towards $A$. The particle $C$ is moving with speed $5 \mathrm {~ms} ^ { - 1 }$ away from $B$ and towards $D$, which is stationary (see diagram). The first collision is between $A$ and $B$ which have masses 0.8 kg and 0.2 kg respectively.

After the particles collide $A$ has speed $4 \mathrm {~ms} ^ { - 1 }$ in its original direction of motion. Calculate the speed of $B$ after the collision. The second collision is between $C$ and $D$ which have masses 0.3 kg and 0.1 kg respectively.
The particles coalesce when they collide. Find the speed of the combined particle after this collision. The third collision is between $B$ and the combined particle, after which no further collisions occur.
Calculate the greatest possible speed of the combined particle after the third collision.

OCR M1 2016 June Q5

5 Three forces act on a particle. The first force has magnitude $P \mathrm {~N}$ and acts horizontally due east. The second force has magnitude 5 N and acts horizontally due west. The third force has magnitude $2 P \mathrm {~N}$ and acts vertically upwards. The resultant of these three forces has magnitude 25 N .

Calculate $P$ and the angle between the resultant and the vertical. The particle has mass 3 kg and rests on a rough horizontal table. The coefficient of friction between the particle and the table is 0.15 .
Find the acceleration of the particle, and state the direction in which it moves.
\includegraphics[max width=\textwidth, alt={}, center]{c6bac5bf-960e-4c3d-b9fa-c52de66ba719-3_485_876_797_598} Two particles $P$ and $Q$ are attached to opposite ends of a light inextensible string which passes over a small smooth pulley at the top of a rough plane inclined at $30 ^ { \circ }$ to the horizontal. $P$ has mass 0.2 kg and is held at rest on the plane. $Q$ has mass 0.2 kg and hangs freely. The string is taut (see diagram). The coefficient of friction between $P$ and the plane is 0.4 . The particle $P$ is released.
State the tension in the string before $P$ is released, and find the tension in the string after $P$ is released.
$Q$ strikes the floor and remains at rest. $P$ continues to move up the plane for a further distance of 0.8 m before it comes to rest. $P$ does not reach the pulley.
Find the speed of the particles immediately before $Q$ strikes the floor.
Calculate the magnitude of the contact force exerted on $P$ by the plane while $P$ is in motion.

OCR M1 2016 June Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{c6bac5bf-960e-4c3d-b9fa-c52de66ba719-4_652_1068_255_500} The diagram shows the ( $t , v$ ) graphs for two particles $A$ and $B$ which move on the same straight line. The units of $v$ and $t$ are $\mathrm { ms } ^ { - 1 }$ and s respectively. Both particles are at the point $S$ on the line when $t = 0$. The particle $A$ is initially at rest, and moves with acceleration $0.18 t \mathrm {~ms} ^ { - 2 }$ until the two particles collide when $t = 16$. The initial velocity of $B$ is $U \mathrm {~ms} ^ { - 1 }$ and $B$ has variable acceleration for the first five seconds of its motion. For the next ten seconds of its motion $B$ has a constant velocity of $9 \mathrm {~ms} ^ { - 1 }$; finally $B$ moves with constant deceleration for one second before it collides with $A$.

Calculate the value of $t$ at which the two particles have the same velocity. For $0 \leqslant t \leqslant 5$ the distance of $B$ from $S$ is $\left( U t + 0.08 t ^ { 3 } \right) \mathrm { m }$.
Calculate $U$ and verify that when $t = 5 , B$ is 25 m from $S$.
Calculate the velocity of $B$ when $t = 16$. \section*{END OF QUESTION PAPER}

OCR M1 2014 June Q1

1 A particle $P$ is projected vertically downwards with initial speed $3.5 \mathrm {~ms} ^ { - 1 }$ from a point $A$ which is 5 m above horizontal ground.

Find the speed of $P$ immediately before it strikes the ground. After striking the ground, $P$ rebounds and moves vertically upwards and 0.87 s after leaving the ground $P$ passes through $A$.
Calculate the speed of $P$ immediately after it leaves the ground. It is given that the mass of $P$ is 0.2 kg .
Calculate the change in the momentum of $P$ as a result of its collision with the ground.

OCR M1 2014 June Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-2_309_520_941_744} A particle rests on a smooth horizontal surface. Three horizontal forces of magnitudes $2.5 \mathrm {~N} , F \mathrm {~N}$ and 2.4 N act on the particle on bearings $\theta ^ { \circ } , 180 ^ { \circ }$ and $270 ^ { \circ }$ respectively (see diagram). The particle is in equilibrium.

Find $\theta$ and $F$. The 2.4 N force suddenly ceases to act on the particle, which has mass 0.2 kg .
Find the magnitude and direction of the acceleration of the particle.

OCR M1 2014 June Q3

3 A particle $P$ travels in a straight line. The velocity of $P$ at time $t$ seconds after it passes through a fixed point $A$ is given by $\left( 0.6 t ^ { 2 } + 3 \right) \mathrm { ms } ^ { - 1 }$. Find

the velocity of $P$ when it passes through $A$,
the displacement of $P$ from $A$ when $t = 1.5$,
the velocity of $P$ when it has acceleration $6 \mathrm {~ms} ^ { - 2 }$.

OCR M1 2014 June Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-3_136_824_260_623} Particles $P$ and $Q$ are moving towards each other with constant speeds $4 \mathrm {~ms} ^ { - 1 }$ and $2 \mathrm {~ms} ^ { - 1 }$ along the same straight line on a smooth horizontal surface (see diagram). $P$ has mass 0.2 kg and $Q$ has mass 0.3 kg . The two particles collide.

Show that $Q$ must change its direction of motion in the collision.
Given that $P$ and $Q$ move with equal speed after the collision, calculate both possible values for their speed after they collide.

OCR M1 2014 June Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-3_652_1675_959_187} A particle $P$ can move in a straight line on a horizontal surface. At time $t$ seconds the displacement of $P$ from a fixed point $A$ on the line is $x \mathrm {~m}$. The diagram shows the $( t , x )$ graph for $P$. In the interval $0 \leqslant t \leqslant 10$, either the speed of $P$ is $4 \mathrm {~ms} ^ { - 1 }$, or $P$ is at rest.

Show by calculation that $T = 1.75$.
State the velocity of $P$ when
(a) $t = 2$,
(b) $t = 8$,
(c) $t = 9$.
Calculate the distance travelled by $P$ in the interval $0 \leqslant t \leqslant 10$. For $t > 10$, the displacement of $P$ from $A$ is given by $x = 20 t - t ^ { 2 } - 96$.
Calculate the value of $t$, where $t > 10$, for which the speed of $P$ is $4 \mathrm {~ms} ^ { - 1 }$.

OCR M1 2014 June Q6

6 A particle $P$ of weight 8 N rests on a horizontal surface. A horizontal force of magnitude 3 N acts on $P$, and $P$ is in limiting equilibrium.

Calculate the coefficient of friction between $P$ and the surface.
Find the magnitude and direction of the contact force exerted by the surface on $P$.
\includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-4_190_579_580_598} The initial 3 N force continues to act on $P$ in its original direction. An additional force of magnitude $T \mathrm {~N}$, acting in the same vertical plane as the 3 N force, is now applied to $P$ at an angle of $\theta ^ { \circ }$ above the horizontal (see diagram). $P$ is again in limiting equilibrium.
(a) Given that $\theta = 0$, find $T$.
(b) Given instead that $\theta = 30$, calculate $T$.

OCR M1 2014 June Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-5_510_1091_269_479}
$A$ and $B$ are points at the upper and lower ends, respectively, of a line of greatest slope on a plane inclined at $30 ^ { \circ }$ to the horizontal. $M$ is the mid-point of $A B$. Two particles $P$ and $Q$, joined by a taut light inextensible string, are placed on the plane at $A$ and $M$ respectively. The particles are simultaneously projected with speed $0.6 \mathrm {~ms} ^ { - 1 }$ down the line of greatest slope (see diagram). The particles move down the plane with acceleration $0.9 \mathrm {~ms} ^ { - 2 }$. At the instant 2 s after projection, $P$ is at $M$ and $Q$ is at $B$. The particle $Q$ subsequently remains at rest at $B$.

Find the distance $A B$. The plane is rough between $A$ and $M$, but smooth between $M$ and $B$.
Calculate the speed of $P$ when it reaches $B$.
$P$ has mass 0.4 kg and $Q$ has mass 0.3 kg .
By considering the motion of $Q$, calculate the tension in the string while both particles are moving down the plane.
Calculate the coefficient of friction between $P$ and the plane between $A$ and $M$. \section*{END OF QUESTION PAPER}

OCR M1 2010 January Q5

Find the value of $t$ when $A$ and $B$ have the same speed.
Calculate the value of $t$ when $B$ overtakes $A$.
On a single diagram, sketch the $( t , x )$ graphs for the two cyclists for the time from $t = 0$ until after $B$ has overtaken $A$.

OCR M1 2015 June Q3

Calculate the distance $A$ cycles, and hence find the period of time for which $B$ walks before finding the bicycle.
Find $T$.
Calculate the distance $A$ and $B$ each travel.

OCR M1 Q1

1
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-02_508_501_274_822} A light inextensible string has its ends attached to two fixed points $A$ and $B$. The point $A$ is vertically above $B$. A smooth ring $R$ of mass $m \mathrm {~kg}$ is threaded on the string and is pulled by a force of magnitude 1.6 N acting upwards at $45 ^ { \circ }$ to the horizontal. The section $A R$ of the string makes an angle of $30 ^ { \circ }$ with the downward vertical and the section $B R$ is horizontal (see diagram). The ring is in equilibrium with the string taut.

Give a reason why the tension in the part $A R$ of the string is the same as that in the part $B R$.
Show that the tension in the string is 0.754 N , correct to 3 significant figures.
Find the value of $m$.

OCR M1 Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-02_643_289_1475_927} Particles $A$ and $B$, of masses 0.2 kg and 0.3 kg respectively, are attached to the ends of a light inextensible string. Particle $A$ is held at rest at a fixed point and $B$ hangs vertically below $A$. Particle $A$ is now released. As the particles fall the air resistance acting on $A$ is 0.4 N and the air resistance acting on $B$ is 0.25 N (see diagram). The downward acceleration of each of the particles is $a \mathrm {~ms} ^ { - 2 }$ and the tension in the string is $T \mathrm {~N}$.

Write down two equations in $a$ and $T$ obtained by applying Newton's second law to $A$ and to $B$.
Find the values of $a$ and $T$. \section*{June 2005}

OCR M1 Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-03_697_579_1238_781} Two small rings $A$ and $B$ are attached to opposite ends of a light inextensible string. The rings are threaded on a rough wire which is fixed vertically. $A$ is above $B$. A horizontal force is applied to a point $P$ of the string. Both parts $A P$ and $B P$ of the string are taut. The system is in equilibrium with angle $B A P = \alpha$ and angle $A B P = \beta$ (see diagram). The weight of $A$ is 2 N and the tensions in the parts $A P$ and $B P$ of the string are 7 N and $T \mathrm {~N}$ respectively. It is given that $\cos \alpha = 0.28$ and $\sin \alpha = 0.96$, and that $A$ is in limiting equilibrium.

Find the coefficient of friction between the wire and the ring $A$.
By considering the forces acting at $P$, show that $T \cos \beta = 1.96$.
Given that there is no frictional force acting on $B$, find the mass of $B$. \section*{June 2005}

OCR M1 Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-04_632_1121_936_511} A car $P$ starts from rest and travels along a straight road for 600 s . The ( $t , v$ ) graph for the journey is shown in the diagram. This graph consists of three straight line segments. Find

the distance travelled by $P$,
the deceleration of $P$ during the interval $500 < t < 600$. Another car $Q$ starts from rest at the same instant as $P$ and travels in the same direction along the same road for 600 s . At time $t \mathrm {~s}$ after starting the velocity of $Q$ is $\left( 600 t ^ { 2 } - t ^ { 3 } \right) \times 10 ^ { - 6 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find an expression in terms of $t$ for the acceleration of $Q$.
Find how much less $Q$ 's deceleration is than $P$ 's when $t = 550$.
Show that $Q$ has its maximum velocity when $t = 400$.
Find how much further $Q$ has travelled than $P$ when $t = 400$. \section*{Jan 2006} 1
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-05_547_455_269_845} Particles $P$ and $Q$, of masses 0.3 kg and 0.4 kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The system is in motion with the string taut and with each of the particles moving vertically. The downward acceleration of $P$ is $a \mathrm {~ms} ^ { - 2 }$ (see diagram).
Show that $a = - 1.4$. Initially $P$ and $Q$ are at the same horizontal level. $P$ 's initial velocity is vertically downwards and has magnitude $2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Assuming that $P$ does not reach the floor and that $Q$ does not reach the pulley, find the time taken for $P$ to return to its initial position. 2
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-05_616_809_1562_667} An object of mass 0.08 kg is attached to one end of a light inextensible string. The other end of the string is attached to the underside of the roof inside a furniture van. The van is moving horizontally with constant acceleration $1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. The string makes a constant angle $\alpha$ with the downward vertical and the tension in the string is $T \mathrm {~N}$ (see diagram).
By applying Newton's second law horizontally to the object, find the value of $T \sin \alpha$.
Find the value of $T$. 3 A motorcyclist starts from rest at a point $O$ and travels in a straight line. His velocity after $t$ seconds is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, for $0 \leqslant t \leqslant T$, where $v = 7.2 t - 0.45 t ^ { 2 }$. The motorcyclist's acceleration is zero when $t = T$.
Find the value of $T$.
Show that $v = 28.8$ when $t = T$. For $t \geqslant T$ the motorcyclist travels in the same direction as before, but with constant speed $28.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find the displacement of the motorcyclist from $O$ when $t = 31$. 4
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-06_225_623_813_762} A block of mass 2 kg is at rest on a rough horizontal plane, acted on by a force of magnitude 12 N at an angle of $15 ^ { \circ }$ upwards from the horizontal (see diagram).
Find the frictional component of the contact force exerted on the block by the plane.
Show that the normal component of the contact force exerted on the block by the plane has magnitude 16.5 N , correct to 3 significant figures. It is given that the block is on the point of sliding.
Find the coefficient of friction between the block and the plane. The force of magnitude 12 N is now replaced by a horizontal force of magnitude 20 N . The block starts to move.
Find the acceleration of the block. 5 A man drives a car on a horizontal straight road. At $t = 0$, where the time $t$ is in seconds, the car runs out of petrol. At this instant the car is moving at $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The car decelerates uniformly, coming to rest when $t = 8$. The man then walks back along the road at $0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ until he reaches a petrol station a distance of 420 m from his car. After his arrival at the petrol station it takes him 250 s to obtain a can of petrol. He is then given a lift back to his car on a motorcycle. The motorcycle starts from rest and accelerates uniformly until its speed is $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$; it then decelerates uniformly, coming to rest at the stationary car at time $t = T$.
Sketch the shape of the $( t , v )$ graph for the man for $0 \leqslant t \leqslant T$. [Your sketch need not be drawn to scale; numerical values need not be shown.]
Find the deceleration of the car for $0 < t < 8$.
Find the value of $T$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-07_540_542_267_799} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A smooth ring $R$ of weight $W \mathrm {~N}$ is threaded on a light inextensible string. The ends of the string are attached to fixed points $A$ and $B$, where $A$ is vertically above $B$. A horizontal force of magnitude $P \mathrm {~N}$ acts on $R$. The system is in equilibrium with the string taut; $A R$ makes an angle $\alpha$ with the downward vertical and $B R$ makes an angle $\beta$ with the upward vertical (see Fig.1).
By considering the vertical components of the forces acting on $R$, show that $\alpha < \beta$.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-07_537_559_1302_833} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} It is given that when $P = 14 , A R = 0.4 \mathrm {~m} , B R = 0.3 \mathrm {~m}$ and the distance of $R$ from the vertical line $A B$ is 0.24 m (see Fig. 2). Find
(a) the tension in the string,
(b) the value of $W$.
For the case when $P = 0$,
(a) describe the position of $R$,
(b) state the tension in the string.
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-08_624_1077_267_536}
$P Q$ is a line of greatest slope, of length 4 m , on a smooth plane inclined at $30 ^ { \circ }$ to the horizontal. Particles $A$ and $B$, of masses 0.15 kg and 0.5 kg respectively, move along $P Q$ with $A$ below $B$. The particles are both moving upwards, $A$ with speed $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $B$ with speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, when they collide at the mid-point of $P Q$ (see diagram). Particle $A$ is instantaneously at rest immediately after the collision.
Show that $B$ does not reach $Q$ in the subsequent motion.
Find the time interval between the instant of $A$ 's arrival at $P$ and the instant of $B$ 's arrival at $P$. \section*{June 2006} 1 Each of two wagons has an unloaded mass of 1200 kg . One of the wagons carries a load of mass $m \mathrm {~kg}$ and the other wagon is unloaded. The wagons are moving towards each other on the same rails, each with speed $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, when they collide. Immediately after the collision the loaded wagon is at rest and the speed of the unloaded wagon is $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find the value of $m$. 2
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-09_622_716_541_715} Forces of magnitudes 6.5 N and 2.5 N act at a point in the directions shown. The resultant of the two forces has magnitude $R \mathrm {~N}$ and acts at right angles to the force of magnitude 2.5 N (see diagram).
Show that $\theta = 22.6 ^ { \circ }$, correct to 3 significant figures.
Find the value of $R$. 3 A man travels 360 m along a straight road. He walks for the first 120 m at $1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, runs the next 180 m at $4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, and then walks the final 60 m at $1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The man's displacement from his starting point after $t$ seconds is $x$ metres.
Sketch the $( t , x )$ graph for the journey, showing the values of $t$ for which $x = 120,300$ and 360 . A woman jogs the same 360 m route at constant speed, starting at the same instant as the man and finishing at the same instant as the man.
Draw a dotted line on your $( t , x )$ graph to represent the woman's journey.
Calculate the value of $t$ at which the man overtakes the woman. June 2006
4 A cyclist travels along a straight road. Her velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, at time $t$ seconds after starting from a point $O$, is given by $$\begin{aligned} & v = 2 \quad \text { for } 0 \leqslant t \leqslant 10 ,
& v = 0.03 t ^ { 2 } - 0.3 t + 2 \quad \text { for } t \geqslant 10 . \end{aligned}$$
Find the displacement of the cyclist from $O$ when $t = 10$.
Show that, for $t \geqslant 10$, the displacement of the cyclist from $O$ is given by the expression $0.01 t ^ { 3 } - 0.15 t ^ { 2 } + 2 t + 5$.
Find the time when the acceleration of the cyclist is $0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. Hence find the displacement of the cyclist from $O$ when her acceleration is $0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. 5 A block of mass $m \mathrm {~kg}$ is at rest on a horizontal plane. The coefficient of friction between the block and the plane is 0.2 .
When a horizontal force of magnitude 5 N acts on the block, the block is on the point of slipping. Find the value of $m$.
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-10_312_711_1244_758} When a force of magnitude $P \mathrm {~N}$ acts downwards on the block at an angle $\alpha$ to the horizontal, as shown in the diagram, the frictional force on the block has magnitude 6 N and the block is again on the point of slipping. Find
(a) the value of $\alpha$ in degrees,
(b) the value of $P$. June 2006
6
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-11_317_999_264_575} A train of total mass 80000 kg consists of an engine $E$ and two trucks $A$ and $B$. The engine $E$ and truck $A$ are connected by a rigid coupling $X$, and trucks $A$ and $B$ are connected by another rigid coupling $Y$. The couplings are light and horizontal. The train is moving along a straight horizontal track. The resistances to motion acting on $E , A$ and $B$ are $10500 \mathrm {~N} , 3000 \mathrm {~N}$ and 1500 N respectively (see diagram).
By modelling the whole train as a single particle, show that it is decelerating when the driving force of the engine is less than 15000 N .
Show that, when the magnitude of the driving force is 35000 N , the acceleration of the train is $0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
Hence find the mass of $E$, given that the tension in the coupling $X$ is 8500 N when the magnitude of the driving force is 35000 N . The driving force is replaced by a braking force of magnitude 15000 N acting on the engine. The force exerted by the coupling $Y$ is zero.
Find the mass of $B$.
Show that the coupling $X$ exerts a forward force of magnitude 1500 N on the engine. 7 A particle of mass 0.1 kg is at rest at a point $A$ on a rough plane inclined at $15 ^ { \circ }$ to the horizontal. The particle is given an initial velocity of $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and starts to move up a line of greatest slope of the plane. The particle comes to instantaneous rest after 1.5 s .
Find the coefficient of friction between the particle and the plane.
Show that, after coming to instantaneous rest, the particle moves down the plane.
Find the speed with which the particle passes through $A$ during its downward motion. 1 A trailer of mass 600 kg is attached to a car of mass 1100 kg by a light rigid horizontal tow-bar. The car and trailer are travelling along a horizontal straight road with acceleration $0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
Given that the force exerted on the trailer by the tow-bar is 700 N , find the resistance to motion of the trailer.
Given also that the driving force of the car is 2100 N , find the resistance to motion of the car. 2
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-12_595_791_671_676} Three horizontal forces of magnitudes $15 \mathrm {~N} , 11 \mathrm {~N}$ and 13 N act on a particle $P$ in the directions shown in the diagram. The angles $\alpha$ and $\beta$ are such that $\sin \alpha = 0.28 , \cos \alpha = 0.96 , \sin \beta = 0.8$ and $\cos \beta = 0.6$.
Show that the component, in the $y$-direction, of the resultant of the three forces is zero.
Find the magnitude of the resultant of the three forces.
State the direction of the resultant of the three forces. 3
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-12_351_716_1804_715} A block $B$ of mass 0.4 kg and a particle $P$ of mass 0.3 kg are connected by a light inextensible string. The string passes over a smooth pulley at the edge of a rough horizontal table. $B$ is in contact with the table and the part of the string between $B$ and the pulley is horizontal. $P$ hangs freely below the pulley (see diagram).
The system is in limiting equilibrium with the string taut and $P$ on the point of moving downwards. Find the coefficient of friction between $B$ and the table.
A horizontal force of magnitude $X \mathrm {~N}$, acting directly away from the pulley, is now applied to $B$. The system is again in limiting equilibrium with the string taut, and with $P$ now on the point of moving upwards. Find the value of $X$. 4
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-13_225_1155_264_497} Three uniform spheres $L , M$ and $N$ have masses $0.8 \mathrm {~kg} , 0.6 \mathrm {~kg}$ and 0.7 kg respectively. The spheres are moving in a straight line on a smooth horizontal table, with $M$ between $L$ and $N$. The sphere $L$ is moving towards $M$ with speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the spheres $M$ and $N$ are moving towards $L$ with speeds $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively (see diagram).
$L$ collides with $M$. As a result of this collision the direction of motion of $M$ is reversed, and its speed remains $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find the speed of $L$ after the collision.
$M$ then collides with $N$.
(a) Find the total momentum of $M$ and $N$ in the direction of $M$ 's motion before this collision takes place, and deduce that the direction of motion of $N$ is reversed as a result of this collision.
(b) Given that $M$ is at rest immediately after this collision, find the speed of $N$ immediately after this collision. 5 A particle starts from rest at a point $A$ at time $t = 0$, where $t$ is in seconds. The particle moves in a straight line. For $0 \leqslant t \leqslant 4$ the acceleration is $1.8 t \mathrm {~m} \mathrm {~s} ^ { - 2 }$, and for $4 \leqslant t \leqslant 7$ the particle has constant acceleration $7.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
Find an expression for the velocity of the particle in terms of $t$, valid for $0 \leqslant t \leqslant 4$.
Show that the displacement of the particle from $A$ is 19.2 m when $t = 4$.
Find the displacement of the particle from $A$ when $t = 7$. \section*{Jan 2007} 6
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-14_558_1373_267_386} The diagram shows the ( $t , v$ ) graph for the motion of a hoist used to deliver materials to different levels at a building site. The hoist moves vertically. The graph consists of straight line segments. In the first stage the hoist travels upwards from ground level for 25 s , coming to rest 8 m above ground level.
Find the greatest speed reached by the hoist during this stage. The second stage consists of a 40 s wait at the level reached during the first stage. In the third stage the hoist continues upwards until it comes to rest 40 m above ground level, arriving 135 s after leaving ground level. The hoist accelerates at $0.02 \mathrm {~ms} ^ { - 2 }$ for the first 40 s of the third stage, reaching a speed of $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find
the value of $V$,
the length of time during the third stage for which the hoist is moving at constant speed,
the deceleration of the hoist in the final part of the third stage. 7 A particle $P$ of mass 0.5 kg moves upwards along a line of greatest slope of a rough plane inclined at an angle of $40 ^ { \circ }$ to the horizontal. $P$ reaches its highest point and then moves back down the plane. The coefficient of friction between $P$ and the plane is 0.6 .
Show that the magnitude of the frictional force acting on $P$ is 2.25 N , correct to 3 significant figures.
Find the acceleration of $P$ when it is moving
(a) up the plane,
(b) down the plane.
When $P$ is moving up the plane, it passes through a point $A$ with speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
(a) Find the length of time before $P$ reaches its highest point.
(b) Find the total length of time for $P$ to travel from the point $A$ to its highest point and back to $A$.

Questions — OCR M1 (141 questions)

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