# Projectile Fired Up A Hill

1. Feb 6, 2008

### cameo_demon

the following problem is from the 5th edition of Thorton and Marion's "Classical Dynamics"
ch.2 problem 14 p.92

The problem statement, all variables and given/known data
A projectile is fired with initial speed v_0 at an elevation angle of alpha up a hill of slope beta (alpha > beta).

(a) how far up the hill will the projectile land?
(b) at what angle alpha will the range be a maximum?
(c) what is the maximum angle?

The attempt at a solution
apparently this has been a stumper in former classical mechanics classes, but here was as far as i got:

i broke down the components of the forces into x and y
x-component:
a_x=0 integrating -->
v_x = v_0 cos(beta) integrating -->
x = v_0 t cos(beta)

a_y = -g integrating -->
v_y = -gt + v_0 sin (alpha - beta) integrating -->
y = ( -gt^2 / 2 ) + v_0 sin (alpha - beta)

the answer in the back of the book for part (a) is:
d = (2 v_0^2 cos(alpha) sin(alpha-beta) ) / (g cos^2(beta))

any idea how to get from the components to the answer?

2. Feb 7, 2008

### Hootenanny

Staff Emeritus
Hint: The distance is the magnitude of the position vector.

3. Sep 19, 2009

### msd213

I too am having trouble with this problem. In the book, they use Taylor expansions in order to find the flight time T and put in back into the range equation. However, they have the luxury of letting y = 0 because the trajectory is a simple parabola. We don't have the same situation here. y doesn't equal zero, y = beta*range.

4. Sep 20, 2009

### msd213

I have no idea how we're supposed to get things like cos($$\beta$$)