Explaining the Physics of a Rolling Cannon

[eraemia]

Homework Statement

Imagine a cannon that is free to roll on wheels. Initially, both the cannon and cannonball are at rest. When the cannonball is fired, though, the momentum transferred to the cannonball comes at the expense of the cannon, so the cannonball and cannon end up with momenta having equal magnitudes but opposite directions. If this is so, why does the cannonball fly away at a large speed while the cannon recoils only very modestly? Explain your response.

Homework Equations

Law of Conservation of Momentum
Newton's Third Law

The Attempt at a Solution

According to the law of conservation of momentum, the momentum of the canon would equal the momentum of the canonball. Since momentum is mv, the cannon's large mass would account for its smaller velocity/recoil, compared with the canonball's smaller mass and greater velocity. Is this right? What am I missing?

 

[Capuchin]

I don't think you're missing anything. Your answer is correct. Except that where you said "the momentum of the canon would equal the momentum of the canonball" you should take care to note that they are in opposite direction and so momentum (a vector) is not equal, but only equal in magnitude.

 

[m2003]

Your understanding of the law of conservation of momentum and Newton's Third Law is correct. The key factor in explaining the difference in recoil between the cannon and cannonball is their masses. The cannon has a much larger mass compared to the cannonball, so even though their momenta are equal, the cannon's velocity will be much smaller due to its larger mass. This is similar to a game of pool, where a cue ball with a smaller mass can transfer its momentum to a heavier pool ball, causing the pool ball to move at a higher velocity. Additionally, the cannonball experiences a greater force due to the explosion of gunpowder, resulting in a higher acceleration and thus a higher velocity. This is in line with Newton's Second Law, which states that the acceleration of an object is directly proportional to the net force acting on it, and inversely proportional to its mass. Therefore, the cannonball will fly away at a higher speed while the cannon will only recoil modestly due to the difference in their masses and the force acting on them.