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Why Maths and Sport?

Part Three: The effects of gravity and the wind

High Jumping

Using strength and energy of motion to increase your potential energy:

potential energy = mgh

where m is the mass of the jumper, g is the acceleration due to gravity, and h is the height the jumper's centre of gravity is raised.

Why use the Fosbury Flop?

Notice how much less of your body you have to raise over the bar to do a Fosbury Flop than to do a Scissors jump.

What happens when someone pole vaults?

How high can you jump on the moon?
Assume g(Moon) = g(Earth)/6
Assume you can clear 2 metres on Earth.

(NB: The answer isn't 12 m. Why not?)

Projectiles

What is the best launch angle?
What about long jumpers?

$\begin{displaymath}x = Ut \end{displaymath}$

$\begin{displaymath}y = Vt - \frac{1}{2}gt^{2} \end{displaymath}$

$\begin{displaymath}tan \theta = \frac{V}{U} \end{displaymath}$

where U is the initial velocity component in the horizontal direction, V is the initial velocity component in the vertical direction, and W is the result of putting these two together; x is the horizontal distance jumped, y is the vertical height jumped.

$\begin{displaymath}W^{2} = v^{2} + U^{2} \end{displaymath}$

A world-class long jumper can run up at about 10m/s (so that gives us a value for U, and can jump upward at about 5m/s (giving us a value for V. Putting these values in the equation for $tan \theta$ we obtain a value for $\theta$ of about 27 degrees. In fact, long jumpers have a launch angle of about 23 degrees. What values of U and V would give a launch angle of 45 degrees, which gives the maximum range possible?

Sprint starting

Should you worry about the speed of sound? Does it make a significant difference to your time if you are in lane 8 rather than lane 1, and the starter is near to lane 1?
Where should the starter stand for the 400m start?
If the velocity of the wind is greater than 2 m/s, the race is said to be "wind assisted" and invalid for records. So is the world record holder the fastest man in the world?

$\begin{displaymath}drag force \propto ([v(runner) - v(wind)]^{2} \end{displaymath}$

Gravity

Weight lifted or launched = Mg

$\begin{displaymath}Maximum horizontal projectile distance = U^{2}/g \end{displaymath}$

$\begin{displaymath}Maximum vertical height = v^{2}/(2g) \end{displaymath}$

All depend on g
You jump longer and higher where g is smaller
You lift larger masses where g is bigger

g varies with latitude

Measured g(equator) < g(poles)
The Earth is not quite spherical
The Earth is rotating

200 kg in Mexico City weights the same as 200.8 kg in Helsinki
8 m long jump in Helskinki is 8.20 m in Mexico City
2 m high jump in Helsinki is 2.05 m in Mexico City

Balancing on a beam

Factors which help:

Be symmetrical about your centre
Distribute mass far away from your centre
Lower your centre of gravity
Widen your base of support