Physics Challenge Problem-Help!!!!?

Question

Problem 11.74
As a hobby, you like to participate in reenactments of Civil War battles. Civil War cannons were "muzzle loaded," meaning that the gunpowder and the cannonball were inserted into the output end of the muzzle, then tamped into place with a long plunger. To recreate the authenticity of muzzle-loaded cannons, but without the danger of real cannons, Civil War buffs have invented a spring-powered cannon that fires a 1.0 kg plastic ball. A spring, with 3000 N/m spring constant , is mounted at the back of the barrel. You place a ball in the barrel, then use a long plunger to press the ball against the spring and lock the spring into place, ready for firing. In order for the latch to catch, the ball has to be moving at a speed of 2.0m/s at least when the spring has been compressed 30 cm. The coefficient of friction of the ball in the barrel is 0.30. The plunger doesn't touch the sides of the barrel.
If you push the plunger with a constant force, what is the minimum force that you must use to compress and latch the spring? You can assume that no effort was required to push the ball down the barrel to where it first contacts the spring.
What is the cannon's muzzle velocity if the ball travels a total distance of 1.5 m to the end of the barrel?

Answer 1

This looks fun:

m=1.0kg
k=3000 N/m
x=0.30 m
u=0.30

barrel length is 1.5 m

I will assume that the barrel is exactly horizontal.
Also, since no mass was given for the plunger, I will assume it is weightless.

Consider a free-body diagram of the ball while being tamped:

The force of the plunger, which is constant
the force of gravity, which is perpendicular to the direction of motion
friction, pushing against the plunger
and the spring pushing against the plunger
I will call the Plunger force F


F-1*.30*9.81-k*x=1.0*a

we know that the speed is 2 m/s. Let's use conservation of energy to compute the force of the plunger

.5*m*v^2+.5*k*x^2+m*g*u*x=F*x
m=1
v=2
k=0.30
x=0.30
g=9.81
k=3000

F=(1/.30)*
(.5*4+.5*3000*0.30^2+
9.81*.3*.3)
=459.61N

For the muzzle velocity, using conservation of energy
since the ball starts from rest:

Energy Will be gained from the spring:
.5*k*x^2
lost to friction
u*m*g*L
where L is the barrel length
the balance will be kinetic energy of the ball
.5*m*v^2=.5*k*x^2-u*m*g*L
v=sqrt((2/m)*(.5*k*x^2-
u*m*g*L))
=16 m/s

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