Conservation of momentum cannon problem

[gaborfk]

A cannon rigidly attached to a carriage, whic can move along horizontal rails but is connected to a post by a large initially unstreched spring with a force constant k=2x10^4N/m. The cannon fires a 200kg projectile at a velocity of 125m/s directed 45 degrees above the horizontal.

a) If the mass of the cannon and its carriage is 5000kg, 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.

d) Consider the system consisting of the cannon, carriage, and shedd. Is the momentum of this system conserved during the firing? Why or why not?

I already know the answer to d, which is no, since the spring is an external force acting on the system.

However, can someone please, help me to get started on a, b, and c?

Thank you

 

[HallsofIvy]

a) Use conservation of momentum (before the spring takes effect). A 200kg projectile at a velocity of 125m/s has momentum vector with magnitude 25000 kgm/s. Since the shell is fired at 45 degrees, the horizontal component is (√(2)/2) times that (the ground will absorb the vertical component). The horizontal component of momentum of the shell is 17678 kgm/s. The momentum of cannon and carriage must be the same but in the opposite direction: 5000 kg *v= -17678 so v= -3.53 m/s (negative meaning opposite direction to flight of shell.)

Knowing the velocity of cannon and carriage, you can calculate their kinetic energy:
(1/2)(5000)(3.53)². The work done by the spring is (1/2)(2x10⁴x² and the spring must do work equal to the kinetic energy to stop the cannon and carriage: solve 10⁴x²= 2500(3.53)².

 

[gaborfk]

Thank you very much!

On part c does the spring exerts only F=-kx?

Can this part be this simple?

 

[Doc Al]

On part c does the spring exerts only F=-kx?

Can this part be this simple?

Yes and yes.