A cannon is rigidly attached to a carriage, which can move along horizontal rail

Question

A cannon is rigidly attached to a carriage, which can move along horizontal rails, but is connected to a post by a large spring, initially unstretched and with force constant k = 2.06 104N/m, as in the figure below. The cannon fires a 200-kg projectile at a velocity of 136 m/s directed 45.0° above the horizontal. (a) If the mass of the cannon and its carriage is 5,000 kg, find the recoil speed of the cannon. (b) Determine the maximum extension of the spring. (c) Find the maximum force the spring exerts on the carriage.

Explanation / Answer

We can apply cons of momentum to the cannon and the projectile in the horizontal direction. the initial momentum is 0, and the final is mvcos(45) - MV
so mvcos(45) - MV =0, or V = m/M vcos(45)

b) now we apply cons of energy to the cannon and the spring. Ei = 1/2MV^2 V from answer a
Ef = 0 + 1/2kx^2 where x is the extension of the spring

Ei = Ef
1/2MV^2 = 1/2 kx^2, solve for x

c) The maximum force exerted by the spring occurs at the max displacment |F| = k x use x from b

d) We can consider the momentum for the cannon, carriage and projectile conserved in the x direction since there are no external forces. However, since there is a normal force and gravity in the y direction, the momentum in the y direction is not conserved.

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