Soccer ball motion approximation

There are two obvious optimisations:

First you need to only calculate cos(angle) and sin(angle) once per kick. You may be able to calculate them together, e.g. some APIs provide a SinCos function which works them both out at once.

Second you do not need to use exp to do drag. Exp arises when you try to write down a precise formula for the position at all times, as it''s the solution to an equation of motion of a particle with linear drag.

But you can intead use numerical techniques. The first equation can be derived from the equation of motion

dv
-- = -Kv/M
dt

where v is the velocity. This can be used to update the horizontal velocity u with the approximation formula

u = u - (K * u / M) * t

where t is the time step between updates.

Then use

x = x + u * t

to update x.

Similarly for the vertical motion use

v = v - (K * v / M + G) * t

and

z = z + v * t

Because u, v, x and z are calculated from previous values they need to be initialised, and for your situation this would be

u = V * cos(angle)
v = V * sin(angle)
x = 0
z = h0

where ''V'' is the launch speed.

It''s important to note that these numerical methods are approximations: over even a short time step u and v are changing, so calculations done with them are incorrect by the end of the time step.

It is therefore important that the time step is as small as possible, and so that updates are done as often as possible. This is usually one of the key decisaions in designing a simulation, and it''s difficult to give precise recommendations, but it''s often best to use an fixed update rate different from the frame rate if it can vary.

Sounds really good. I''ve tried in Excel and it is just perfect.
Thanks.