Model Solutions

Age

16 to 18

Challenge level

How do these modelling assumption affect the solutions?

Thinking mathematically tags

Mathematical mindsets tags

Being curious Being resourceful Being resilient Being collaborative

Problem

A shot-putter puts her 4kg shot at some fixed angle and a known, fixed speed; it flies through the air and lands.

The motion is modelled several times according to each of the following assumptions:

g is constant, there is no air resistance, the shot is modelled as a point mass.
g is constant, it is a still day with low humidity and the radius of the physical shot is taken into consideration.
g is constant, it is a misty, still day and the radius of the physical shot is taken into consideration.
g is variable, there is no air resistance and the radius of the physical shot is taken into consideration.
g is variable, there is no air resistance, the shot is a point mass.
g is constant, it is quite blustery and the radius of the physical shot is taken into consideration.

In each case it is assumed that the ground is flat and horizontal.

Based on these modelling assumptions, can you put into order the distances travelled by the shot in each case before it first strikes the ground? What about the times of travel? Are there any that you cannot put in order without more information?

Can you think of any other modelling assumptions which might affect the results?

If you wanted to break the world record for the shot put, in what conditions would you try to do it?

Does the relative importance of these modelling assumptions change for striking a golf ball or hitting a table tennis ball?

NOTES AND BACKGROUND

Mechanics is all about things moving. In reality, any moving object is subject to a bewildering complexity of forces and is composed of a similarly bewildering complexity of constituent pieces. So, we make modelling assumptions. Rather beautifully, these modelling assumptions can reduce aspects of physics to a very simple set of equations which nevertheless produce strikingly accurate and predictive results. Knowing which modelling assumptions are safe and reasonable to make is a skill, and to develop this skill we need to understand how modelling assumptions impact the solution to a problem.

Read our article on modelling assumptions if you are interested in finding out more mathematical details.

Student Solutions

Matthew produced this fine solution

Extra information needed

Whether the shot-put always has the same initial speed or if it depends on its mass, and if so what the masses of the shots in those scenarios where it is not stated are (affects situations 4 and 6).

More details of the wind conditions - whether it is to be assumed it is of constant speed and direction, or if the velocity varies depending on position and if so (affects 6).

Order

Without this information, the following order for the range of the shot-put in the different situations can be established:

5 (greatest)

1

2

3 (least)

The following is a brief qualitative description of how the various factors affect the range or the shot-put, for a more in depth numerical analysis see the Modelling Assumptions article.

Variable g

Gravitational force follows an inverse square relationship with separation, therefore assuming constant g is defined as that at ground level, a object subject to variable g will travel very slightly further than one under constant g (and all other conditions the same) as the accelerations due to gravity during its motion will be slightly lower. Over the heights that could be achievably reached by a shot-put in reality, the difference would be so small it would be impossible to measure.

Air resistance

Air resistance always acts to oppose motion and so will slow both the horizontal and vertical speeds of the shot-put, at the velocities likely to be encountered here the drag force being modelled as proportional to the relative velocity between the object and fluid (Stokes' drag). Although the actual mathematics involved in showing how this will affect the overall range is quite complicated (see Modelling article for details), intutition and real-world experience suggest that including air resistance will decrease range and that the more viscous the air conditions (e.g. misty vs. dry) the greater the reduction in range will be.

Below is an animation showing how Stokes' drag affects the modelled trajectory of a projected object.

Released under Creative Commons Attribution Share Alike 3.0 Licence Author: AllenMcC

Both objects are thrown at the same angle and with the same initial velocity. The blue object doesn't experience any drag and moves along a parabola. The black object experiences air resistance (Stokes' drag).

Wind

A simple way of considering the effect of wind is in terms of how it affects the air resistance force. Here air resistance is being modelled as proportional in magnitude to (and opposite in direction to) the relative velocity between the shot-put and the fluid it is moving through. If the fluid is moving (wind) this will affect the relative velocity and so also the drag force.

If we consider only cases where the wind velocity is along the line of the shot-put's horizontal velocity (with respect to the ground) then if the wind is acting in the same direction as the shot-put's motion, the relative velocity between fluid and shot will decrease and so correspondingly the drag force and retardation of motion. If the wind speed is greater than that of the shot, the drag force will actually assist the motion of the shot (with respect to the ground). Conversely if the wind is opposing the shot's motion it will increase the relative velocity and so also the drag force.

Teachers' Resources

Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.

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.

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