CIE A Level Maths: Mechanics

Topic Questions

1.1 Quantities, Units & Modelling

1
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6 marks

Given that scalar quantities have magnitude (size), but not direction, and vector quantities have both direction and magnitude.

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

(i)
A motorbike accelerates at 3.4 space straight m space straight s to the power of negative 2 end exponent .

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

(iii)
A unicorn runs at an average speed of 128.4 space straight m space straight s to the power of negative 1 end exponent

(iv)
A room is 3.95 space straight m wide.

(v)
A model boat travels at 2 space straight m space straight s to the power of negative 1 end exponenton a bearing of 035 degree.

(vi)
A golf ball rolls 154 space cm before stopping.

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2
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3 marks

Standardised (S.I.) units for length, time and mass are meters (m), seconds (s) and kilograms (kg) respectively.

Convert the following into S.I. units:

(i)   34.5 space km (ii)   4 space hours space 17 space minutes  (iii)  860 space g

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3
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3 marks

S.I. units can be combined to form derived compound units for quantities such as velocity open parentheses straight m space straight s to the power of negative 1 end exponent close parentheses, acceleration open parentheses straight m space straight s to the power of negative 2 end exponent close parentheses and force open parentheses straight N equals kg space straight m space straight s to the power of negative 2 end exponent close parenthesesetc.

Convert the following into S.I. units:

(i)
24 space cm space per space minute

(ii)
6.4 space km space straight s to the power of negative 2 end exponent

(iii)
38 space straight g space straight m space straight s to the power of negative 2 end exponent

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4
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2 marks

A film crew record a cheetah running for 2.5 space minutes, it covers a distance of 2.4 space km.

Calculate the cheetah’s average speed in straight m space straight s to the power of negative 1 end exponent?

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5a
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2 marks

Label the following diagrams with the names of the forces given in each case.

A toy being pulled along on a string: tension, friction, weight, normal reaction.

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

5b
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2 marks
A paper plane flying through the air: air resistance (drag), upward thrust (lift), thrust, weight.

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

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6
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4 marks

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

Explain what effect the following assumptions have on the model described above.

(i)
The train is modelled as a particle.

(ii)
The string is inextensible.

(iii)
The floor is flat.

(iv)
The surface is of the carpet is rough.

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7
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6 marks

Draw a simple diagram to represent each of the following models. Label your diagrams, where appropriate, with any of the following words: 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
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6 marks

List three assumptions you may need to make in order to create a simple model for each of the following. It may help to think about things that might be affecting the way the object moves, the sorts of external forces that might be acting on the object and also about what factors 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
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2 marks

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

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

Starting at A it 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 the following are 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
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3 marks

Convert the units in the following statements into S.I. units.

(i)
A hot air balloon descends at 5.94 space cross times space 10 to the power of 5 space cm space straight h to the power of negative 2 end exponent.
(ii)
The density of oil is about 0.95 space straight g space cm to the power of negative 3 end exponent.
(iii)
The average spike ball player can hit the ball at a velocity of 43.5 space km space straight h to the power of negative 1 end exponent.

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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 space km space straight s to the power of negative 2 end exponent.

   B: A cheetah accelerates at 28.8 space km space min to the power of negative 2 end exponent

   C: A race car accelerates at 108 space straight m per square min.

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3a
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3 marks

Define each of the following and give an example of how they could be used in a mathematical model, include any related assumptions which can be made.

A lamina and a non-uniform rod.

3b
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3 marks

A bead and wire. 

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4a
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2 marks

Label the following diagrams with the appropriate forces.

A speed boat travelling through water.

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

4b
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2 marks

A climber abseiling down a cliff.

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

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5
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6 marks

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

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

Explain what effect the following assumptions would have on the model described above and whether or not they are 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
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8 marks

Draw a simple diagram to represent each of the following models. Label your diagrams with the appropriate forces involved and detail any assumptions you make about your model.

(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
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6 marks

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

(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
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4 marks

A yo-yo moves in a straight line up and down a 1 m string. As it travels down the string it has a velocity of 0.15 space straight m space straight s to the power of negative 1 end exponent and a velocity of 0.12 space straight m space straight s to the power of negative 1 end exponent when returning towards the hand. In order to do certain tricks, the yo-yo stops at different places along the string as shown in the diagram below. The stops of one particular trick, ‘The Mechanic’, are as follows: Hand-2-1-3-1-Hand

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

Stop 1 is begin mathsize 14px style 1 fourth end style of the way along the string, Stop 2 is begin mathsize 14px style 4 over 5 end style of the way along the string. When the yo-yo reaches Stop 3 the string is fully extended. 

By indicating your chosen positive direction clearly on the diagram above, state the following in relation to the yo-yo:

(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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1
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6 marks

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

(i)
A golf ball is hit with a force of 17793 space straight N.

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

(iii)
A hot air balloon rises vertically at 2.8 space straight m space straight s to the power of negative 2 end exponent.

(iv)
The density of water is about 1 space straight g space cm to the power of negative 3 end exponent.

(v)
Gravity on Earth is approximated to 10 space straight m space straight s to the power of negative 2 end exponent.

(vi)
The average human has a body temperature of 37 degree C.

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2
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4 marks

Convert the following into S.I. units:

(i)
2.7 space km per hour       

(ii)
460 space km space straight h to the power of negative 2 end exponent

(iii)
3.2 space cross times space 10 to the power of 5 space kg space cm to the power of negative 2 end exponent

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3
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6 marks

State the S.I. units which would be used to measure the following.

(i)
The displacement of a particle.

(ii)
The acceleration of a car.

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

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

(v)
A force acting on a particle.

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4
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3 marks

Car A is travelling at 95 space km space straight h to the power of negative 1 end exponent, Car B travels 5.2 space km in 5 minutes.  Find the difference in their average speeds. Giving your answer in straight m space straight s to the power of negative 1 end exponent, correct to three significant figures.

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5a
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2 marks

Label the following diagrams with the appropriate forces.

A snowboarder sliding down a ski slope.

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

5b
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3 marks

A parachuting penguin.

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

 

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6
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5 marks

Two blocks, A and B are attached by means of a light inextensible string running over a smooth pulley, as shown in the diagram below. Block A is accelerating along a smooth horizontal surface in the direction shown, block B is moving towards the ground. Both A and B are modelled as particles.

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

Explain what effect each of the following assumptions have on the model described above.

 

(i)
A and B are both particles.

(ii)
The string is light.

(iii)
The string is inextensible.

(iv)
The pulley is smooth and light.

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

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7
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6 marks

Draw a simple diagram to represent each of the following models. Labelling your diagrams with the appropriate forces involved.
 

(i)
A child swinging on a rope swing.

(ii)
A bowling ball just as it hits a skittle.

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8
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6 marks

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

(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
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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 space km space straight h to the power of negative 1 end exponent on stretches of the canal that are over 5 space km long and 4 km h-1 on stretches that are under 5 space km long.

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

Taking the positive direction as shown in the diagram, state the following in relation to the canal barge:

(i)
the velocity from home to Lock A

(ii)
the displacement from home once Lock A is reached

(iii)
the velocity from Lock A to home

(iv)
the velocity from home to Lock B

(v)
the displacement from home once Lock B is reached

(vi)
the velocity from Lock B to home

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

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1
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5 marks

A speedboat accelerates from rest to 90 mph in 5 seconds. The distance travelled, d meters, in time t seconds, can be modelled by a quadratic equation in the form d equals space k t squared. When t equals 2 the speedboat has travelled a distance of 80.4 space straight m.

(i)
Use the model to find the distance travelled when the boat reaches 90 space mph.

(ii)
For what range of values is t valid for this model. Give a reason for your answer.

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2a
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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 m, at time t seconds after it is thrown is modelled by a quadratic equation in the form h space equals space a t squared plus b t plus c.

Explain why the value of a must be negative and what the variable c must indicate.

2b
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3 marks
(i)
Explain why the model may not be valid for h less than 0.
(ii)
Describe a situation where the model might only be valid for h greater or equal than 0.

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3
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3 marks

The diagram below shows a child holding onto a kite flying in the wind. Label the diagram with the appropriate forces, explain any assumptions you make about the diagram.

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

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4
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6 marks

Define each of the following and give an example of how they could be used in a mathematical model, include any related assumptions which can be made.

(i)
A rough surface and a smooth pulley.
(ii)
A peg and inextensible string.

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5a
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6 marks

The diagram below shows a basketball player taking a shot.

Label the forces on the ball at the three different 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, hitting the back of the metal rim.

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

5b
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3 marks

List any assumptions you made for each, or all, stages of the throw A, B and C in order to answer part (a).

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6a
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4 marks

Draw a simple diagram to represent each of the following models. Label your diagrams with the appropriate forces involved and detail any assumptions you make about each model.

A boxer hitting a punch bag suspended from the ceiling.

6b
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4 marks

The forces acting on a cat batting at a piece of string held by its owner.

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7
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3 marks

A particle is attached to the end of a rod. One end of the rod is fixed to a wall using a hinge, the other end is held using a piece of string before it is let go.

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

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