# Trajectory equation needed

1. Jan 6, 2008

### vidhatanand

The General Information about the game:
The law of the Football game says that the ball should be a sphere of leather or plastic,
filled with air. Its circumference should be 68-70 cm and weight should be 410-450 g,
inflated to a pressure of 8-12 psi. The L * W dimension of the ground should be 90-120m
* 85-90m and the L*W*H dimension of the goal post should be 8 * 4 * 3 m³
When a player hits the ball, the trajectory of the ball gets affected by the wind flowing
around. The energy generated by the wind is proportional to the cube of wind velocity.
The general formula for wind energy is:
Energy = ½ (Density of air) * (Swept area) * (Wind velocity cubed)
Energy = ½.ñ.A.v³
If the velocity (v) is in m/s, then at sea level (where the density of air is 1.2 kg/m³) the
power in the wind is:
Energy = 0.6 x v³ Watts per m² of Swept area
This means that the energy density in the wind will range from 10W/m² at 2.5m/s (a light
breeze) to 41,000W/m² at 40m/s (a hurricane) at sea level.
The ball which is inflated to 10psi would bounce back to 50% of the height where it hit
the ground from, making same angle with the normal to the ground with which it hit the
ground, but in the other direction.
Assumptions
1) The wind transfers 80% of its energy to the ball when it is in the air.
2) The player is allowed to hit the ball in lower spherical half of the ball.
3) The line of force applied to the ball by the player passes through the centre of
gravity of the ball.
4) The maximum force that a player can apply to the ball should be sufficient
enough to take it to the Goal post from ¾ of the total ground length with no wind
flowing around.
5) If wind is flowing, it will always be parallel to the ground.
Your objective is to make these two teams play in the manner explained above and
decide the winner. Also show the trajectory and final position of the ball when it’s hit
from the Half-Way line. Following minimum inputs would be given by the user.
1) Force applied by the player
2) Direction of the line of force with respect to ground and with respect to Goal Post
as well.(Assume that goal post is in the North direction)
3) The position of the ball on the Half-Way line
4) Direction w.r.t. North and Speed of the wind flowing around.
Any other required parameter can be assumed as per convenience.