Question

how does the total kinetic energy of a cannon and cannonball system change as the mass of the cannon increases and the energy of the blast remains the same? (1 point)
the total kinetic energy remains the same.
the total kinetic energy cannot be determined.
the total kinetic energy decreases.
the total kinetic energy increases.

Explanation:

Step1: Recall conservation of momentum

In a cannon - cannonball system, the initial momentum is zero (at rest). After the blast, $m_{cannon}v_{cannon}+m_{cannonball}v_{cannonball}=0$, so $m_{cannon}v_{cannon}=-m_{cannonball}v_{cannonball}$.

Step2: Express kinetic energy

The total kinetic energy $K = \frac{1}{2}m_{cannon}v_{cannon}^2+\frac{1}{2}m_{cannonball}v_{cannonball}^2$. From momentum conservation, $v_{cannon}=-\frac{m_{cannonball}v_{cannonball}}{m_{cannon}}$. Substitute into the kinetic - energy formula: $K=\frac{1}{2}m_{cannon}(\frac{m_{cannonball}^2v_{cannonball}^2}{m_{cannon}^2})+\frac{1}{2}m_{cannonball}v_{cannonball}^2=\frac{m_{cannonball}^2v_{cannonball}^2}{2m_{cannon}}+\frac{1}{2}m_{cannonball}v_{cannonball}^2$.

Step3: Analyze the effect of increasing cannon mass

As $m_{cannon}$ increases while the energy of the blast (which is related to the total kinetic energy) remains the same initially, the first term $\frac{m_{cannonball}^2v_{cannonball}^2}{2m_{cannon}}$ decreases. Since the energy of the blast is conserved and is the total kinetic energy of the system, and the first term decreases with increasing $m_{cannon}$, the total kinetic energy remains the same because the energy is just redistributed between the cannon and the cannon - ball.

Answer:

The total kinetic energy remains the same.