Is there a formula I can use as an approximation to the following equation for velocity of a projectile when $\sigma$ is very small?
$\dot{x}[n] =v_0\prod_{i=0}^{n}(1-\sigma i)$ $\sigma\approx\frac{C_1}{1,000,000};C_1\in[0,100]$
I am trying to compute the distance travelled (programmatically) when it is shot with initial velocity $v_0$.
The sum would be:
$x[n]=\sum_{j=0}^{n}\dot{x}$
or $x[n]=v_0\sum_{j=0}^{n}\prod_{i=0}^{j}(1-\sigma i).$
The solution for the sum according to a wolfram alpha computation is very complicated and computationally expensive. I derived this formula from the following function which updates the velocity every tick (10 milliseconds).
fire() { frictiontimer = 1000000; } // This is called every 10ms update() { if (xspeed) { int workfriction = frictiontimer / 1000; xspeed = xspeed * workfriction / 1000; frictiontimer -= FRICTION_CONSTANT; if (frictiontimer < 0) frictiontimer = 0; } }
Is there is a simplification I can use somewhere?