Conservation Laws and Projectile Motion

[physicspupil]

Homework Statement

A cannon in a fort overlooking the ocean fires a shell of mass M at an elevation angle, theta and muzzle velocity, v0. At the highest point, the shell explodes into two fragments (masses m1 + m2 = M), with an additional energy E, traveling in the original horizontal direction. Find the distance separating the two fragments when they land in the ocean. For simplicity, assume the cannon is at sea level.

Homework Equations

Conservation of Momentum:

px: Mv0cos(theta) = m1v1cos(theta) + m2v2cos(theta)

py: Mv0sin(theta) = m1v1sin(theta) + m2v2sin(theta)

Conservation of Energy

Mv0^2/2 + E = m1v1^2/2 + m2v2^2/2

Maybe

y = y0 + v0tsin(theta) – gt^2/2 and x = x0 + v0tcos(theta) at the end…

The Attempt at a Solution

First I used the kinematic equations to find the time when the two broken things will hit the ocean (they hit at the same time, I think)… t = 2v0sin(theta)/g

Then I tried to find v2 in terms of v1. I did this by squaring px and py above…

(M^2)(v0^2)(sin^2 + cos^2) = (m1^2)(v1^2) (sin^2 + cos^2) + etc.

I got v0^2 = (m1v1 + m2v2)^2/M^2 and plugged this into the conservation of energy equation…

If I can ever get my algebra correct, I was thinking about plugging this result back into the above equations and messing with things until I can get both v1 and v2 in terms of v0…

Then I can just use the x component of the kinematic equations to find their distance from the cannon, right? And take the difference?.. It seems simple enough, but its just not working out for me. Is there a better/easier way to do this? Thanks!

 

[Mindscrape]

You are on the right track. The energy equation you have, however, is not in the most convenient form. Can you express vo in terms of the original energy (rhetorical question)?

Yeah, you have the equations down though, and the rest of the problem is just algebraic manipulation, and a lot of it at that.

 

[physicspupil]

Thanks, Mindscrape I figured out how to write my v1 and v2 in terms of v0, so I think I can do it now.