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### Number and algebra

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# Model Solutions

### Why do this problem?

A firm understanding of the modelling assumptions made in a
mechanics problem is important if mechanics is to go beyond a set
of pointless technical manipulations. This
problem will allow learners to consider the effects of
modelling assumptions from an intuitive perspective and it is ideal
for use either at the start of a mechanics course, where the
discussion can be more intuitive, or towards the end when students
will be able to back up some of the discussion with the use of
equations. In the latter case, it shows that beyond a certain
point, equations become difficult to solve and highlights the sorts
of problems that will be encountered at university level.
### Possible approach

### Key questions

### Possible extension

### Possible support

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Age 16 to 18

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

This is an ideal problem for discussion, with the goal of
drawing students into the understanding that some effects retard
the motion and some effects enhance the motion. It will also show
how equations of motion are constructed based on the modelling
assumptions and Newton's 2nd law of motion. This will give insights
into the structure of the resulting differential equations.

It is important to stress that a REAL shot put will definitely
follow SOME path: the task of the modeller is to formulate
equations which take us as close to that path as possible and to
understand the circumstances in which our idealised path is likely
to most differ from the reality.

Key is the concept that the equation of motion for an object
can be constructed from the forces which act on it. Some of these
forces are constant, some vary, some depend on the velocity, cross
section and so on.

Depending on the technical skill of the students, equations of
motion could be created for each case, with suggestions from the
class as to the best way to model friction or drag. Solution of
these equations is very difficult for all but the simplest of
cases.
Our article on modelling assumptions might well be of interest
to those keen on persuing applied mathematics in some guise at
university.

The last part of the question involving ideal world-record
conditions might well lead into other questions of modelling
assumptions; creativity in this area should be encouraged.

How might each of the effects either retard or enhance the
motion?

Which effects feel most important to you?

Are there any cases where you could definitely construct an
equation of motion?

How might you model each of the effects in an equation?

Work out some figures for the first case and then make
estimates for some of the others.

Think of another situation in mechanics and consider creating
a similar set of modelling assumptions. Share with friends to
determine in which order the different situations would come.

Some very mathematically competent students initially find
concepts of mathematical modelling stressful because they are
perceived as vague or woolly. Encourage such students to leave
their comfort zone and try to impose structure on the modelling
context by realising that statements such as IF (the following
assumptions are made) AND (I take as an axiom the following
physical laws) THEN (the following mathematical conclusions follow)
are clear and unambiguous.

Encourage all students to use their common sense.

A ball whooshes down a slide and hits another ball which flies off the slide horizontally as a projectile. How far does it go?

How fast would you have to throw a ball upwards so that it would never land?