Related Collections

Mechanics Mapping Document

Ball Bearings

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

Overarch 2

Bricks are 20cm long and 10cm high. How high could an arch be built without mortar on a flat horizontal surface, to overhang by 1 metre? How big an overhang is it possible to make like this?

Cushion Ball

The shortest path between any two points on a snooker table is the straight line between them but what if the ball must bounce off one wall, or 2 walls, or 3 walls?

Model Solutions

Age 16 to 18
Challenge Level

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

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.

Key questions

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?

Possible extension

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.

Possible support

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.