# Projectile Motion: Range proof

1. ### zanazzi78

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.

Last edited: Nov 1, 2005

### Staff: Mentor

Somewhere there seems to be an 'R' missing.

Also, one of the latex expression needs \ in front of frac.

3. ### daniel_i_l

866
delta x = cos(alpha) ??

4. ### zanazzi78

110
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.

5. ### Fermat

876
This is where the R is missing. They should be,

$$\triangle x = Rcos \alpha$$and $$\triangle y = Rsin \alpha$$

6. ### zanazzi78

110
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?

7. ### Fermat

876
You're almost there. But the R in the last term should be R²