[Maximum number: 4]

A ball of mass 2.7 g is released from rest from a height of 28 m above horizontal ground.

(a)

The graph shows the variation with time t of the speed v of the ball from the instant it is released until it impacts the ground.

[ 2 ]

(i)

Determine k.

[ 2 ]

(b)

The ball rebounds from the ground with speed $7.8 \mathrm{~ms}^{-1}$ . The ball is in contact with the ground for a time T. The average resultant force on the ball during this time is 1.1 N .
Determine T.

[ 2 ]

[Maximum number: 1]

A group of students is trying to determine the density and the viscosity of a liquid.

To determine the density, they use a balance to read the mass m of a sphere in air and immersed in the liquid.

They use a sphere of volume $V=1.827 \times 10^{-7} \mathrm{~m}^{3}$ .
The readings are $m_{\text {air }}=1.427 \mathrm{~g}$ in air and $m_{\text {lmmersed }}=1.208 \mathrm{~g}$ in the liquid.
The readings are different due to buoyancy. The buoyancy force $F_{\mathrm{b}}$ is given by

F_{\mathrm{b}}=\rho V g

where V is the volume of the sphere and $\rho$ is the density of the liquid.

(a)

Show that $F_{\mathrm{b}}$ is about 2 mN .

[ 1 ]

[Maximum number: 9]

A spherical oil droplet is released from rest at the bottom of a column of water.

The graph shows the variation with time t of the vertical velocity of the oil droplet.

(a)

The following data are available:

\begin{aligned} \text { radius of the oil droplet } & =3.5 \mathrm{~mm} \\ \text { weight of the oil droplet } & =1.6 \times 10^{-3} \mathrm{~N} \\ \text { density of the water } & =1000 \mathrm{~kg} \mathrm{~m}^{-3} \\ \text { viscosity of the water } & =1.1 \times 10^{-3} \mathrm{Pas} \end{aligned}

[ 3 ]

(i)

Calculate the initial acceleration of the oil droplet.

[ 3 ]

(b)

Describe why the acceleration of the oil droplet changes.

[ 2 ]

(c)

(i)

Explain why the velocity of the oil droplet is constant for $t>3 \mathrm{~s}$ .

[ 2 ]

(ii)

Deduce the velocity of the oil droplet for $t>3 \mathrm{~s}$ .

[ 2 ]

[Maximum number: 3]

A box of mass 1.2 kg is lying at rest on a surface. The coefficient of static friction between the box and the surface is 0.36 and the coefficient of dynamic friction between the box and the surface is 0.28 .

(a)

Outline why the coefficients of friction have no units.

[ 1 ]

(b)

Show that the minimum force needed to accelerate the box is about 4 N .

[ 2 ]

[Maximum number: 5]

A student strikes a tennis ball that is initially at rest so that it leaves the racquet at a speed of $64 \mathrm{~m} \mathrm{~s}^{-1}$ . The ball has a mass of 0.058 kg and the contact between the ball and the racquet lasts for 25 ms .

(a)

Calculate the

[ 2 ]

(i)

average force exerted by the racquet on the ball.

[ 2 ]

(b)

The student models the bounce of the tennis ball to predict the angle $\theta$ at which the ball leaves a surface of clay and a surface of grass.

The model assumes
- during contact with the surface the ball slides.
- the sliding time is the same for both surfaces.
- the sliding frictional force is greater for clay than grass.
- the normal reaction force is the same for both surfaces.
Predict for the student's model, without calculation, whether $\theta$ is greater for a clay surface or for a grass surface.

[ 3 ]

[Maximum number: 9]

A student uses a load to pull a box up a ramp inclined at $30^{\circ}$ . A string of constant length and negligible mass connects the box to the load that falls vertically. The string passes over a pulley that runs on a frictionless axle. Friction acts between the base of the box and the ramp. Air resistance is negligible.

The load has a mass of 3.5 kg and is initially 0.95 m above the floor. The mass of the box is 1.5 kg .
The load is released and accelerates downwards.

(a)

Outline two differences between the momentum of the box and the momentum of the load at the same instant.

[ 2 ]

(b)

The vertical acceleration of the load downwards is $2.4 \mathrm{~ms}^{-2}$ .

Calculate the tension in the string.

[ 2 ]

(c)

(i)

The radius of the pulley is 2.5 cm . Calculate the angular speed of rotation of the pulley as the load hits the floor. State your answer to an appropriate number of significant figures.

[ 2 ]

(d)

The student then makes the ramp horizontal and applies a constant horizontal force to the box. The force is just large enough to start the box moving. The force continues to be applied after the box begins to move.

Explain, with reference to the frictional force acting, why the box accelerates once it has started to move.

[ 3 ]

[Maximum number: 6]

A stationary ball is hanging from a light string. A pellet from an air rifle is travelling horizontally and becomes embedded in the ball. The velocity of the pellet when it strikes the ball is $160 \mathrm{~ms}^{-1}$ .

The following data are given.

\begin{gathered} \text { Mass of the ball }=250 \mathrm{~g} \\ \text { Mass of the pellet }=2.0 \mathrm{~g} \end{gathered}

(a)

Calculate the speed of the ball and the pellet immediately after the impact.

[ 2 ]

(b)

Suggest why the combined kinetic energy of the ball and the pellet after the impact is less than the initial kinetic energy of the pellet.

[ 2 ]

(c)

The ball with the embedded pellet rises to a maximum vertical height h.

Draw and label the free-body diagram for the ball at height h.

[ 2 ]

[Maximum number: 9]

The diagram below shows part of a downhill ski course which starts at point $\mathrm{A}, 50 \mathrm{~m}$ above level ground. Point B is 20 m above level ground.

(a)

(i)

The dot on the following diagram represents the skier as she passes point B . Draw and label the vertical forces acting on the skier.

[ 2 ]

(b)

The skier reaches point C with a speed of $8.2 \mathrm{~m} \mathrm{~s}^{-1}$ . She stops after a distance of 24 m at point D .

Determine the coefficient of dynamic friction between the base of the skis and the snow. Assume that the frictional force is constant and that air resistance can be neglected.

[ 3 ]

(c)

At the side of the course flexible safety nets are used. Another skier of mass 76 kg falls normally into the safety net with speed $9.6 \mathrm{~ms}^{-1}$ .

[ 4 ]

(i)

Calculate the impulse required from the net to stop the skier and state an appropriate unit for your answer.

[ 2 ]

(ii)

Explain, with reference to change in momentum, why a flexible safety net is less likely to harm the skier than a rigid barrier.

[ 2 ]

(a)

In a "loop-the-loop" toy, a car of mass 0.12 kg is released from rest. The initial position of the car is 45 cm above level ground. The radius of the circular loop is 15 cm . The car reaches the top of the loop at position P. Frictional and air resistance forces are negligible.

[ 1 ]

(i)

State why the car stays in contact with the loop.

[ 1 ]

(b)

At point A the car collides with a block of mass 0.18 kg and sticks to it. After the collision, the car and the block move together with speed $1.2 \mathrm{~ms}^{-1}$ .

[ 3 ]

(i)

The surface from A to B is rough and the combined car and block come to rest at B. The distance A B is 0.20 m . Determine the rate of change of momentum of the combined car and block from A to B.

[ 3 ]

[Maximum number: 1]

A girl rides a bicycle that is powered by an electric motor. A battery transfers energy to the electric motor. The emf of the battery is 16 V and it can deliver a charge of 43 kC when discharging completely from a full charge.

(a)

The bicycle and the girl have a total mass of 66 kg . The girl rides up a slope that is at an angle of $3.0^{\circ}$ to the horizontal.

[ 1 ]

(i)

Calculate the component of weight for the bicycle and girl acting down the slope.

[ 1 ]

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