# OCR AS/A Level Physics A

# Gravitational fields

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## Introduction

### Overview

Delivery guides are designed to represent a body of knowledge about teaching a particular topic and contain:

- Curriculum Content: A clear outline of the content covered by the delivery guide
- Thinking Conceptually: Expert guidance on the key concepts involved, common difficulties students may have, approaches to teaching that can help students understand these concepts and how this topic links conceptually to other areas of the subject
- Thinking Contextually: A range of suggested teaching activities using a variety of themes so that different activities can be selected which best suit particular classes, learning styles or teaching approaches.

## Curriculum Content

### Overview

**Content (from A Level)**

**5.4 Gravitational fields**

This section provides knowledge and understanding of Newton’s law of gravitation, planetary motion and gravitational potential and energy.

Newton’s law of gravitation can be used to predict the motion of orbiting satellites, planets and even why some objects in our Solar system have very little atmosphere with the opportunity to analyse evidence and look at causal relationships (HSW1,2,5,7).

Geostationary satellites have done much to improve telecommunications around the world. They are expensive; governments and industry have to make difficult decisions when building new ones. Learners have the opportunity to discuss the societal benefits of satellites and the risks they pose when accidents do occur (HSW9,10).

**5.4.1 Point and spherical masses**

**(a)** gravitational fields are due to objects having mass

**(b)** modelling the mass of a spherical object as a point mass at its centre

**(c)** gravitational field lines to map gravitational fields

**(d)** gravitational field strength; \(\displaystyle g=\frac{F}{m}\)

**(e)** the concept of gravitational fields as being one of a number of forms of field giving rise to a force.

**5.4.2 Newton's law of gravitation**

**(a)** Newton's law of gravitation; \(\displaystyle F=\textbf{–}\frac{GMm}{r^2}\) for the force between two masses

**(b)** gravitational field strength \(\displaystyle g=\textbf{–}\frac{GM}{r^2}\) for a point mass

**(c)** gravitational field strength is uniform close to the surface of the Earth and numerically equal to the acceleration of free fall.

**5.4.3 Planetary motion**

**(a)** Kepler’s three laws of planetary motion

**(b)** the centripetal force on a planet is provided by the gravitational force between it and the Sun

**(c)** the equation \(\displaystyle T^2=\left(\frac{4\pi^2}{GM}\right)r^3\)

**(d)** the relationship for Kepler's third law \(\displaystyle T^2 \propto r^3\) applied to systems other than our solar system

**(e)** geostationary orbit; uses of geostationary satellites.

**5.4.4 Gravitational potential and energy**

**(a)** gravitational potential at a point as the work done in bringing unit mass from infinity to the point; gravitational potential is zero at infinity

**(b)** gravitational potential \(\displaystyle V_g=\textbf{–}\frac{GM}{r}\) at a distance \(\displaystyle r\) from a point mass \(\displaystyle M\);

changes in gravitational potential

**(c)** force–distance graph for a point or spherical mass; work done is area under graph

**(d)** gravitational potential energy \(\displaystyle E=mV_g=\textbf{–}\frac{GMm}{r}\) at a distance \(\displaystyle r\) from a point mass \(\displaystyle M\)

**(e)** escape velocity.

## Thinking Conceptually

### Overview

**Approaches to teaching the content**

What actually is a ‘gravitational field’? Students should be aware of the historical development of the concept, which can show how ideas are formed, developed and rejected as experimental evidence becomes available.

If we start with the idea of a field, which can be thought of as a volume of space in which one object is affected by a force arising from a particular property of another object, then what is the ‘particular property’ of the objects that give rise to the force?

In the case of gravitational fields, this property is mass. The actual nature of mass itself is an area of active research, and a complex treatment is beyond this specification, but more able students (and their teachers) are directed towards the work of Professor Peter Higgs and others that resulted in the award of the 2013 Nobel Prize in Physics.

The idea of ‘amount’ is as old as civilisation itself. Aristotle held that the downward motion of objects was due to their ‘nature’, which caused them to move towards their ‘natural place’. There was therefore some form of *gravitas*, or ‘heaviness’ associated with objects, and this is where the name ‘gravity’ comes from. Brahmagupta, the Indian mathematician and philosopher, suggested some 1400 years ago that the Earth was spherical and that it attracted things, and Islamic philosophers such as al-Hamdani and al-Biruni took these ideas further around three hundred years later.

Some 500 years ago, Galileo is popularly considered to have disproved the theory of Aristotle in his famous experiment at the Leaning Tower of Pisa, where he showed that objects of different mass fall at the same rate. Whether he actually carried out this experiment in Pisa or not is a matter of debate, but something similar to it was definitely carried out in Delft (in what is now The Netherlands) in 1586 by Simon Stevin. It was shown that objects fall at a rate independently of their mass as long as the effects of air resistance are neglected, and so Aristotle’s ideas needed replacement.

Galileo showed that all objects fall at the same rate towards the Earth, and Newton took this work further when he published his Law of Gravitation, along with his Three Laws of Motion, in the *Principia* of 1687. He wasn’t the first to observe the idea of an inverse square law linking gravitational force and distance, as Hooke (and others) had observed this a quarter of a century before, but by now, the idea of ‘action at a distance’ was firmly established, and Newton linked the ideas of mass and force together in his famous equation \(\displaystyle F = GM_1M_2/r^2\).

The concept of gravitational fields existing around massive objects was now a fundamental one, and it followed that any object that has mass will interact with a gravitational field around another object.

**Point and spherical masses**

On the celestial scale, where the distance between objects is very large compared to the sizes of the objects themselves, then the objects can be treated as point masses, with the mass concentrated at their centres.

The strength of the gravitational field due to the Earth decreases from a value of 9.81 N kg-1 at the surface to zero at the centre of the Earth, and students should be encouraged to find out and explain why this is so.

Lines of force arising from gravitational fields are visually represented by field lines, in much the same way as electrical and magnetic fields are shown. The lines show the direction of the force acting, and the spacing of the lines represent its strength.

Gravitational fields are radial fields, with a large point mass at the centre.

Gravitational forces act along the straight line axis between the two objects in question, and they are equal and opposite – it may come as a surprise to students that the Earth is being attracted to them just as much as they are attracted to it! The reason why they detect no motion is, of course, due to the larger mass of the Earth when compared to the student – the difference is usually of the order of 10^{22}.

So what is the value of the gravitational field strength, \(\displaystyle g\)? Students should be familiar with the idea of \(\displaystyle g\), having encountered it when considering the difference between mass and weight earlier in their study of Physics, and a value for \(\displaystyle g\) of 10 N kg^{-1} (or 10 m s^{-2}) will probably have been used in simple calculations. At A level, the accepted experimentally determined value is 9.81 N kg^{-1} (or m s^{-2}), and this value should be used in problem-solving unless indicated otherwise.

Newton’s Second Law tells us that \(\displaystyle F = \text{ma} \) and so the force acting on an object of mass \(\displaystyle \text{m} \) in a gravitational field strength \(\displaystyle g\) can be expressed as \(\displaystyle F = \text{m}g\) or \(\displaystyle g = \frac {F}{\text{m}}\).

This specification enables the student to learn about a number of forms of field, amongst them electrical, magnetic and gravitational. It is easy to demonstrate that gravitational fields are the weakest of these three on the laboratory scale, but gravitational fields act over huge distances due to the large masses of stars, planets and other objects found in space.

However, all three fields involve an inverse square law with respect to distance, and the concepts of field, field strength and potential are common to all three forms of field. Students should not therefore be deterred by the mathematical requirements of a proper study of fields, since the skills learned in the study of one field will be directly applicable to the others.

There are several quantities involved in gravitation that need to be learned. Exercise 1 is a student exercise that allows quantities and definitions to be matched, and the magnitudes of some of the numerical quantities involved are also included for completeness.

**Newton’s Law of Gravitation**

In 1687, Newton published his three Laws of Motion, along with the Law of Gravitation, in the *Principia*. In 1963, Richard Feynman described the Law of Gravitation as ‘elegantly simple’, and provided such a clear and accessible description that his treatment is still highly recommended for students and teachers some fifty years later.

However, Newton probably didn’t simply come up with the Law of Gravitation after watching an apple fall, despite the legend. He built upon the work of Nicolas Copernicus (who rediscovered that planets went around the Sun, an idea originally postulated during ancient times), that of Tycho Brahe (a Danish nobleman who kept meticulous observations of the paths of the planets and stars over several decades), the work of Johannes Kepler (who worked with Brahe at the end of the Dane’s life, and who turned his observations into three beautifully simple Laws, which are dealt with in the following section), and the degree to which he drew on the earlier work of Robert Hooke is still the subject of controversy. So he did indeed ‘see further through standing on the shoulders of giants’, even if he didn’t always acknowledge the names or credit the work of those giants.

Newton’s Law of Gravitation stated that the gravitational force between two objects is proportional to the product of their masses, and inversely proportional to the square of the distance between them (if they are considered as point masses). The essential test was that it explained the orbits of the planets as described by Kepler, which is a classic demonstration of theory following experimental data.

However, Newton had no way of actually measuring the magnitude of the forces in question, especially between objects of relatively small mass on Earth, and he certainly had no way of measuring \(\displaystyle G\), the Universal Gravitational Constant.

Measurement of \(\displaystyle G\) by Cavendish came a century later in one of the most delicate and elegant experiments ever undertaken, and a study of this experiment by students is highly recommended so that they may appreciate the technical brilliance and painstaking attention to detail shown by Cavendish using the laboratory equipment available at the end of the eighteenth century.

Following Cavendish’s determination of \(\displaystyle G\), confirmation of the value of the gravitational field strength \(\displaystyle g\) then became possible. The value of \(\displaystyle g\) can be estimated by many experimental methods, such as by free fall, by measuring the oscillations of a pendulum of known length and by the study of the motion of projectiles, and some of these methods are referenced in the Thinking Contextually section so that students may use them to measure their own values of \(\displaystyle g\).

**Planetary motion**

Tycho Brahe had spent decades recording the positions of the known planets, although an explanation of their motion eluded him. The Ancients had named them ‘planets’ after the Greek word for ‘wanderer’ for good reason – their motion in the heavens was impossible to explain using the models of antiquity, and a very complex system of celestial spheres had to be continually refined to take account of their motion.

Johannes Kepler was able to make sense of the motion of the planets using the large amount of data produced by Tycho Brahe. Copernicus had postulated that the planets orbit the Sun in 1543, and Tycho Brahe mapped the planets positions over a quarter of a century from 1576. Kepler took nearly a decade to analyse the planetary motion data, and his findings rank amongst the most important ever reported. After finally disentangling the retrograde motion of Mars, he postulated the first two of the three Laws that now bear his name.

The First Law is that the planets move in elliptical orbits with the Sun at one focus of the ellipse. Feynman lucidly describes how to form an ellipse using two pencils and string, and this exercise is highly recommended for students.

The Second Law is that as a planet moves along its elliptical path, it sweeps out an equal area in an equal time. This showed that planets move more quickly when nearer the Sun than when further away, and is ultimately due to a faster acceleration when it is nearer the Sun’s large mass. This showed that the centripetal acceleration of the planet is provided by the gravitational force between it and the Sun.

The first two of Kepler’s Laws dealt with the motion of a single planet and the Sun. His Third Law related the square of the time taken to cover one elliptical orbit and the cube of the radius of that orbit, and the two bodies involved do not have to include the Sun.

The equation can be simplified as \(\displaystyle T^2 = \text{k }r^3\), with \(\displaystyle \text{k}=\left(\frac{4\pi^2}{GM}\right)\), where \(\displaystyle M\) is the mass of the larger object.

This Law can not only be used for orbits of planets around the Sun, but also of the Moon around the Earth, satellites around the Earth (and other planets) and in studies of exoplanets around distant stars, which is why the ESA Kepler mission was so aptly named. Interestingly, Kepler had no idea why his Laws worked, just that they *did*.

An example of the application of Kepler’s Third Law is vital for the placement of geostationary satellites. If \(\displaystyle T\) is to be 24 hours (or 86 400 seconds), the value of \(\displaystyle G\) has been established and the mass \(\displaystyle M\) of the Earth is known (6.0 x 10^{24} kg is a generally accepted value), then the value of \(\displaystyle r\) is easily calculated, and it is recommended that students do it. This of course assumes that the Earth is a point mass, and the distance \(\displaystyle r\) will be from the centre of the Earth, not its surface.

Geostationary satellites are crucial to everyday life in the early 21st century, as they enable reliable communication (such as the Astra cluster for TV, Inmarsat and Eutelsat for navigation and telephone/data communications), accurate determination of position on the Earth’s surface (through the GPS and Galileo systems), disaster evaluation and response planning (such as the UK-coordinated DMC system) and the real-time monitoring of conditions on the Earth’s surface (such as Meteosat).

A study of one or more of these areas by Learners will provide them with a better understanding of the technological basis of many space-based services that are of great value to today’s (and tomorrow’s) society. A study of how GPS systems work is suggested in **PAG 12** of the specification. Such a study could interest the Learner in a career in space science and technology, which is a major employer in many countries.

**Gravitational potential and energy**

The concept of gravitational potential energy should be familiar to students from earlier study of the energy changes of falling objects. A short refresher may however be required before the related Altering the position of a mass in a gravitational field from a position with one value of \(\displaystyle g\) to another position with a different value of \(\displaystyle g\) requires energy to be transferred, and so work will be done. This is a useful concept in calculating the energy transfers required to change the orbital altitude of satellites, but it is particularly important in concept of gravitational potential is taught.

Altering the position of a mass in a gravitational field from a position with one value of \(\displaystyle g\) to another position with a different value of \(\displaystyle g\) requires energy to be transferred, and so work will be done. This is a useful concept in calculating the energy transfers required to change the orbital altitude of satellites, but it is particularly important in calculating the escape velocity required for a body to escape from the influence of an object’s gravitational field.

There is therefore a need for the concept of gravitational potential at a point, which is defined as the work done (or energy transferred) in bringing unit mass from infinity to that point, and it is independent of the path followed (it may be straight or curved).

The gravitational potential at infinity is defined as zero. The gravitational potential \(\displaystyle V_g\) of an object of unit mass at a distance r from a point mass \(\displaystyle M\) is given by the equation \(\displaystyle V_g = - \frac{GM}{r}\). It is negative because a gravitational force can only be attractive.

As the value of \(\displaystyle V_g\) reduces, then the force required to move an object against the gravitational field will also reduce. If a force-distance graph is plotted for such a process, then the work done in the process of moving the object will be the area under such a graph (in terms of units, force x distance = \(\displaystyle N\) x \(\displaystyle \text{m}\), = \(\displaystyle \text{J}\)).

The student can now review their knowledge of the idea of gravitational energy. At earlier levels it may be expressed as GPE = weight x height, but this can now be extended to GPE = mass x gravitational potential, i.e. \(\displaystyle E_p = mV_g = - \frac{GM}{r}\) for an object of mass \(\displaystyle m\) at a distance \(\displaystyle r\) from a larger object of mass \(\displaystyle M\).

The escape speed of a projectile, such as a satellite, can be calculated using the concept of gravitational potential and GPE. An object launched will follow a parabolic path – this should be familiar from earlier study. The faster the object is launched for a given angle of projection, the further away from the starting point it will land. Eventually, a launch speed is reached where the curve of the parabolic path is equal to the curve of the planet, so that the Earth’s surface effectively falls away at the same rate as the object falls, and so the object will then orbit the Earth.

If air resistance is negligible, then the initial speed needed for a body to escape from the surface of a spherical mass \(\displaystyle M\) with radius \(\displaystyle r\) is equal to ** v_{1}** \(\displaystyle = \sqrt\frac{2GM}{r}\).

With this equation in mind, ** v_{1}** can be calculated for Earth at approximately 1.12 x 10

^{4}m s

^{-1}. It also explains the positioning of satellite launch sites near to the Equator (e.g. the ESA facility in French Guiana), since the speed of an object at the Earth’s surface at the Equator is already some 74 m s

^{-1}before launch, and Learners should be asked to calculate this figure.

It also explains the multi-stage design of rockets that carry large loads into orbit, since it is only the load that needs to achieve the required escape speed, not the carrier rocket, and each stage can be used to progressively accelerate the load until it achieves the required escape velocity.

Explanation of this could be used as a research exercise for students. The calculation of escape speeds from other planets is suggested as an activity in HSW2.

**Common misconceptions or difficulties students may have.**

It is a common student misconception that the strength of the Earth’s gravitational field increases with depth, and this misconception could arise because of the correct notion that if it decreases with increasing height, then surely it must increase with depth.

Students should be familiar with the difference between mass and weight, as this is covered at lower level study. However, a quick reminder will be of value, since understanding the difference is an important foundation of the study of gravitational fields.

Students may have the misconception that the value of \(\displaystyle g\) drops rapidly above the surface of the Earth until it becomes zero at the altitude of orbiting satellites (resulting in the erroneous belief that they orbit because they have escaped from the Earth’s gravitational field), but it does not and this misconception needs to be addressed as soon as possible.

The value of \(\displaystyle g\) at the Earth’s surface can be calculated using values for the mass of the Earth of 6.0 x 10^{24} kg, \(\displaystyle G\) = 6.67 x 10^{-11} Nm^{-2}s^{-2} and r = 6.38 x 10^{6} m, and it is easy to show that this does not change greatly for low Earth orbit altitudes of a few hundred kilometres. It is a useful exercise for students to calculate the value of \(\displaystyle g\) at various distances from the surface so that they can understand this, and Exercise 2 has been provided for this purpose.

The units of \(\displaystyle g\) is a potential source of confusion for students. The two units are \(\displaystyle N\) kg^{-1} (this should be familiar from a treatment of weight and mass) and m s^{-2} (from a consideration of F = m a). The numerical value for each is the same, namely 9.81. Students should be encouraged to prove the equivalence of the two units for \(\displaystyle g\).

Learners are very likely to come into the course with the misconception that orbits are circular. Some time will be needed on the teaching of Kepler’s Laws to address this misconception, and an exercise in drawing ellipses is strongly advised.

**Conceptual links to other areas of the specification – useful ways to approach this topic to set students up for topics later in the course****e.g. links to circular motion, energy, kinematics astrophysics and cosmology.**

Circular motion is closely related to a study of gravitational fields, since the motion of a bung on a string when (safely) whirled above the head is a useful introductory analogy to planets orbiting the Sun. There must be a force acting perpendicularly to the direction of the planet’s travel, but there is no string connecting the Earth to the Sun. This should initiate a discussion about ideas of force and ‘gravity’.

‘Energy’ itself is a concept which some students find difficult throughout their learning careers. It is recommended that they are made aware of the explanation given by Richard Feynman (in printed and video form) to help them increase their understanding of the crucial and central idea of energy.

Further study of astrophysics and cosmology will require a firm understanding of the Law of Gravitation. Calculating escape velocities from different planets is a suggested activity in HSW2. A deeper study of black holes requires an understanding of escape speed, and a study of Einstein’s modifications to include relativity requires an understanding of mass-energy equivalence as well as Newton’s Law of Gravitation itself.

As in all parts of the course dealing with the collection and analysis of experimental data, the terms ‘range’, ‘uncertainty’ and ‘percentage uncertainty’ can cause confusion, as can the rules for the correct combination of percentage uncertainties from measured quantities into a single percentage figure for a calculated quantity. There may also be potential issues with the correct plotting of data, the correct calculation of the gradient of a graph, and the expression of the correct unit for the gradient of a graph. The OCR Practical Skills Handbook available on OCR Interchange in the Science Coordinator materials section gives further guidance on these and many other matters.

Newton’s ‘*Principia*’ of 1687 is rather inaccessible by today’s standards, but Richard Feynman’s treatment of the theory of gravitation, given in chapter 5 of ‘Six Easy Pieces’, Penguin Books, 1998 (ISBN 978-0-140-27666-4) is highly recommended for all Learners and their teachers.

Feynman’s descriptions of energy are also very clear and explain a conceptually difficult topic – these are in chapter 2 of the above book, on the Calfornia Institute of Technology website and on YouTube.

Professor Peter Higgs’ work on mass, that resulted in a share of the 2013 Nobel Prize in Physics, is beyond the scope of this specification, but more able Learners will gain from the description given here.

A good description of the development of the theory of gravitation, starting from Aristotle, is given here.

The websites at NASA and ESA contain much information on current space science, including the background to GPS systems. The European Space Education Resource Office (ESERO) website also has much material of use of teachers and Learners and can be found here.

## Thinking Contextually

### Overview

After a brief introduction to the concept of gravity and the terminology involved (Learner Resource - Gravitational Exercise 1 will help new learners with this), the best place to embark on a learning journey about gravitational fields is possibly to envisage what would happen if there were no gravitational field (answers are provided by Learner Resource - Gravitational Exercise 1: Answers - see 'Whole topic words' activity below).

The presence of a gravitational field can be demonstrated using falling objects, and it may come as a surprise to Learners that objects of different masses will fall at the same rate if the effects of air resistance are ignored – the ‘guinea and feather’ experiment will prove the point in a memorable manner.

The Learner can then move on to the concept of a gravitational field, where lines are used to map the direction and strength of forces. The strength of the gravitational field is given by \(\displaystyle g\), and the experimental determination of g can be undertaken using free fall experiments (as described in the PAG practicals) and using the oscillation of a pendulum. Measuring \(\displaystyle g\) using observations of the time period \(\displaystyle T\) of a simple pendulum of length l, followed by use of the equation \(\displaystyle T = 2\pi\sqrt\frac{l}{g}\) will give the Learner much useful experience in the recording of results, presentation of the data in a graphical form and in the analysis of experimental uncertainties.

Depending on the material already covered in the course, the concept of a gravitational field can be compared to electric and magnetic fields. Learners should be aware that the mathematical treatment of all these fields is broadly similar, since they all follow inverse square laws with respect to distances between objects.

The Learner should then move on to Newton’s Law of Gravitation. The equation itself may seem daunting at first, but use of a visual simulation may help show that the force between two objects depends on the product of their masses and is inversely proportional to the square of the distance between them. It is very important that Learners can manipulate the equation correctly and confidently, and use the correct units throughout.

Combination of Newton’s Law of Gravitation with his Second Law yields an expression for the value of \(\displaystyle g\), namely \(\displaystyle g = - \frac{GM}{r^2}\). It is important that the negative sign is included, as gravitational forces are always attractive. Learners should be able to work out the SI units of \(\displaystyle g\) from this equation, and should also be able to show that 1 N kg^{-1} is equal to 1 ms^{-2}.

Learner Resource - Gravitational Excercise 2 (see activity below) allows learners to calculate \(\displaystyle g\) at the surface of the Earth, and for a number of different altitudes above the Earth (the answers are provided by Learner Resource - Gravitational Exercise 2: Answers). A plot of \(\displaystyle g\) against \(\displaystyle r\) will yield a characteristic curve, and one of \(\displaystyle g\) against \(\displaystyle \frac{1}{r^2}\) should give a straight line, and Learners should be encouraged to explain the shape of the curve and gradient of the straight line with reference to Newton’s law of Gravitation.

Cavendish’s experiment to measure \(\displaystyle G\) is one which is elegant and yet painstakingly detailed, and Learners should study it to understand the complexities of measuring very small effects using simple techniques.

Kepler’s Three Laws may then be studied, and these are at the heart of orbit theory and practice. Learners should be familiar with the form of an ellipse, and they should create one using the simple technique described by Feynman. Simulations of solar system construction and orbits are very useful tools to visualise the forms of orbits, and Learners will enjoy testing their skills and understanding using these.

Calculation of the positioning of geostationary satellites using Kepler’s Third Law will lead the learner into understanding the uses of such satellites, and a research project into GPS satellites and their utility in today’s society will enhance their knowledge of the use of Physics to serve communication and observation needs. Materials are available from NASA and ESA which act as a starting point for such research.

The Learner then moves on to the last part of the study of gravitational fields in this specification, which is also conceptually the most difficult part of this section. The definition of gravitational potential is important, and the concept of energy transfer (and therefore work done) in bringing a unit mass from infinity (where the gravitational potential is, by definition, zero) to a certain point is important. The equation for gravitational potential, \(\displaystyle V_g=\textbf{–}\frac{GM}{r}\), is sometimes confused with that for g ( \(\displaystyle g = - \frac{GM}{r^2}\) ), and care must be taken to differentiate between the two. A force-distance graph for a unit mass at varying distances point or spherical mass should be plotted so that Learners can recognise its characteristic shape, and Learners must also recognise that the area underneath such a graph is equivalent to work done.

Learners will recognise gravitational potential energy, \(\displaystyle E_p\), as equal to weight x height from previous study, and this should be developed to \(\displaystyle E = mV_g = - \frac{GMm}{r}\) at a distance \(\displaystyle r\) from a point mass \(\displaystyle M\).

Finally, the calculation of escape velocity from various objects should be covered as an exercise (HSW2), and to extend the learner’s knowledge of the factors that affect the launch of satellites and other missions into space. A good place to start is visualising an object fired from a cannon on a tower – it’s clear from previous work on projectiles that the faster the muzzle velocity is, the larger the range of the projectile, but how quickly does it need to be fired so that its curved path matches the curvature of the Earth? This will introduce Learners to the idea of escape velocity.

Learners could consider how the escape velocity from the Moon and/or Mars differs from that from Earth, and what effect this could have on a multi-step manned mission to Mars (i.e. construction of spacecraft in Moon orbit from parts launched from Earth, then smaller escape velocity needed to boost craft towards Mars).

### Whole topic words

### Gravitational exercise 2

*g*at the surface of the earth and for a number of different altitudes above the Earth.

### Gravity: TAP, Nuffield and YouTube video clips

### What is gravity?

### The guinea and feather experiment

### Mapping gravitational fields

### Measurement of g using free fall

*g*using free fall is described in the PAG practicals and descriptions of the method may be found on the Nuffield Foundation's website (see links 'Investigating free fall with a light gate' and 'Multiflash photographs of free fall').

### Measuring g from a pendulum

*g*using a simple pendulum is straightforward and is described on the Nuffield Foundation's website (see link 'Investigation of a simple pendulum').

### Newton's law of gravitation

### Cavendish's determination of G

### Apollo 11 mission data

### Kepler's laws of planetary motion

Descriptions of Kepler’s laws of planetary motion can be found on the following websites:

YouTube ('Kepler's 1st law')

The making of an ellipse is described here and by Feynman (see previous section).

### Orbits and solar systems

### Geostationary satellite use

NASA, ESA and ESERO provide materials for starting research into geostationary satellite use.

ESERO - European Space Agency Resource Office

Also, there are ESA satellite imaging resources at their teachers' corner (see link 'Earth and the environment').

ESA for kids has some useful information and activities for all ages.

### Gravitational potential energy

### Potential gradient

### Firing a cannon to put an projectile into orbit

### Escape velocity calculation and simulation

## Acknowledgements

### Overview

OCR’s resources are provided to support the teaching of OCR specifications, but in no way constitute an endorsed teaching method that is required by the Board and the decision to use them lies with the individual teacher. Whilst every effort is made to ensure the accuracy of the content, OCR cannot be held responsible for any errors or omissions within these resources. We update our resources on a regular basis, so please check the OCR website to ensure you have the most up to date version.

© OCR 2015 - This resource may be freely copied and distributed, as long as the OCR logo and this message remain intact and OCR is acknowledged as the originator of this work.