A LevelMathsEdexcelMechanicsExam QuestionsQuantities & Units in MechanicsQuantities, Units & Modelling

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

Exam code: 9MA0

3 hours34 questions
Easy
Medium
Hard
Very Hard
1
6 marks

Scalar quantities have magnitude (size) but no direction. Vector quantities have both magnitude and direction.

State whether the quantity described in each of the following statements is a scalar or a vector.

(i) A motorbike accelerates at 3.4\text{ m s}^{-2}.

(ii) A concert lasts 2 hours and 53 minutes.

(iii) A unicorn runs at an average speed of 128.4\text{ m s}^{-1}.

(iv) A room is 3.95\text{ m} wide.

(v) A model boat travels at 2\text{ m s}^{-1} on a bearing of 035^{\circ}.

(vi) A golf ball rolls 154\text{ cm} before stopping.

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2
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3 marks

The standard international (S.I.) units of length, time and mass are metres (\text{m}), seconds (\text{s}) and kilograms (\text{kg}) respectively.

Convert each of the following into S.I. units.

(i) 34.5\text{ km}

(ii) 4 hours and 17 minutes

(iii) 860\text{ g}

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3
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3 marks

S.I. units can be combined to form derived units for quantities such as velocity (\text{m s}^{-1}), acceleration (\text{m s}^{-2}) and force (\text{N} = \text{kg m s}^{-2}).

Convert each of the following into S.I. units.

(i) 24\text{ cm} per minute

(ii) 6.4\text{ km s}^{-2}

(iii) 38\text{ g m s}^{-2}

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4
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2 marks

A film crew record a cheetah running for 2.5 minutes; in this time the cheetah covers a distance of 2.4\text{ km}.

Calculate the cheetah's average speed in \text{m s}^{-1}.

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5a
2 marks

Label the diagram below with the names of the forces acting in this situation: tension, friction, weight, normal reaction.

A toy is pulled along a flat surface using a horizontal string.

edexcel-al-maths-mechanics-topic-1-1-easy-q5a

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5b
2 marks

Label the diagram below with the names of the forces acting in this situation: air resistance (drag), lift (upward thrust), thrust, weight.

A paper plane flies horizontally through the air.

edexcel-al-maths-mechanics-topic-1-1-easy-q5b

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6
4 marks

A toy train is being pulled over a carpeted floor by a string.

A toy train is pulled across a carpeted floor by a string.

Explain the effect of each of the following assumptions on the model of this situation.

(i) The train is modelled as a particle.

(ii) The string is inextensible.

(iii) The floor is flat.

(iv) The surface of the carpet is rough.

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7
6 marks

Draw a simple labelled diagram to represent each of the following models. Where appropriate, label your diagrams with the relevant forces from this list: air or water resistance (drag), buoyancy, friction, normal reaction, tension, forward thrust, weight.

(i) A child being pulled along on a sledge.

(ii) A duck paddling forwards on a pond.

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8
6 marks

List three assumptions you may need to make in order to create a simple model for each of the following situations. (You may want to think about external forces acting on the object, factors that could affect its motion, and factors that may be ignored.)

(i) The motion of an ice hockey puck after being hit across the ice rink.

(ii) The motion of a skydiver after jumping from a plane, before their parachute is deployed.

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9
2 marks

Each day a train travels in a straight line between three stations A, B and C, as shown in the diagram below.

edexcel-al-maths-mechanics-topic-1-1-easy-q9

Starting at A, the train travels directly to C. It then travels back to B, before returning to its starting position at station A.

Taking the positive direction as shown in the diagram, state whether each of the following is positive or negative.

(i) The displacement from A to C.

(ii) The displacement from B to A.

(iii) The velocity from C to B.

(iv) The velocity from A to C.

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1
6 marks

State whether the quantity described in each of the following statements is a scalar or a vector.

(i) A golf ball is hit with a force of 17\,793\text{ N}.

(ii) A bath tub holds 42 gallons of water.

(iii) A hot air balloon rises vertically at 2.8\text{ m s}^{-2}.

(iv) The density of water is approximately 1\text{ g cm}^{-3}.

(v) The acceleration due to gravity on Earth is approximately 9.8\text{ m s}^{-2}.

(vi) The average human body temperature is 37^{\circ}\text{C}.

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2
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4 marks

Convert each of the following into S.I. units.

(i) 2.7 km per hour

(ii) 460\text{ km h}^{-2}

(iii) 3.2 \times 10^{5}\text{ kg cm}^{-2}

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3
5 marks

State the S.I. unit which would be used to measure each of the following quantities.

(i) The displacement of a particle.

(ii) The acceleration of a car.

(iii) The velocity of a yo-yo travelling down its string.

(iv) The velocity of the yo-yo from part (iii) travelling back up its string.

(v) A force acting on a particle.

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4
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3 marks

Car A travels at an average speed of 95\text{ km h}^{-1}. Car B travels 5.2\text{ km} in 5 minutes.

Find the difference in their average speeds, giving your answer in \text{m s}^{-1} to three significant figures.

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5a
2 marks

A snowboarder slides down a ski slope.

Label the diagram below with the appropriate forces.

edexcel-al-maths-mechanics-topic-1-1-m-q5

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5b
3 marks

A penguin descends through the air with an open parachute.

Label the diagram below with the appropriate forces.

edexcel-al-maths-mechanics-topic-1-1-m-q5b

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6
5 marks

Two blocks A and B are attached by a light, inextensible string running over a smooth pulley, as shown in the diagram below. Block A accelerates along a smooth horizontal surface in the direction shown, and block B moves toward the ground. Both A and B are modelled as particles.

edexcel-al-maths-mechanics-topic-1-1-m-q6

Explain the effect of each of the following assumptions on the model.

(i) A and B are both particles.

(ii) The string is light.

(iii) The string is inextensible.

(iv) The pulley is smooth.

(v) The surface A is moving along is smooth.

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7
6 marks

Draw a simple labelled diagram to represent each of the following models, including all the appropriate forces involved.

(i) A child swinging on a rope swing.

(ii) A bowling ball at the moment it strikes a skittle.

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8
6 marks

List any assumptions you may need to make in order to create a simple model for each of the following situations.

(i) The motion of a dog on a skateboard rolling down a hill.

(ii) The motion of a hot air balloon rising to its maximum height.

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9
4 marks

A canal barge travels in a straight line from home to two locks, A and B, as shown in the diagram below. The barge travels at a speed of 6\text{ km h}^{-1} on stretches of the canal that are over 5\text{ km} long, and at 4\text{ km h}^{-1} on stretches that are under 5\text{ km} long.

edexcel-al-maths-mechanics-topic-1-1-m-q9

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

(i) The velocity from home to Lock A.

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

(iii) The velocity from Lock A back to home.

(iv) The velocity from home to Lock B.

(v) The displacement from home once the barge reaches Lock B.

(vi) The velocity from Lock B back to home.

(vii) The displacement once the barge has returned home.

How did you do?

10
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3 marks

Convert each of the following into S.I. units.

(i) A hot air balloon descends at 5.94 \times 10^{5}\text{ cm h}^{-2}.

(ii) The density of oil is approximately 0.95\text{ g cm}^{-3}.

(iii) The average spike ball player can hit the ball at a velocity of 43.5\text{ km h}^{-1}.

How did you do?

1a
2 marks

Roger throws a frisbee upwards. He releases the frisbee at a height of 127\text{ cm} above horizontal ground.

The motion of the frisbee is modelled by the equation

h(x) = H + 1.7x - 0.3x^2, \qquad x \ge 0

where h is the height of the frisbee above the ground, in metres, and x is the horizontal distance travelled by the frisbee, in metres.

Write down the value of the constant H and explain why the model is only valid for x \ge 0.

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1b
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3 marks

Use the model to find the horizontal distance the frisbee will have travelled by the time it reaches the ground. Give your answer to three significant figures.

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1c
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3 marks

According to the model, find the maximum height that the frisbee reaches. Give your answer to three significant figures.

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2
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4 marks

By converting to S.I. units, compare which of the following accelerates quickest.

A: A motorbike accelerates at 0.0035\text{ km s}^{-2}.

B: A cheetah accelerates at 28.8\text{ km min}^{-2}.

C: A race car accelerates at 108\text{ m min}^{-2}.

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3a
3 marks

Define each of the following terms, give an example of how each could be used in a mathematical model, and include any related modelling assumptions.

A lamina and a non-uniform rod.

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3b
3 marks

Define each of the following terms, give an example of how each could be used in a mathematical model, and include any related modelling assumptions.

A bead and a wire.

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4a
2 marks

A speedboat travels through water.

Label the diagram below with the appropriate forces.

edexcel-al-maths-mechanics-topic-1-1-h-q5

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4b
2 marks

A climber abseils down a vertical cliff.

Label the diagram below with the appropriate forces.

edexcel-al-maths-mechanics-topic-1-1-h-q5b

How did you do?

5
6 marks

Fishing company Fin-tastic Rods are designing a new fishing pole. They set up the model shown in the diagram below to consider the forces involved in catching a fish.

edexcel-al-maths-mechanics-topic-1-1-h-q6

For each of the following assumptions, explain its effect on the model and discuss whether the assumption is suitable.

(i) The fishing line is inextensible and light.

(ii) The fishing pole is a light rod.

(iii) The fish is modelled as a particle.

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6
8 marks

Draw a simple labelled diagram to represent each of the following models. Label your diagrams with the appropriate forces involved, and state any modelling assumptions you make.

(i) A lawnmower being pushed along to mow an uneven lawn.

(ii) A bucket on a rope being raised using a pulley.

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7
6 marks

List any assumptions you could make in order to create a simple model for each of the following situations.

(i) The motion of a pencil rolling along a table.

(ii) The motion of a ball being hit by a bat.

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8
4 marks

A yo-yo moves up and down a string of length 1\text{ m}. As the yo-yo travels down the string it has a speed of 0.15\text{ m s}^{-1}, and as it returns toward the hand it has a speed of 0.12\text{ m s}^{-1}.

To perform certain tricks, the yo-yo stops at different positions along the string. The stops for the trick called "The Mechanic" are:

\text{Hand} \to \text{Stop 2} \to \text{Stop 1} \to \text{Stop 3} \to \text{Stop 1} \to \text{Hand}

edexcel-al-maths-mechanics-topic-1-1-h-q9

Stop 1 is \tfrac{1}{4} of the way along the string and Stop 2 is \tfrac{4}{5} of the way along the string. When the yo-yo reaches Stop 3, the string is fully extended.

Taking the downward direction along the string (away from the hand) as positive, state the following.

(i) The displacement from the hand once Stop 2 is reached.

(ii) The velocity when travelling between Stop 2 and Stop 1.

(iii) The velocity when travelling between Stop 1 and Stop 3.

(iv) The maximum displacement from the hand.

(v) The displacement from Stop 3 to Stop 1.

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9
3 marks

The diagram below shows a child holding onto a kite which is flying in the wind. Label the diagram with the appropriate forces, and state any modelling assumptions you make.

edexcel-al-maths-mechanics-topic-1-1-vh-q4

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1
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5 marks

A speedboat accelerates from rest to 90\text{ mph} in 5 seconds.

The distance travelled by the speedboat, d metres, t seconds after it starts accelerating, is modelled by the equation

d = kt^2

where k is a constant.

Given that the speedboat travels a distance of 80.4\text{ m} in the first 2 seconds:

(i) Use the model to find the distance travelled by the speedboat at the moment it reaches 90\text{ mph}.

(ii) State, giving a reason, the range of values of t for which this model is valid.

How did you do?

2a
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4 marks

Melody throws a netball into a net. The path of the netball, from the moment it leaves Melody's hand to the moment it passes through the net, is modelled by the equation

h(x) = 1.8 + 1.2x - 0.2x^2, \qquad x \ge 0

where h is the height of the netball above the ground in metres, and x is the horizontal distance travelled by the netball in metres.

Find the height of the netball when it is horizontally halfway between being thrown and reaching its maximum height.

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2b
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4 marks

The model is valid for 0 \le x \le k, where k metres is the horizontal distance of the net from Melody.

The standard height of a netball hoop is 3.05\text{ m}.

Find the value of k, giving your answer to three significant figures.

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2c
1 mark

Explain why the model is not valid for x > k.

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3a
2 marks

A stone is thrown from the edge of a lake into the water.

The height of the stone above the water level, h metres, t seconds after it is thrown, is modelled by the equation

h = at^2 + bt + c

where a, b and c are constants.

Explain why a must be a negative constant, and state the physical meaning of c in this model.

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3b
2 marks

(i) Explain why the model may not be valid for h < 0.

(ii) Describe a situation where the model would only be valid for h \ge 0.

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4
6 marks

Define each of the following terms, give an example of how each could be used in a mathematical model, and include any related modelling assumptions.

(i) A rough surface and a smooth pulley.

(ii) A peg and an inextensible string.

How did you do?

5a
6 marks

The diagram below shows a basketball player taking a shot.

Label the forces acting on the ball at each of the three stages of the throw.

A: As the player takes the shot, with their hands still in contact with the ball.

B: When the ball is in flight at its maximum height.

C: When the ball reaches the basket and hits the back of the metal rim.

edexcel-al-maths-mechanics-topic-1-1-vh-q6

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5b
3 marks

List the modelling assumptions you made for each (or all) of the stages A, B and C in part (a).

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6
3 marks

A particle is attached to the end of a rod. One end of the rod is fixed to a wall by a hinge, and the other end is held in position by a piece of string before it is released.

State all the modelling assumptions that would need to be made to model this situation. You may draw a diagram to support your answer.

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