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# Model 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:

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.

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

#### Wind

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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).

5 (greatest)

12

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.

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

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).

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.

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?