I was wondering how to derive a formula for a length of projectiles path in the air (2 dimensions). I figured out the relationship of x and y, y=x*tg(a) - gx2/2v02cos2(a)
Then i combined some logic with the mean value theorem and got that lenght of path is integral from zero to x=v02sin(2a)/g of sqrt(1 + (tg(a) - xg/v02cos2(a))2) dx
When we raise (tg(a) - xg/v02*cos2(a))2 and simplify a little bit we get integral from zero to x=v02sin(2a)/g of sqrt((v04cos2(a) - 2v02sin2(a)*xg + x2g)/v04cos4(a)) dx . Any ideas how to solve it? Thank you!