Questions M1 - A-Level Maths

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 $A$.

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