Kinetic and potential energy in firing a cannon.

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1. Nov 4, 2014

timnswede

1. The problem statement, all variables and given/known data
A 20.0 kg cannonball is fired from a cannon with a muzzle speed of 1000 m/s at an angle of 37.0° with the horizontal. A second ball is fired at an angle of 90.0°.

(a) Use the isolated system model to find the maximum height reached by each ball.

(b) What is the total mechanical energy of the ball-Earth system at the maximum height for each ball? Let y = 0 at the cannon.

2. Relevant equations
0=ΔKsys+ΔUsys
Emech=Usys+Ksys

3. The attempt at a solution
I understand how to do part a). The ball fired straight up has initial kinetic energy of (½)(20(1000^2)=(20)(9.8)h and h=51020m. Same idea for the second ball and h=18479m. I'm really confused about part b) though, apparently the total Mechanical energy for each ball is equal to the initial kinetic energy, which is (½)(20)(1000^2). But for the ball fired straight up I thought it would be Emech=-(½)(20)(1000^2)+(20)(9.8)(51020), which is wrong. I don't really understand how to find Emech.

2. Nov 4, 2014

Staff: Mentor

Perhaps they want you to consider the recoil/rebound of the Earth? The cannon balls hurtles away in one direction, the Earthmoves away in another. Both carry K.E.

Does the textbook give you the answer?

3. Nov 4, 2014

timnswede

Yes, and that is the answer I found when I looked it up online. This is the answer on the guide for the problem:

4. Nov 4, 2014

Staff: Mentor

So they are saying the mechanical energy of the ball remains constant throughout its flight, and mechanical energy is the sum of kinetic and gravitational P.E.

5. Nov 4, 2014

timnswede

I got the right answer if I assume there to be no initial kinetic or potential energy at the cannon. So for the one at the angle the initial kinetic would be (.5)(20)(1000cos37^2) + mgh. And the one straight up would just have potential at the height. I'm not sure if that is the correct way of thinking of it, and it is also pretty much the opposite of what the guide has.

6. Nov 4, 2014

Staff: Mentor

The total energy of both balls is identical. They each begin their flight with the same muzzle velocity, some of it transforms into P.E., but there are no losses. The vertically-directed ball at some stage converts all of its K.E. into gravitational P.E. The ball fired at an inclination converts only some of its K.E. into P.E.