Spring Constant problem - gun?

Question

The spring of a spring gun has force constant k = 400 N/m and negligible mass. The spring is compressed 6.00 cm and a ball with mass 0.0300 kg is placed in the horizontal barrel against the compressed spring. The spring is then released, and the ball is propelled out the barrel of the gun. The barrel is 6.00 cm long, so the ball leaves the barrel at the same point that it loses contact with the spring. The gun is held so the barrel is horizontal.

A) Calculate the speed with which the ball leaves the barrel if you can ignore friction.
B) Calculate the speed of the ball as it leaves the barrel if a constant resisting force of 6.00 N acts on the ball as it moves along the barrel.
C)For the situation in part (b), at what position along the barrel does the ball have the greatest speed? (In this case, the maximum speed does not occur at the end of the barrel.)
D) What is that greatest speed?

Thank You

Answer 1

P(spring)=.5 k x^2 (see ref) Potential energy of a spring
Ke(ball)=.5 m v^2 Kinetic energy of the ball
P(spring)=Ke(ball)

A) Velocity v=Sqrt(2Ke/m)
v=Sqrt(2(.5 k x^2)/m)
v=Sqrt(k x^2)/m)
v=V(400 N/m (.06)^2 /0.0300 )
v=6.93 m/s

B) Some equation
F(spring) - ma - F(resistance)=0 [forces involved]
a = ((Fs - Fr)/m
d=.5at^2
t=sqrt(2d/a)
v=at=a sqrt(2d/a)
v=sqrt(2 d a) since a=((Fs - Fr)/m
v=sqrt(2d ((Fs - Fr)/m)

C) Vmax will take place at Ft=0
Ft = Fs - Fr=0
kx-Fr=0
x=Fr/k=6/400=0.015m (1.5cm) before leaving the barrel.

D ) Same as A) but now the X=6-1.5 =4.5cm

Have fun

Answer 2

A) The max spring force is 400*.06 = 24 N, so the stored energy is 12*.06 = .72 N.m This equals the kinetic energy of the ball, so ½mV² = .72 → V = √48 m/s

B) Since the resisting force will cause the force vs distance line to drop by 6N (to ¾ of Fmax), the line will cross the distance axis at 4.5 cm and have a value of -6 N at the end of the barrel. Energy considerations then demand that V² = (1/.06)(18*.045 - 6*.015), so V = √12 m/s

C) ¾ down the length of the barrel, or 4.5 cm

D) √(18*.045/.06) = √13.5 m/s