MST209_5 Mathematical methods and models

Modelling with first order differential equations

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978-1-4730-1564-7 (.kdl)
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Contents

- Introduction

- Learning outcomes

- 1 Modelling with first order differential equations

- Conclusion

- Keep on learning

- Acknowledgements

Introduction

This course lays the foundations of Newtonian mechanics and in particular the procedure for solving dynamics problems. The
prerequisite skills needed for this course are the ability to solve first- and second-order differential equations, a knowledge
of vectors, and an understanding of the concept of a force.

This OpenLearn course provides a sample of level 2 study in Mathematics.

Learning outcomes

After studying this course, you should be able to:

- understand and use the basic terms for the description of the motion of particles: position, velocity and acceleration

- understand, use and differentiate vector functions

- understand the fundamental laws of Newtonian mechanics

- solve mechanics problems in one dimension by drawing a sketch, choosing a suitable x-axis and origin, drawing a force diagram,
applying Newton’s second law, taking the x-component, and making suitable substitutions

- solve mechanics problems in one dimension that involve one or more of the forces of gravity, friction, air resistance and
water resistance.

1 Modelling with first order differential equations

The main teaching text of this course is provided in the workbook below. The answers to the exercises that you'll find throughout
the workbook are given in the answer book. You can access it by clicking on the link under the workbook. When prompted after
exercise 2.2 to watch the video for this course, return to this page and watch the four clips below. After you've watched
the clips, return to the workbook.

Click the link below to open the workbook (PDF, 1.0 MB).

Workbook

Click the link below to open the answerbook (PDF, 0.4 MB).

Answer book

Workbook contents

- Introduction

- 1 Describing motion

- 1.1 The motion of a particle

- 1.2 One-dimensional motion

- 2 A theory of motion

- 3 Predicting motion

- 4 Some more force models

- 4.1 Friction

- 4.2 Air resistance

- 4.3 Water resistance

Click play on the blank screen below to watch the low resolution version of 'A question of Newton', or 'Launch in a separate
player' for the high resolution version.

Video content is not available in this format.

A question of Newton - Part 1

View transcript - A question of Newton - Part 1

Click play or 'Launch in a separate window' to continue watching 'A question of Newton'.

Video content is not available in this format.

A question of Newton - Part 2

View transcript - A question of Newton - Part 2

Click play or 'Launch in a separate window' to see the third part of the video.

Video content is not available in this format.

A question of Newton - Part 3

View transcript - A question of Newton - Part 3

Click play or 'Launch in a separate window' to watch the final part of the video.

Video content is not available in this format.

A question of Newton - Part 4

View transcript - A question of Newton - Part 4

Conclusion

This free course provided an introduction to studying Mathematics. It took you through a series of exercises designed to develop
your approach to study and learning at a distance and helped to improve your confidence as an independent learner.

Keep on learning

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Acknowledgements

The content acknowledged below is Proprietary (see terms and conditions) and is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 Licence

Course image: ▓▒░ TORLEY ░▒▓ in Flickr made available under Creative Commons Attribution-ShareAlike 2.0 Licence.

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A question of Newton - Part 1

Transcript

MARK NEWLAND-SMITH (pilot)

We1re ready to go, let's load up the bombs.

JOY MANNERS

Test your intuition about motion on this problem. Sometimes gliding enthusiasts practise flour bombing an airfield target.
First they have to load up with flour, and then get pulled up into the sky to become airborne. There's little wind around
to affect the path or'the flour bomb once it's been dropped, so the question is at what point should the flour bomb be released
to hit a particular target. Many peoples intuition suggests releasing the flour bomb when the glider is almost directly over
the target, but that doesn't work. In fact, you have to release the flour bomb well before the glider is over the target.

MARK NEWLAND-SMITH

... 3 ... 2 .... 1 ... drop.

JOY MANNERS

Let's see that again. If you consider the case of a glider travelling at 50 kph at a height of say 50m, then the flour bomb
would have to be released at a point at least 40m before the target.

MARK NEWLAND-SMITH

... 3 ... 2 ... 1 ... drop.

JOY MANNERS

Our intuition about motion can easily let us down. Here's a simple experiment. This pink beanbag is twice as heavy as each
of these two blue ones. So, if l take the heavy beanbag and one of the light ones, and let them fall from the same height
at the same moment, which do you think will hit the ground first. [DROPS BAGS] Well, contrary to what intuition might suggest,
they both hit the ground at the same time. Possibly that surprised you. Many of our intuitive ideas about motion are wrong,
and yet they were embodied in the theories of one of the greatest thinkers in history, Aristotle. Aristotle lived in Greece
in the 4th Century BC. He was a pupil of Plato, and tutor to Alexander the Great.

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A question of Newton - Part 2

Transcript

JOY MANNERS

Aristotle thought it was obvious that a stone dropped from a moving chariot would fall behind the chariot. From this he deduced
that the Earth is not moving, because he knew that a stone thrown straight up in the air wouldn't get left behind. He believed
that he had· no need to experiment, because logical thought would lead him to the correct conclusions. However, the modem
scientific method holds the view that all theory must be tested by experiment, and we know that the Earth is rotating. Aristotle's
deduction was wrong, because it was based on a false premise. Ifhe had been prepared to experiment he would have seen that
he was wrong. Well here's something Aristotle might have found interesting. I've got a small cannon here, with a firing mechanism
which can fire a small ball like this vertically upwards.. So I'll just load it into the cannon, and when I fire the mechanism
with this remote switch ..... the ball travels straight up and down landing back at the cannon. Now, what would happen to
the path the ball takes if the cannon were moving at a constant speed when it's fired? Well in fact my cannon is sitting on
a device called a linear air track, which enables it to glide on a cushion of air. If I push the glider... it maintains a
constant velocity down the track. To perform the experiment I'll set the glider in motion, and remotely fire the cannon. What
do you think will happen? So Aristotle was wrong. The ball still lands by the cannon. And the same thing can happen on the
moving Earth. I can throw a ball up, and still catch it! Aristotle provided cogent and persuasive arguments,. but the experiment
shows that one of his basic premises was wrong.

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A question of Newton - Part 3

Transcript

JOY MANNERS

Well, Galileo was fairly convincing in his argument that the ball rolling on a horizontal friction free surface, 'as smooth
as polished silver' he said, that the ball would neither speed up, nor slow down, and must therefore maintain a constant speed.
What he would have liked, would have been to do the experiment with a truly frictionless surface. Well in fact, our linear
air track gives a very good approximation to friction free motion. The glider is supported on a cushion of air, which is forced
through these tiny holes. So, when it moves along, no solid surfaces rub together. It's better even than a puck sliding over
smooth ice. To measure the velocity at various points, we have some timers like this along the track. Each timer is connected
to a light beam, which passes between a pair of posts that hang down on either side of the track. The timing begins - and
ends - every time the beam of light is broken.

So, when a card like this is attached to a glider, as it moves past the timer, the first leading edge will cut the beam and
the timing commences. And the next leading edge, which is positioned IO cm along, cuts the beam again to stop the timing.
In other words, the time measured is the time that the glider takes to travel a distance of 0 .. 1 meters, and so for a measured
time 't', the velocity is given by (0 .. 1 )/t metres per second, assuming that the velocity is constant over that short distance.
So let's see what happens to the velocity along the whole length of the track. If the glider moves at constant velocity, each
of the timers will show the same time. Pretty good agreement to an accuracy of a 100th of a second. Well in case you think
there's a slope which is compensating for friction, I'll repeat the experiment in the other direction, and check the constant
velocity again . . 19, .18, and .19.

So again we get very good agreement. So this experiment confinns what Galileo thought - that an object moving horizontally
with no resultant force on it will move at a constant velocity. But the story's not yet complete. Galileo has rightly separated
motion into its horizontal and vertical components. But this was his difficulty. The concepts of horizontal and vertical are
intimately tied up with the fact of standing on the surface of the Earth. So Galileo was unable to extend his understanding
to the motion of the planets .. What was needed, was a synthesis to understand motion in any circumstances. The man who knitted
these ideas together into a coherent theory was born in 1642, the year that Galileo died and his name was Isaac Newton.

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A question of Newton - Part 4

Transcript

JOY MANNERS

So Newton formulated what are quite general laws of motion. His First Law - that a body will continue in its state of rest
or uniform motion in a straight line, unless acted upon by an external force - is actually what we demonstrated earlier on
with the air track. In our case, the downward force of gravity is exactly compensated by the upward force of the air, and
there aren't any horizontal fore.es after the initial push, because it's essentially friction free, and this track isn't long
enough to run into trouble with the curvature of the Earth. How can we test Newton's second law?

Does a constant force produce a constant acceleration? Many experiments can be performed with a linear air track to verify
this. If you give the air track a gradient, then gravity will provide a constant force down the track. In this case, the 4
meter track has been raised at one end by a couple of centimetres. Here's the motion of the glider down the slope.

There are six timings, t1 up to t6

Remember, the velocity V at each ti~ng point can be calculated given the time 't' that the glider records on the timer. Using
this equation, the velocity v1 at the first timer, is 0.23 meters per second, v2 becomes 0.36 meters per second - and so on.
By plotting the velocities against the timing to the experiment that each velocity was measured, you'll see how the velocity
of the glider changes with time. Let's replay the experiment in slow motion, and plot the values ofv1 to Vo. From release,
it takes around 3 seconds for the glider to reach the first timing point, where it's velocity is 0.23 meters per second.

Then the second timing comes here ... and soon. So the graph of velocity against time is a straight line. In summary, what
you've seen is that a constant force produces a constant acceleration. In fact with more experimentation we would show that
the component of force along the track, is equal to the mass times the component of acceleration along the track. But remember
the cannon experiment.

That showed that the components of the motion can be treated separately. There, the horizontal motion of the ball was unaffected
by its vertical acceleration. It maintains the same horizontal velocity as the cannon. So a second component would satisfy
the same form of the equation ... as would a third component. These scalar equations can be amalgamated into one vector equation.
More usually written as vector F equals M times vector A.

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