A-Level Maths (Edexcel 9MA0) · Mechanics Year 1 · Chapter 10: Forces and motion · Lesson 2

Forces and acceleration

121 marks

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Date

Do Now

Quick questions to start. Some use earlier lessons; some set a slip that costs marks.

A1
Needed today
A cyclist increases her speed uniformly from $3 m s^{- 1}$ to $15 m s^{- 1}$ in $4$ s. Find her acceleration and the distance she travels in this time.
[3 marks]
Answer
A2
Last lesson
A box of mass $4$ kg rests on a horizontal table. A light string pulls vertically upwards on the box with tension $15$ N. Find the normal reaction from the table.
[2 marks]
Answer
A3
Common slip
A car slows down uniformly from $20 m s^{- 1}$ to rest in a distance of $50$ m. Taking the direction of motion as positive, find the acceleration of the car.
[2 marks]
Answer
A4
Common slip
A particle of mass $250$ g is acted on by a resultant force of $0.6$ N. Find its acceleration.
[2 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

A box of mass $8$ kg is pushed along a horizontal floor by a horizontal force of $50$ N. The resistance to motion is $18$ N. Find the acceleration of the box and the normal reaction from the floor.

Hint Draw the four forces and the acceleration arrow. In

F = ma

F

is the RESULTANT force in the direction of the acceleration, not the push alone.

B1
A car of mass $1200$ kg moves along a straight horizontal road. The engine gives a constant driving force of $3000$ N and the resistance to motion is constant. The speed of the car increases from $4 m s^{- 1}$ to $16 m s^{- 1}$ in $8$ s. Find the resistance to motion.
[4 marks]
Answer

Example 2

A load of mass $50$ kg is raised by a vertical rope with an upward acceleration of $0.6 m s^{- 2}$ . Find the tension in the rope. Later the load is lowered with a downward acceleration of $0.6 m s^{- 2}$ . Find the tension now.

Hint The weight

50 g

N always acts down. Take the direction of the acceleration as positive and write

F = ma

with every force in that direction.

B2
A crate of mass $80$ kg is lowered on a vertical rope. It starts from rest and moves down $4.5$ m in $3$ s with constant acceleration. Find the tension in the rope.
[5 marks]
Answer

Example 3

A woman of mass $60$ kg stands in a lift. Find the force exerted on her by the floor when the lift (a) moves upwards with acceleration $0.8 m s^{- 2}$ , (b) moves upwards and slows down with deceleration $0.5 m s^{- 2}$ .

Hint Write the equation of motion for the WOMAN only: her weight down and the reaction

R

up. The lift's cable does not act on her.

B3
A man stands on weighing scales in a lift. When the lift accelerates upwards at $1.2 m s^{- 2}$ the scales show a force of $880$ N. Find the mass of the man. Find the force shown by the scales when the lift is moving downwards and slowing down at $0.6 m s^{- 2}$ .
[5 marks]
Answer

Example 4

A particle of mass $2$ kg is pulled from rest across a horizontal floor by a horizontal force of $14$ N. The resistance to motion is a constant $4$ N. After $3$ s the pulling force is removed. Find the total distance the particle travels.

Hint Two stages, two accelerations. Find the speed at the end of stage 1: it is the initial speed of stage 2. In stage 2 only the resistance acts.

B4
A sledge of mass $25$ kg is pulled from rest across horizontal ice by a horizontal rope with tension $75$ N. The resistance to motion is a constant $30$ N. After $10$ s the rope is released. Find the total distance the sledge travels.
[6 marks]
Answer

Example 5

A ball of mass $0.5$ kg is thrown vertically upwards with speed $14 m s^{- 1}$ . Air resistance is modelled as a constant force of $1.1$ N. Find the greatest height the ball reaches above the point of projection, and its speed when it returns to that point.

Hint Air resistance opposes the motion: down while the ball rises, up while it falls. So the accelerations up and down are different.

B5
A stone of mass $0.4$ kg is thrown vertically upwards with speed $21 m s^{- 1}$ . Air resistance is modelled as a constant force of $0.56$ N. Find the greatest height the stone reaches above the point of projection, and the total time until it returns to that point.
[7 marks]
Answer

Practice

A-Level

C1
A bucket of mass $6$ kg is raised by a vertical rope with an upward acceleration of $0.4 m s^{- 2}$ . Ravi writes $T = 6 \times 0.4 = 2.4$ N. Explain his mistake and find the tension.
[4 marks]
Answer

C2
A man of mass $70$ kg stands in a lift that is moving downwards and slowing down at $0.7 m s^{- 2}$ . Pia says: "The lift is going down, so the floor pushes on him with less than his weight." Explain why Pia is wrong and find the force the floor exerts on him.
[4 marks]
Answer
C3
A van of mass $2000$ kg moves on a straight horizontal road. The engine gives a driving force of $5000$ N and the resistance to motion is $1400$ N. Ben writes $a = \frac{5000}{2000} = 2.5 m s^{- 2}$ . Correct his working, and find the time the van takes to increase its speed from $5 m s^{- 1}$ to $23 m s^{- 1}$ .
[4 marks]
Answer
C4
A boat of mass $1500$ kg moves at a constant speed of $6 m s^{- 1}$ in a straight line. The engine gives a forward force of $2400$ N. Zoe writes $2400 - R = 1500 \times 6$ . Explain her mistake and find the resistance $R$ . The engine force is then increased to $3000$ N with the resistance unchanged. Find how far the boat travels while its speed increases to $10 m s^{- 1}$ .
[5 marks]
Answer
C5
A box is pulled along a horizontal floor by a horizontal force. The resistance to motion is a constant $F$ N. When the force is $40$ N the acceleration is $3 m s^{- 2}$ , and when the force is $64$ N the acceleration is $5 m s^{- 2}$ . Find the mass of the box and $F$ .
[4 marks]
Answer

Exam-style questions

Edexcel exam style

D1
A lift of mass $500$ kg carries a woman of mass $65$ kg. The lift is raised by a vertical cable. The velocity-time graph shows its motion: it starts from rest, moves upwards with constant acceleration for $2$ s, moves at a constant speed of $3 m s^{- 1}$ , then slows down at a constant rate and comes to rest at $t = 10$ s. The cable is modelled as light and inextensible.
(a)
Find the tension in the cable during the first $2$ s.
[3 marks]
Answer
(b)
Find the force exerted on the woman by the floor of the lift during the last $3$ s.
[3 marks]
Answer
(c)
Find the total distance the lift rises.
[2 marks]
Answer
(d)
Explain why the mass $565$ kg is used in part (a) but the mass $65$ kg is used in part (b).
[2 marks]
Total for D1: 10 marks

D2
A stone of mass $0.25$ kg is thrown vertically upwards with speed $18 m s^{- 1}$ . Air resistance is modelled as a constant force of $0.7$ N. Kim writes: "Going up, $0.25 a = 0.25 g - 0.7$ , so the deceleration is $7 m s^{- 2}$ ."
Kim is wrong.
(a)
Explain Kim's mistake.
[2 marks]
(b)
Find the greatest height of the stone above the point of projection.
[3 marks]
Answer
(c)
Find the total time for which the stone is more than $5$ m above the point of projection.
[5 marks]
Answer
Total for D2: 10 marks
D3
A stone of mass $0.6$ kg is dropped from rest from a point $2.5$ m above soft ground. It falls freely, hits the ground and sinks $4$ cm into the ground before it comes to rest. The ground is modelled as giving a constant resistance to the motion of the stone, and air resistance is ignored. Find the magnitude of the resistance from the ground.
[7 marks]
Answer
D4
A crate of mass $120$ kg is raised from rest on the ground by a vertical cable. For the first $4$ s the tension in the cable is $1296$ N. The cable is modelled as light and inextensible and air resistance is ignored.
(a)
Find the acceleration of the crate during the first $4$ s.
[2 marks]
Answer
(b)
Find the speed of the crate and its height above the ground at $t = 4$ s.
[2 marks]
Answer
(c)
The tension is then changed so that the crate rises at constant speed. Find the new tension.
[2 marks]
Answer
(d)
After rising at constant speed for $3$ s, the cable breaks. Find the time from the moment the cable breaks until the crate hits the ground.
[4 marks]
Answer
Total for D4: 10 marks
D5
A particle $P$ of mass $3$ kg is pulled from rest across a horizontal floor by a constant horizontal force of $21$ N. The resistance to motion is a constant $F$ N. In the first $4$ s, $P$ moves $16$ m.
(a)
Find $F$ .
[4 marks]
Answer
(b)
At $t = 4$ s the pulling force is removed. Find the time $P$ takes to come to rest and the total distance it moves.
[3 marks]
Answer
(c)
Sketch a velocity-time graph for the motion of $P$ from $t = 0$ until it comes to rest, showing the values where the lines meet the axes and the greatest speed.
[2 marks]
Total for D5: 9 marks
D6
A firework of mass $0.5$ kg is launched vertically upwards from rest on the ground. For the first $2$ s the engine gives a constant upward thrust of $9$ N. After $2$ s the thrust stops. The firework is modelled as a particle of constant mass and air resistance is ignored.
(a)
Find the acceleration of the firework while the engine works.
[2 marks]
Answer
(b)
Find the speed and the height of the firework at $t = 2$ s.
[2 marks]
Answer
(c)
Find the greatest height of the firework above the ground.
[3 marks]
Answer
(d)
State one way in which the model of constant mass is not realistic, and its effect on the answer to part (c).
[2 marks]
Total for D6: 9 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
A woman stands on weighing scales in a lift. When the lift moves upwards with acceleration $a m s^{- 2}$ the scales show $693$ N. When the lift moves upwards and slows down with deceleration $a m s^{- 2}$ the scales show $483$ N. Find the mass of the woman and the value of $a$ . On one journey the lift starts from rest, accelerates at $a m s^{- 2}$ for $T$ s, moves at constant speed, then slows down at $a m s^{- 2}$ for $T$ s and comes to rest. The journey takes $10$ s and the lift rises $28$ m. Find $T$ and the greatest speed of the lift.
[9 marks]
Hint 1 · What to try
Write two equations of motion for the woman: $693 - m g = ma$ and $483 - m g = - ma$ . Add them.
Hint 2 · The first line
Sketch the velocity-time graph. The greatest speed is $a T$ and the time at constant speed is $10 - 2 T$ . The area is $28$ .
Hint 3 · The full method
The equation in $T$ is a quadratic with two roots. Check each against the fact that the two slowing and speeding stages fit inside $10$ s.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can write an equation of motion F = ma using the resultant force, with the weight mg when the motion is vertical.
I can find the tension in a rope or the reaction of a lift floor when the load accelerates up or down.
I can combine F = ma with the constant acceleration formulae, including when a force changes part way.
I can solve problems where air resistance gives different accelerations going up and coming down.

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