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STEP 1 - FBD: Draw a SINGLE free body diagram (FBD) of the system of cart + cannon + cannonball.
STEP 2 - Kinetics: Write down the impulse/momentum equation in the horizontal direction (x-direction) for the the system of cart + cannon + cannonball. Based on the above FBD, is the momentum conserved in the x-direction for that system?
STEP 3 - Kinematics
STEP 4 - Solve. Solve for the velocity of the cart + cannon.
QUESTION: The above analysis allows you to find the answer to the first part of the problem. Unfortunately, it is not useful for the find the answer to the second part of the problem where you want to find the force on the cart/cannon. What do you need to change in the analysis to find this force?