# Astronaut firing a Canon ball

1. Oct 25, 2014

### Satvik Pandey

1. The problem statement, all variables and given/known data

An astronaut in space weights 100 kg and is holding a cannon which can shoot a 10kg cannonball. The astronaut is moving at 10 m/s. The astronaut wishes to fire the cannonball such that he turns the maximum possible angle. Find this angle in degrees.

2. Relevant equations

3. The attempt at a solution

If we consider astronaut and cannon ball to be a system then on that system no external force and torque acts on it so we can use equations for conservation of momentum and angular momentum and energy.

I have a confusion, when astronaut fires the cannon an impulsive force(due to recoiling) acts on it, due to which the astronaut rotates. Also as no other torque acts on it afterwards so I think it will go on rotating forever.

However I proceeded. I think for turning through maximum angle I think the initial kinetic energy of should get converted to the rotational kinetic energy. For finding rotational energy I need to calculate moment of inertia of astronaut first. But how? Also we can not consider as a point mass. Can we?

2. Oct 25, 2014

### Orodruin

Staff Emeritus
I believe what the problem intends to ask is the angle for which the direction of travel after the firing deviates the most from the direction of travel before (naturally in the frame where the astronaut is travelling at 10 m/s). Assuming the astronaut fires the cannon in such a way that the force is directed through the centre of mass, the astronaut will not actually gain any angular momentum.

3. Oct 25, 2014

### Staff: Mentor

I agree with the conservation of momenta, but there's another repository of energy in the system that would have to be taken into account here if you plan to use conservation of energy. Think about what makes a canon fire.

However, I think you might be able to get away without having to apply conservation of energy here.
I think the "rotation" here is meant to be the angle through which the trajectory changes, not a rotation of the spacecraft itself.

<<EDIT: Ah! Orodruin beat me to it!>>

4. Oct 25, 2014

### Orodruin

Staff Emeritus
Only because I did not take the time to draw a picture ...

5. Oct 25, 2014

### Staff: Mentor

By the way Satvik, were you given information about the speed for the canon ball after firing?

6. Oct 25, 2014

### Satvik Pandey

No, velocity of canon is not given.

7. Oct 25, 2014

### Satvik Pandey

In order to use conservation of momentum I should know the direction and magnitude of the speed of the canon. Should I assume that canon is fired at angle $\theta$ (please refer to figure) with the horizontal.

8. Oct 25, 2014

### Staff: Mentor

Well that makes things easier Suppose the speed of the canon ball after firing is such that its momentum is exactly equal to the initial momentum of the astronaut plus canon ball before firing. What does that tell you about the new velocity of the astronaut? What if the speed of the canon ball were even larger? What then is the range of possible turning angles?

9. Oct 25, 2014

### Staff: Mentor

The recoil angles of the two objects will not necessarily be equal. You can say though that by conservation of momentum the net momentum in the X and Y directions are separately conserved.

10. Oct 25, 2014

### Satvik Pandey

Then its speed should be zero.

Then the velocity of astronaut should be opposite to the velocity of the canon.

It is $0 to 180$?

11. Oct 25, 2014

### Satvik Pandey

I was too thinking that. But if we assume that the velocity vector of canon makes angle $\phi$ with the horizontal and velocity of astronaut makes $\theta$ with the horizontal then by using conservation of momentum in X and Y direction and by conservation of energy I can make three equations but there are 4 variables.

12. Oct 25, 2014

### Staff: Mentor

Right.

If the canon ball's velocity is not limited then the turning angle can be anything you want, up to and including 180°.

13. Oct 25, 2014

### Satvik Pandey

14. Oct 25, 2014

### Staff: Mentor

Conservation of energy is a dubious thing to use here since the firing of the canon is analogous to an inelastic collision happening in reverse. Kinetic energy is not conserved across an inelastic collision.

If the canon ball's speed is not specified then perhaps a better approach is to simply assume that the firing of the canon imparts a change of momentum ΔP of some magnitude and direction. The canon ball carries off momentum ΔP and the astronaut suffers a change of -ΔP (since momentum is conserved).

Without knowing what the magnitude of ΔP can be you can't really say anything more specific.

15. Oct 25, 2014

### Staff: Mentor

What do you think?

If the complete problem statement is as you've presented it then you can't determine a particular angle other than by considering extremes of the unspecified variables.

You could come up with a set of expressions for different cases where the |ΔP| is less than, equal to, or greater than that of the initial momentum. But that seems like a lot of work for such a loosely phrased problem. How does this problem's complexity compare to others in the same problem set?

16. Oct 25, 2014

### Satvik Pandey

Answer of this question is 27.
You can find this problem on (https://brilliant.org/community-problem/astronaut-in-space/?group=hjkNbxhQsI1z&ref_id=455645 [Broken])

Last edited by a moderator: May 7, 2017
17. Oct 25, 2014

### Staff: Mentor

The problem statement does not contain enough information to arrive at a specific answer. The problem is flawed.

Last edited by a moderator: May 7, 2017
18. Oct 25, 2014

### Satvik Pandey

Thank you gneill for helping. Sorry I posted incomplete question.:s

19. Oct 25, 2014

### Orodruin

Staff Emeritus
Don't be sorry, it is the fault of brilliant.org (or rather people posting problems there). I generally do not trust those problems as it seems nobody is checking them. I still have a problem with some ants on a tetrahedron in vivid memory ... Solvable, but not with the alleged answer of the poster, and definitely not using maths as simple as the poster imagined.

20. Oct 26, 2014

### Satvik Pandey

Yes I remember the problem 'Ants on tetrahedron'.
Thank you Orodruin and gneill for helping.
Congratulation Orodruin. You are a staff now.:w