Practice Paper 1 (Cambridge International A-Level Mathematics)

A car of mass $3 m$ kg tows a trailer of mass $m$ kg along a straight horizontal road, using a light rigid tow-bar parallel to the road. A constant resistance to motion of $R$ N acts on the trailer, and a resistance to motion of \left(R + k v\right) N acts on the car, where $v\quad \text{m s}^{-1}$ is the speed of the car and $k$ is a constant.

The greatest speed of the car and trailer on this road is 30 $\text{m s}^{-1}$, reached with the engine working at its maximum power of 48.84 kW. When the engine is working at a constant rate of 21 kW, the car and trailer move at a constant speed of 14 $\text{m s}^{-1}$.

Find

(i) the value of $k$,

(ii) the total resistance to the motion of the car and trailer, in terms of $v$.

The car now tows the trailer up a straight road inclined at an angle $α$ to the horizontal, along a line of greatest slope, where $\text{sin} α = \frac{1}{19}$. The resistances to motion on the car and on the trailer are unchanged. The engine is working at a constant rate of 45.3 kW, and at the instant when $v = 12$ the acceleration of the car and trailer is 1.1 $\text{m s}^{-2}$.

Find the mass of the car.

A car is moving up a straight hill, along the line of greatest slope, inclined at $α$° to the horizontal where $\text{sin} α = \frac{3}{5}$. The non-gravitational resistance acting on the car parallel to its motion has magnitude $0 . 115 R$ N, where $R$ is the magnitude of the normal contact force exerted on the car by the ground.

The car travels at constant speed for 1 km, and the work done by the engine during this time is 5250 kJ.

Find the combined mass of the car and its driver.

The car is travelling at 12 $\text{m s}^{-1}$ when the driver sees a small dog run into the road ahead and applies the brakes. The total work done against the resistance and the braking force is 21400 J. The braking force acts parallel to the motion of the car and cannot be assumed to be constant.

Find the distance the car travels before coming to rest.

A large steel wrecking ball of weight 24500 N is being lifted by a crane. The tension in the lifting cable is T N. Other than the weight of the ball and the tension in the cable, all other forces acting on the ball may be ignored.

The wrecking ball moves vertically upwards with constant velocity.

Find the value of T, explaining your reasoning.

The tension in the cable is now increased by 3000 N.

Describe the resulting motion of the wrecking ball, explaining the physical reasoning behind your answer.

A locomotive of mass 10000 kg pulls a number of identical carriages along a straight horizontal track. The locomotive and the carriages are connected in a line by light rigid rods, which are parallel to the track. The locomotive's engine produces a driving force of 20000 N, and the resistance to motion of the locomotive is a constant 3000 N. Each carriage has mass $M$ kg, and the resistance to motion of each carriage is a constant 1000 N.

When three carriages are attached, the tension in the rod connecting the last carriage to the carriage in front of it is 437.8 N greater than the tension in the corresponding rod when four carriages are attached.

Find the value of $M$.

A small object is placed on a rough plane whose angle of inclination to the horizontal can be adjusted. The coefficient of friction between the object and the plane is $μ$.

When the plane is inclined at an angle $θ_{1}$ to the horizontal, the object is released from rest and remains at rest. When the plane is inclined at a larger angle $θ_{2}$ to the horizontal, where $θ_{2} < 90 ^{\circ}$, the object is released from rest and slides down the plane.

Show that \text{tan} \; \theta_{1} \leqslant \mu < \text{tan} \; \theta_{2}.

A ball is projected vertically upwards from horizontal ground. The ball hits the ground 1.6 seconds after reaching its greatest height.

Find the greatest height reached by the ball and the speed of projection.

A ferry leaves a port O and travels in a straight line to its destination C. Its journey is shown on the displacement-time graph below, where displacement is measured in metres and time in seconds.

Find the velocity of the ferry during the fastest part of its journey.

The ferry company would like to reduce the total journey time from O to C by 5 minutes. The velocity of the ferry can be increased only between A and B.

Find the increase in velocity required.

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 book of mass 0.6 kg rests in equilibrium on a horizontal table.

(i) Name the two forces acting on the book, and state the direction of each.

(ii) The table exerts a normal contact force on the book. State the force which forms a Newton's third law pair with this force, saying what exerts it, what it acts on, and in which direction it acts.

Two particles $A$ and $B$, of masses $7 m$ kg and $4 m$ kg respectively, are moving towards each other along the same straight line on a smooth horizontal plane. Immediately before they collide, $A$ has speed $3 u \text{m s}^{-1}$ and $B$ has speed $5 u \text{m s}^{-1}$. The particles collide directly. Immediately after the collision, the direction of motion of $A$ is reversed, $A$ has speed $v_{A} \text{m s}^{-1}$ and $B$ has speed $v_{B} \text{m s}^{-1}$.

Explain why the direction of motion of $B$ must also be reversed by the collision, and show that $v_{B} > \frac{u}{4}$.

Given that $v_{B} = 2 v_{A}$, find $v_{A}$ and $v_{B}$ in terms of $u$.