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