Quantities, Units & Modelling (Cambridge (CIE) A Level Maths: Mechanics): Exam Questions

(i) A puck slides across a horizontal ice rink after being hit. The ice is modelled as a smooth surface.

State the force that this assumption removes, name the forces that then act on the puck, and describe the motion that the model predicts.

(ii) A skydiver falls vertically after jumping from a plane, before the parachute is opened. A first model ignores air resistance.

Explain why this model is not suitable for the whole of the fall.

Car A travels at a constant speed of 26 ${\text{m s}}^{-1}$.

Car B travels 5200 m in a time of 300 s.

Find the difference between the average speeds of the two cars, giving your answer correct to three significant figures.

A snowboarder slides down a rough slope. The diagram shows the three forces acting, labelled A, B and C.

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

A parachutist descends vertically with an open parachute. The diagram shows the two forces acting, labelled D and E.

(i) Label each of the forces D and E with its name.

(ii) The parachutist descends at a constant speed. State what this tells you about the two forces.

The slope in part (a) is now modelled as smooth.

(i) State which of the forces A, B and C is removed by this assumption, and describe the motion of the snowboarder that the model then predicts.

(ii) Give one reason why modelling a real snow slope as smooth may not be suitable.

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.

Draw a diagram to represent each of the following situations, labelling it with all the forces acting on the body in italics. In both cases air resistance is to be ignored.

Choose the force names from: friction, normal contact force, tension, thrust, weight.

(i) A child swinging on a rope swing, at the lowest point of the swing.

(ii) A bowling ball rolling along a horizontal lane, just as it strikes a skittle.

(i) A dog on a skateboard rolls down a hill. In a first model the hill is treated as a smooth slope and the dog and skateboard together are treated as a particle.

State what the smooth assumption allows you to assume about the forces, and explain why the model becomes less reliable the longer the dog rolls.

(ii) A hot air balloon rises vertically towards its maximum height. A first model ignores air resistance.

Explain why ignoring air resistance is a poorer assumption for the balloon than it would be for a small dense object, and state what the balloon reaches that the model cannot predict.

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.

An astronaut and her equipment have a combined mass of 80 kg.

On Earth the acceleration due to gravity is taken to be 10 ${\text{m s}}^{-2}$. On the Moon it is 1.6 ${\text{m s}}^{-2}$.

(i) Find the weight of the astronaut and her equipment on Earth, and their weight on the Moon.

(ii) State which of the mass and the weight is unchanged when the astronaut travels from the Earth to the Moon, and explain why.

A runner runs in a straight line from a point $O$ to a point $P$, a distance of 600 m, taking 200 s. She then turns and runs straight back from $P$ to $O$, taking 300 s.

Taking the direction from $O$ to $P$ as positive, find for the whole journey:

(i) the total distance travelled and the displacement from $O$;

(ii) the average speed and the average velocity.

A particle rests in equilibrium on a smooth plane inclined at an angle to the horizontal. It is held in place by a string lying along a line of greatest slope, as shown.

(i) Label each of the forces P, Q and R with its name.

(ii) Explain why no frictional force appears on the diagram.

A climber abseils down a vertical cliff face. The climber's feet remain in contact with the cliff and the rope is held taut.

Label each of the forces W, X, Y and Z with its name.

A car tows a trailer along a straight horizontal road. In one model the car and trailer are connected by a light rigid tow-bar; in another they are connected by a light rope. In both models the car and the trailer are treated as particles.

(i) State what modelling the tow-bar as light allows you to assume about the force in it, and what modelling the car and trailer as particles allows you to assume.

(ii) While the car is accelerating, state the direction of the force which the tow-bar exerts on the trailer, and name that force.

(iii) The driver now brakes sharply. Explain why the rigid tow-bar can slow the trailer down but a rope cannot.

Draw a diagram to represent each of the following situations, labelling it with all the forces acting on the body in italics, and stating one modelling assumption you have made in each case.

Choose the force names from: air resistance, friction, normal contact force, tension, thrust, weight.

(i) A lawnmower being pushed along a horizontal lawn by a handle inclined to the horizontal.

(ii) A bucket being raised at a constant speed by a rope which passes over a fixed pulley.

A yo-yo moves in a straight line up and down a string of length 1 m. As it travels down the string it moves at 0.15 ${\text{m s}}^{-1}$, and as it returns towards the hand it moves at 0.12 ${\text{m s}}^{-1}$.

To perform tricks the yo-yo stops at three points along the string. Stop 1 is a quarter of the way along the string and Stop 2 is four fifths of the way along it. At Stop 3 the string is fully extended. The trick "The Mechanic" runs Hand, Stop 2, Stop 1, Stop 3, Stop 1, Hand.

Taking the direction from the hand down the string as positive, state the velocity of the yo-yo:

(i) as it travels from Stop 2 to Stop 1;

(ii) as it travels from Stop 1 to Stop 3.

Using the same positive direction, find:

(i) the displacement of the yo-yo from Stop 3 to Stop 1;

(ii) the total distance travelled by the yo-yo during the whole trick.

A person stands on the horizontal floor of a lift. Two of the forces in this situation are:

• $P$, the normal contact force exerted by the floor on the person;

• $Q$, the weight of the person.

(i) Explain why $P$ and $Q$ are not a Newton's third law pair.

(ii) State the force which does form a Newton's third law pair with $P$, and state its direction.

(iii) The lift now accelerates upwards. State what happens to the magnitude of $P$ compared with $Q$, and state whether the pair you named in part (ii) remains equal in magnitude to $P$.

A block rests on a rough plane inclined to the horizontal. The angle of the plane is slowly increased until the block is on the point of slipping down the plane.

(i) Label each of the forces J, K and L with its name.

(ii) State what "on the point of slipping" tells you about the frictional force.

(iii) State one modelling assumption that has been made about the block.

(i) Explain what is meant by a rough surface and by a smooth pulley, and give an example of a situation in which both would be used in the same model.

(ii) Explain what is meant by a light rigid tow-bar and by a light inextensible string, and state what each allows you to assume about the force it carries.

(iii) A rigid tow-bar can exert a thrust on the body it is attached to, but a string can never do so. Explain why.

A basketball player takes a shot. Consider the ball at the following two instants.

At A, as the player takes the shot, with their hands still in contact with the ball.

At C, as the ball strikes the back of the metal rim of the basket.

Name all the forces acting on the ball at each of the two instants, stating the direction of each.

State three modelling assumptions you have made in answering part (a), and for each one state what it allows you to assume.

A boxer strikes a punch bag which hangs from the ceiling on a chain.

Draw a diagram showing all the forces acting on the punch bag at the moment the glove strikes it, labelling each force and choosing the names from: air resistance, friction, normal contact force, tension, thrust, weight.

State also one modelling assumption you have made, and what it allows you to assume.

A cat bats at a piece of string which its owner holds at the upper end, letting the lower end hang freely.

Draw a diagram showing all the forces acting on the string at the moment the cat's paw touches it, labelling each force from the same list as in part (a).

State also one modelling assumption you have made, and what it allows you to assume.

A stone is thrown vertically upwards from the edge of a lake. Its height above the water level, $h$ m, at time $t$ s after it is thrown is modelled by

$h=a{t}^2+bt+c$

where $a$, $b$ and $c$ are constants. Air resistance is ignored.

(i) State what the constant $c$ represents in this model.

(ii) Show that $a=-5$.

Explain why the model should not be used once $h$ becomes negative.