Practice Paper 2 (Cambridge International A-Level Mathematics)

A runner of mass 65 kg works at a constant rate of $P$ W. Any resistance to motion is ignored. At an instant when the runner is moving at 4 $\text{m s}^{-1}$ on horizontal ground, the acceleration of the runner is twice the acceleration at an instant when the runner is moving at 4 $\text{m s}^{-1}$ up a road inclined at 1.5° to the horizontal, along a line of greatest slope.

Find the value of $P$.

The runner, still working at $P$ W, runs up a road inclined at 8° to the horizontal, along a line of greatest slope.

Find the acceleration of the runner at the instant when the speed is 2 $\text{m s}^{-1}$.

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 $A$.

The diagram shows the four forces acting on a particle which is in equilibrium.

Find the values of P and Q.

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 0.9 kg is at rest on a rough horizontal plane. A force of magnitude $P$ N acts on the particle, directed downwards at an angle of 40° below the horizontal, as shown in the diagram. The coefficient of friction between the particle and the plane is 0.3.

Given that the particle is about to move, find the value of $P$.

Two stations, A and B, are 8.2 km apart on a straight track. A train leaves A from rest and moves towards B with constant acceleration 0.2 $\text{m s}^{-2}$. At the same instant, a second train leaves B from rest and moves towards A along a parallel track with constant acceleration 0.16 $\text{m s}^{-2}$. The trains are modelled as particles.

Find the distance from A at which the trains pass each other.

A cable car crosses a river, running horizontally at a fixed height. It leaves one side of the river from rest with constant acceleration $3 \text{m s}^{-2}$ for 5 seconds. It then travels at constant speed until it has covered a further 45 m. The cable car then decelerates uniformly, coming to rest on the other side of the river 10 seconds later.

Sketch the velocity-time graph for the journey of the cable car across the river.

A zip-wire runs in a straight horizontal line between two trees in a children's park. A child on the zip-wire leaves the first tree from rest at time $t = 0$. The velocity-time graph shows the motion of the child, where $v \text{m s}^{-1}$ is the velocity of the child at time $t$ seconds.

For $0 \leq t \leq 4$, the velocity is given by $v = 5 t^{\frac{1}{2}}$. The child reaches the second tree at $t = 16$ and rebounds from it, then moves back towards the first tree, coming to rest at $t = 20$.

(i) Find the distance between the two trees.

(ii) Find the distance of the child from the second tree when the child comes to rest.

Find the acceleration of the child when $t = 1$.

Two blocks A and B are attached by a light inextensible string which runs over a smooth pulley fixed at the edge of a table, as shown in the diagram. Block A accelerates along the smooth horizontal surface of the table in the direction shown, while block B moves towards the ground.

State what each of the following assumptions allows you to assume in this model.

(i) A and B are both particles.

(ii) The surface of the table is smooth.

State what each of the following assumptions allows you to assume about the tension in the string, or about the motion of the two blocks.

(i) The string is light.

(ii) The string is inextensible.

(iii) The pulley is smooth.

Two particles $P$ and $Q$, of masses 10 kg and 16 kg respectively, are moving in the same direction along the same straight line on a smooth horizontal surface when they collide directly. Immediately before they collide, $P$ is moving with speed 13 $\text{m s}^{-1}$ and $Q$ is moving with speed 1 $\text{m s}^{-1}$. Immediately after they collide, $Q$ is moving with speed 10 $\text{m s}^{-1}$.

Explain how you know that the direction of motion of $Q$ does not change as a result of the collision.

(i) Find the speed of $P$ immediately after the collision.

(ii) Explain how you know that the direction of motion of $P$ has been reversed.

A motorcycle and its rider have a combined mass of 250 kg and travel along a straight horizontal road. The resistance to motion is $k v$ N, where $v\quad \text{m s}^{-1}$ is the speed of the motorcycle and $k$ is a constant.

Show that, when the motorcycle is travelling at a constant speed, the power $P$ W developed by the engine is given by $P = k v^{2}$. Justify your answer.

The greatest speed that the motorcycle can reach is 50 $\text{m s}^{-1}$, and the greatest power that its engine can develop is 30 kW.

Find the value of $k$.

Find the maximum speed of the motorcycle when its engine is working at a constant rate of 12 kW.

A stone of mass 80 g is catapulted horizontally towards a leaf on a tree, and the leaf is 0.2 cm thick. When the stone hits the leaf it is travelling horizontally with speed 30 $\text{m s}^{-1}$. The stone passes horizontally through the leaf, which exerts a constant resistive force of 1000 N on it.

Find

(i) the work done by the resistive force of the leaf on the stone,

(ii) the kinetic energy of the stone at the moment it hits the leaf,

(iii) the speed at which the stone emerges from the leaf.

After the stone emerges from the leaf it falls to the ground, and its gravitational potential energy decreases by 8 joules.

Find the vertical distance between the leaf and the ground.

The diagram shows the four forces acting on a particle which is in equilibrium.

Given that $5 F_{4} = 9 F_{2}$ and $3 F_{1} + 5 F_{3} = 450$, find the values of $F_{1}$, $F_{2}$, $F_{3}$ and $F_{4}$.

A hydraulic lift in a warehouse is used to raise a crate of mass 1200 kg, which rests on the horizontal platform of the lift. The platform has mass 300 kg. The lifting force is transmitted to the platform by a light vertical rod, as shown in the diagram.

The platform and crate start from rest and move vertically upwards with constant acceleration, reaching a speed of 3 $\text{m s}^{-1}$ after 2 seconds.

Find the thrust in the rod.

Find the magnitude and direction of the force exerted by the crate on the platform.