Practice Paper 2 (Cambridge International AS Mathematics)

A lorry of mass 27000 kg travels at a constant speed of 12 $\text{m s}^{-1}$ up a straight road inclined at 8° to the horizontal, along a line of greatest slope. The engine of the lorry is working at a constant rate of 462 kW. The driving force of the lorry is $F$ N and there is a constant resistance to motion of $R$ N.

The diagram shows some of the forces acting on the lorry.

Complete the diagram to show the weight of the lorry, its speed and its acceleration.

Show that $F = 38500$.

Find $R$.

Two loads $A$ and $B$, of masses $m$ kg and $3 m$ kg respectively, are attached to the ends of a light inextensible string passing over a fixed smooth pulley. $A$ is at the bottom of a rough plane inclined at $α$° to the horizontal, where $\text{sin} α = 0 . 6$, and $B$ hangs freely at a height of 1.3 m above the level of the bottom of the plane, as shown in the diagram.

The system is released from rest, and in the subsequent motion $A$ moves up the plane along the line of greatest slope.

Given that the net loss in gravitational potential energy of the system when $B$ hits the floor is 12.48 J, find the value of $m$.

The speed of the loads at the point when $B$ hits the ground is exactly 2 $\text{m s}^{-1}$ faster than their speed at the instant when $B$ is 1 m above the ground.

Find

(i) the speed with which $B$ hits the ground,

(ii) the magnitude of the frictional force acting on load $A$.

A car of weight 12000 N is moving along a horizontal straight road. The car's engine provides a forward driving force of 8500 N. The total resistance to the car's motion is modelled as a constant force of 1000 N. All other forces acting on the car may be ignored.

(i) Write down the magnitude and direction of the normal reaction force exerted on the car by the road.

(ii) The car is modelled as a particle. Draw a diagram showing the forces acting on the car.

Find the magnitude and direction of the resultant force acting on the car.

A sled of mass 5.6 kg is pulled along a straight horizontal path by a horizontal rope attached to its front. A constant resistance to motion of magnitude 3.2 N acts on the sled. The sled starts from rest and moves with constant acceleration, reaching a speed of 1.5 $\text{m s}^{-1}$ after 6 seconds.

Find the acceleration of the sled.

Find the tension in the rope.

A particle of mass 0.8 kg is released from rest on a rough plane inclined at an angle $θ$ to the horizontal, where $\text{tan} θ = \frac{2}{7}$. The particle slides down a line of greatest slope, and 4 s after it is released its speed is 1.35 $\text{m s}^{-1}$.

Find the coefficient of friction between the particle and the plane.

A stone is released from rest at the top of a cliff and falls vertically. At the instant when the speed of the stone is 18.8 $\text{m s}^{-1}$, it has not yet reached the sea below.

Find the distance the stone has fallen at this instant. Give your answer correct to 3 significant figures.

The velocity-time graph below shows the motion of a particle moving in a straight line. Time is measured in seconds and velocity is measured in metres per second.

Write down

(i) the initial speed of the particle

(ii) the speed of the particle after 13 seconds

(iii) the time at which the particle is instantaneously at rest.

Find

(i) the acceleration of the particle during the first 4 seconds

(ii) the displacement of the particle during the first 6 seconds

(iii) the displacement of the particle during the last 7 seconds of its motion.

A go-kart manufacturer is testing a new model on a straight horizontal road. The go-kart starts from rest, and its velocity $v \text{m s}^{-1}$ at time $t$ seconds is modelled by

$v = \frac{1}{10} t ( 36 - t )$

for $t \geq 0$.

Find the maximum velocity of the go-kart and the time at which it occurs. Justify that this is a maximum.

The test ends when the go-kart next comes to rest. Find the time at which the test ends.

A box is pulled along a rough horizontal floor by a string. The diagram shows the four forces acting on the box, labelled A, B, C and D.

Label each of the forces A, B, C and D with its name, choosing from: tension, friction, weight, normal contact force.

Two marbles $M$ and $N$, of masses 3 kg and $m$ kg respectively, are rolled towards each other along a straight line on a smooth horizontal table. The marbles are modelled as particles. Immediately before they collide, the speeds of $M$ and $N$ are 3 $\text{m s}^{-1}$ and 5 $\text{m s}^{-1}$ respectively. The marbles collide directly and $M$ is brought to rest. Immediately after the collision, $N$ moves with speed 4 $\text{m s}^{-1}$.

Find the value of $m$.

A bus of mass 2400 kg moves along a straight horizontal road against a constant resistance to motion of 2000 N. The maximum speed of the bus on this road is 30 $\text{m s}^{-1}$.

Find the maximum power of the engine of the bus.

The engine of the bus is working at its maximum power.

Find the acceleration of the bus at the instant when its speed is 10 $\text{m s}^{-1}$.

A horse rider moves their horse from rest to a gentle trot of 5 $\text{m s}^{-1}$ on flat horizontal ground.

Treating the horse and rider as a single particle of total mass $m$ kg, find an expression for their increase in kinetic energy.

Given that the increase in kinetic energy is 7500 joules, use your answer to part (a) to find the total mass of the horse and rider.

A stack of n identical books, each of weight 12 N, rests on a horizontal table. The stack is in equilibrium.

(i) Find, in terms of n, the magnitude of the normal reaction exerted by the table on the bottom book of the stack.

(ii) The normal reaction exerted by the table on the bottom book is 96 N. Find the value of n.

(iii) State the magnitude and direction of the force exerted on the table by the bottom book, and name the law which justifies your answer.

Two railway carriages, each of mass 3000 kg, are at rest on a straight horizontal track. They are connected by a light rigid rod which is parallel to the track. The resistance to motion of each carriage is a constant force of 5800 N.

A constant horizontal force of 12500 N is applied to the rear carriage, pushing both carriages forwards along the track.

By considering the two carriages as a single system, find the acceleration of the carriages.

By considering the front carriage alone, find the thrust in the rod.

A particle of mass 5 kg is projected up a line of greatest slope of a rough plane inclined at 30° to the horizontal. The coefficient of friction between the particle and the plane is 0.3. The particle comes to instantaneous rest and then slides back down the plane.

Find the magnitude of the frictional force acting on the particle while it moves up the plane, and state its direction.

Find the deceleration of the particle while it moves up the plane.

State the magnitude and direction of the frictional force acting on the particle while it slides back down the plane.

Find the acceleration of the particle while it slides down the plane.