8 Newton’s laws of motion
Stretch and challenge
AQA Physics
Rockets
Specification references
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3.4.1.5 Newton’s laws of motion
3.4.1.6 Momentum
M0.1, M0.2, M0.5
M1.1
M2.2, M2.3
Introduction
You learnt about Newton’s second law of motion at GCSE, in the form F  m a
(force  mass  acceleration). But Newton actually wrote this law in a form similar
to ‘the rate of change of momentum of an object is proportional to the resultant
force on it’.
In this worksheet you will apply all of Newton’s laws to questions about rockets.
Learning objectives
After completing the worksheet you should be able to:
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calculate the acceleration of a rocket given the mass and velocity of gases
ejected
apply Newton’s laws to the different stages of a rocket’s flight.
Background
What is a rocket?
A rocket is any spacecraft or projectile that contains a ‘rocket engine’. A rocket
engine burns solid or liquid fuel to release a gas which is enclosed in a chamber,
under pressure. A small opening at one end of the chamber allows the gas to
escape and this provides a thrust that propels the rocket. The action is similar to that
of a balloon – if you allow the air to escape, the balloon is propelled in the opposite
direction to the motion of the escaping air.
Applying Newton's first law
If a rocket is at rest on the launch pad, the weight of the rocket (m g) and the
reaction force of the launch pad on the rocket are balanced. As the engines are
ignited, the thrust force causes the forces on the rocket to become unbalanced and
the rocket to move upwards.
Applying Newton's second law
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Newton’s second law of motion states that the rate of change of momentum of
an object is proportional to the resultant force on it.
For a rocket, the thrust is equal to the rate of change of momentum of the gas.
total mass of rocket  mass of rocket itself  mass of fuel
© Oxford University Press 2015
www.oxfordsecondary.co.uk/acknowledgements
This resource sheet may have been changed from the original
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8 Newton’s laws of motion
Stretch and challenge
AQA Physics
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The largest part of the rocket's mass is its fuel. This mass reduces as the
engines fire since the engines expel the used fuel in the exhaust gases.
In order for F  m a to still balance, if the thrust remains constant the
acceleration of the rocket must increase as its mass decreases.
The increasing acceleration is modelled by the following equation.
increase in velocity of the rocket  velocity of gases  ln
initial mass of rocket
final mass of rocket
o Note ln  loge (you need to use the ‘ln’ button on your calculator)
 A rocket starts off moving slowly and goes faster and faster as it climbs into
space.
 For a rocket to climb into low-Earth orbit, it must achieve a speed of over
28 000 km per hour.
 For a rocket to leave Earth and travel out into deep space, it must achieve a
speed of over 40 250 km per hour, this is called the escape velocity.
Applying Newton's third law
Newton's third law of motion can be stated as: every action has an equal and
opposite reaction.
A rocket can lift off from a launch pad only when it expels gas out of its engine. The
rocket pushes on the gas, and the gas in turn pushes on the rocket. The action is
the expulsion of the gas. The reaction is the movement of the rocket in the opposite
direction. To enable a rocket to lift off from the launch pad, the action, or thrust, from
the engine must be greater than the weight of the rocket. In space, however, even
tiny thrusts will cause the rocket to change direction.
Summary
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An unbalanced force must be exerted for a rocket to lift off from a launch pad or
for a craft in space to change speed or direction (application of Newton’s first
law).
The amount of thrust produced by a rocket engine will be determined by the
mass of rocket fuel that is burned and how fast the gas escapes the rocket
(application of Newton’s second law).
The reaction of the rocket is equal to and in the opposite direction to the action,
or thrust, from the engine (application of Newton’s third law).
The law of conservation of momentum applies, such that
backward momentum of gases  forward momentum of rocket
© Oxford University Press 2015
www.oxfordsecondary.co.uk/acknowledgements
This resource sheet may have been changed from the original
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8 Newton’s laws of motion
Stretch and challenge
AQA Physics
Worked example 1
Question
A rocket ejects gases at a speed of 4.5  103 m s−1 and a rate of 6.0 kg m s−1.
Calculate the thrust.
Answer
thrust  rate of change of momentum of gases
consider the change of momentum over 1 s
thrust 
v  Δm
1
(4.5  10 3 )  6.0
1
 2.7  104 N

Worked example 2
Question
At lift-off, the mass of a rocket is 1.5  105 kg and the thrust developed by the
engines is 2  106 N. Calculate the initial acceleration of the rocket.
Answer
Step 1
Calculate the resultant force acting on the rocket.
resultant force, F  thrust, FT − weight of rocket, FW
 FT – m g
 (2  106) − (9.81  1.5  105)
 5.29  105 N
 5.3  105 N (to two significant figures)
Step 2
Use F  m a to find the acceleration.
F  ma
a
F
m
5.3  10 5
1.5  10 5
 3.52 m s−2

 3.5 m s−2 (to two significant figures)
© Oxford University Press 2015
www.oxfordsecondary.co.uk/acknowledgements
This resource sheet may have been changed from the original
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8 Newton’s laws of motion
Stretch and challenge
AQA Physics
Worked example 3
Question
A rocket of mass 2  104 kg emits fuel at a speed of 3.5  104 m s−1.
Calculate the change in velocity of the rocket when it has used up 500 kg of fuel.
Answer
increase in velocity of the rocket  velocity of gases  ln
 (3.5  104)  ln
initial mass of rocket
final mass of rocket
2  10 4
(2  10 4 ) – (5  10 2 )
 (3.5  104)  ln (1.03)
 8.86  102 m s−1
 8.9  102 m s−1 (to two significant figures)
Questions
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A rocket engine ejects 200 kg of hot gases at a speed of 3000 m s−1 and a rate of
5.0 kg s−1. Calculate the thrust the engine exerts on the rocket.
At lift-off, the mass of a rocket is 2.45  106 kg, while at burn-out, all of the
rocket’s fuel is spent and its mass has fallen to 7.5  104 kg. Assuming that the
rocket’s engines developed a constant thrust of 3.3  107 N and g  9.81 m s–2
throughout, calculate
a the acceleration of the rocket at lift-off
b the acceleration of the rocket just before burn-out.
A space rocket of total mass, including fuel, of 1.9  103 kg stands on a launch
pad. On ignition, gas is ejected from the rocket at a speed of 2.4  103 m s−1 and
at a constant rate of 7.9 kg s−1.
a Calculate the thrust of the rocket.
b Draw a diagram to show the values of the forces acting on the rocket.
State if there is a delay before lift-off.
c If there is 1500 kg of fuel on board, calculate how long the fuel will last.
A spacecraft of mass 500 tonnes is powered by rockets which emit fuel at a
speed of 4000 m s−1. What is the increase in speed of the spacecraft when it has
used up
a 100 tonnes of fuel
b 300 tonnes of fuel.
A spacecraft leaves the outskirts of our Solar System with a mass of 500 tonnes
and a speed of 12 000 m s−1 and heads towards the star Alpha Centauri. It uses
up 300 tonnes of fuel, with the gases emitted at 3.5  103 m s−1, to increase its
speed, and then travels at constant speed. If Alpha Centauri is 1.37  108 light
seconds away, how long does it take the spacecraft to get there?
© Oxford University Press 2015
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This resource sheet may have been changed from the original
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