# Projectile Motion: Range proof

by zanazzi78
Tags: motion, projectile, proof, range
 P: 110 Q. A projectile is fired with an initial speed $$V_0$$ at an angle $$\beta$$ to the horitontal. Show that it's range alnog a plane which it's self is inclined at an angle $$\alpha$$ to the horitontal $$( \beta > \alpha)$$ is given by: $$R = \frac{(2{V_0}^2 cos \beta sin(\beta - \alpha )}{g {cos}^2\alpha}$$ A.So Ive started off with $$\triangle x = (V_0 cos \beta) t$$ and $$\triangle y = (V_0 sin \beta) t - frac{1}{2} g t^2$$ $$\triangle x = cos \alpha$$and $$\triangle y = sin \alpha$$ so i rearranged $$\triangle x$$ to get $$t = \frac{cos \alpha}{V_0 cos \beta}$$ and sub it into $$\triangle y$$ now i have $$\triangle y = (V_0 sin \beta)\frac{cos \alpha}{V_0 cos \beta} - \frac{1}{2} g ( \frac{cos \alpha}{V_0 cos \beta} )$$ $$\triangle y= \frac {V_0 sin \beta cos \alpha}{V_0 cos \beta} - \frac{1}{2} g ( \frac{cos \alpha}{V_0 cos \beta})$$ $$sin \alpha = tan \beta cos \alpha - \frac {g {cos}^2 \alpha}{2 {V_0}^2 {cos}^2 \beta}$$ now im stuck ... Any hints/tips would be great. Cheers.
 Admin P: 21,637 Somewhere there seems to be an 'R' missing. Also, one of the latex expression needs \ in front of frac.
 PF Gold P: 867 delta x = cos(alpha) ??
P: 110

## Projectile Motion: Range proof

 Quote by daniel_i_l delta x = cos(alpha) ??
alpha is the angle of the inclinded plane. it forms a right triangle with the horizontal and the point at which the projectile meets the inclinded plane. therefore delta x is equal to cos alpha.
HW Helper
P: 879
 Quote by zanazzi78 ... $$\triangle x = cos \alpha$$and $$\triangle y = sin \alpha$$ ...
This is where the R is missing. They should be,

$$\triangle x = Rcos \alpha$$and $$\triangle y = Rsin \alpha$$
P: 110
 Quote by Fermat This is where the R is missing. They should be, $$\triangle x = Rcos \alpha$$and $$\triangle y = Rsin \alpha$$
ok so ive put the 'R' s in and get

$$R Sin \alpha = R tan \beta cos \alpha - \frac {R g {cos}^2 \alpha}{2 {V_0}^2 {cos}^2 \beta}$$

but i don`t see how this helps?
 HW Helper P: 879 You're almost there. But the R in the last term should be R²

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