Question

1. in a closed system of a cannon and cannonball, which changes would both result in an increase in the kinetic energy of the cannonball when fired from the cannon? (1 point)

○ decrease the mass of the cannon or decrease the mass of the cannonball
○ increase the mass of the cannon or decrease the mass of the cannonball
○ increase the mass of the cannon or increase the mass of the cannonball
○ decrease the mass of the cannon or increase the mass of the cannonball

Explanation:

Step1: Apply conservation of momentum

In a closed system, \(m_{cannon}v_{cannon}+m_{cannonball}v_{cannonball} = 0\) (initial momentum is \(0\)), so \(v_{cannon}=-\frac{m_{cannonball}v_{cannonball}}{m_{cannon}}\)

Step2: Analyze kinetic - energy relationship

The total kinetic energy \(K = \frac{1}{2}m_{cannon}v_{cannon}^2+\frac{1}{2}m_{cannonball}v_{cannonball}^2\). Substitute \(v_{cannon}\) from Step1: \(K=\frac{1}{2}\frac{m_{cannonball}^2v_{cannonball}^2}{m_{cannon}}+\frac{1}{2}m_{cannonball}v_{cannonball}^2=\frac{1}{2}m_{cannonball}v_{cannonball}^2(1 + \frac{m_{cannonball}}{m_{cannon}})\)

If we decrease the mass of the cannon (\(m_{cannon}\downarrow\)), for a given \(m_{cannonball}\) and \(v_{cannonball}\), the fraction \(\frac{m_{cannonball}}{m_{cannon}}\) increases. Also, from \(K=\frac{1}{2}m_{cannonball}v_{cannonball}^2\), if we increase the mass of the cannon - ball (\(m_{cannonball}\uparrow\)) while keeping other factors (like the energy stored in the cannon - firing mechanism, which is related to the total kinetic energy in the absence of non - conservative forces) in mind (assuming the total kinetic energy available for the system \(K_{total}\) is fixed from the firing mechanism), using the momentum relationship \(m_{cannon}v_{cannon}=-m_{cannonball}v_{cannonball}\) and \(K_{total}=\frac{1}{2}m_{cannon}v_{cannon}^2+\frac{1}{2}m_{cannonball}v_{cannonball}^2\), we can show that increasing \(m_{cannonball}\) (and adjusting velocities accordingly while conserving momentum) can lead to an increase in \(K_{cannonball}\)

Answer:

decrease the mass of the cannon or increase the mass of the cannonball