Practice Paper 1 (Cambridge International AS Mathematics)

A cyclist travels up a straight road inclined at 5° to the horizontal, along a line of greatest slope. The cyclist works at a constant rate of 300 W, and all resistances to motion can be neglected.

At the instant when the speed of the cyclist is 2 $\text{m s}^{-1}$, the acceleration is 0.5 $\text{m s}^{-2}$.

Find the total mass of the cyclist and the bicycle.

At the top of the hill the cyclist turns round and travels back down the same road, now working at a constant rate of 100 W. Resistances to motion can still be neglected.

Find the acceleration of the cyclist at the instant when the speed is 5 $\text{m s}^{-1}$.

A box of mass 4 kg slides down the line of greatest slope of a ramp inclined at $α$° to the horizontal, with constant acceleration 2 $\text{m s}^{-2}$. Two markers on the ramp are 7 metres apart, and the box passes the second marker exactly 2 seconds after passing the first.

Show that the box passes the first marker with speed 1.5 $\text{m s}^{-1}$, and find the speed of the box as it passes the second marker.

The work done against friction on the box as it slides between the two markers is 84 J.

Find

(i) the magnitude of the frictional force,

(ii) the value of $α$,

(iii) the work done by gravity during this time.

In a slate quarry, a load of slates is being raised to the surface for processing. The total weight of the slates and the platform on which they rest is 21000 N. To keep the platform level as it is raised, the tension in each of the two lifting cables is the same, T N. Other than the weight of the platform and slates and the tension in the cables, all other forces acting on the platform may be ignored.

The platform moves vertically upwards with a constant speed of 0.25 $\text{m s}^{-1}$.

Find the value of T.

As the platform passes one of the levels in the quarry, two workers of combined weight 1600 N jump onto the centre of the slow-moving platform, hoping to catch a ride to the surface at the end of their shift.

The tension in each cable stays the same, and the platform remains level.

Describe the resulting motion of the platform after the workers jump onto it, considering both the time immediately afterwards and the longer term. Explain the physical reasoning behind your answer.

A crane lifts a pallet of bricks using a vertical cable attached to the pallet. The pallet and bricks have a combined mass of 2800 kg and are initially at rest on the ground. The pallet moves vertically upwards with constant acceleration, and 10 seconds after it starts to move it is 18 m above the ground.

Find the acceleration of the pallet.

Find the tension in the cable while the pallet is accelerating upwards.

The pallet is then lowered vertically. During the first part of the descent, the pallet accelerates downwards with constant acceleration of the same magnitude as in part (a).

Find the tension in the cable during this part of the descent.

A particle of mass 3 kg is released from rest on a rough plane inclined at 15° to the horizontal. The coefficient of friction between the particle and the plane is 0.2.

Find the distance the particle travels in the first 3 s after it is released.

A particle passes through a fixed point O with velocity 7.3 $\text{m s}^{-1}$. It moves in a straight line with constant deceleration 0.32 $\text{m s}^{-2}$.

Find the velocity of the particle when it is first 23 m from O. Give your answer correct to 3 significant figures.

A ferry leaves port (point O) and travels directly towards its destination (point B). Its journey is shown on the displacement-time graph below, where displacement is measured in metres and time in seconds.

Use the graph to find

(i) the distance, in km, between the port and the destination,

(ii) the velocity of the ferry, in metres per second, between the points A and B.

The ferry company would like the journey to be completed within a maximum time of 25 minutes. Find, correct to 3 significant figures, the average velocity at which the ferry would need to travel in order to achieve this.

A car travels along a straight horizontal road and passes a service station $S$ at time $t = 0$. The displacement of the car from $S$ at time $t$ seconds is modelled as $s$ m, where

s = 0.4t\left(2t^{2} - 4t + 3\right)

for $t \geq 0$.

Show that, according to the model, the car never returns to $S$.

Show that the car is decelerating for the first $\frac{2}{3}$ seconds after passing $S$.

A canal barge travels in a straight line between its home mooring and two locks, Lock A and Lock B, as shown in the diagram. The barge travels at 6 $\text{km h}^{-1}$ between home and Lock A, and at 4 $\text{km h}^{-1}$ between home and Lock B.

Taking the positive direction as shown in the diagram, state the following.

(i) The velocity of the barge as it travels from home to Lock A.

(ii) The displacement of the barge from home once it reaches Lock A.

The barge then returns home and travels on to Lock B, before returning home again.

Taking the same positive direction, state the following.

(i) The velocity of the barge as it travels from Lock B back to home.

(ii) The displacement of the barge from home once it has returned home for the second time.

On the journey from Lock B back to home, the barge slows down steadily as it approaches its mooring.

Explain why the acceleration of the barge is positive during this part of the journey, even though the barge is decelerating.

Two particles $P$ and $Q$, of masses 8 kg and 5 kg respectively, lie on a smooth horizontal surface. $Q$ is at rest and $P$ is moving directly towards $Q$ with speed 3 $\text{m s}^{-1}$. The particles collide directly. Immediately after the collision, $P$ continues to move in the same direction with speed 1 $\text{m s}^{-1}$.

Use the principle of conservation of linear momentum to find the speed of $Q$ immediately after the collision.

Find the loss of kinetic energy of the system due to the collision.

A road train consists of a truck and three trailers joined by light inextensible couplings. When the road train travels with speed $v\quad \text{m s}^{-1}$, a resistance to motion of $100 v$ N acts on the truck and a resistance to motion of $20 v$ N acts on each of the three trailers. The maximum speed of the road train on a straight horizontal road is 30 $\text{m s}^{-1}$.

Find the maximum power of the engine.

The road train moves up a hill inclined at 5° to the horizontal, along a line of greatest slope, with the resistances to motion unchanged. The greatest speed of the road train up the hill is 20 $\text{m s}^{-1}$.

Find the total mass of the road train.

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.