Question

The Great Sandini is a 60 kg circus performer who is shot from a cannon (actually a spring gun). You don't find many men of his caliber, so you help him design a new gun. This new gun has a very large spring with a very small mass and a force constant of 1300 N/m that he will compress with a force of 6500 N. The inside of the gun barrel is coated with Teflon, so the average friction force will be only 50 N during the 5.0 mm he moves in the barrel.

Required:
At what speed will he emerge from the end of the barrel, 2.5 mabove his initial rest position?

Answer 1

22m/s

Step-by-step explanation:

Mass, m=60 kg

Force constant, k=1300N/m

Restoring force, Fx=6500 N

Average friction force, f=50 N

Length of barrel, l=5m

y=2.5 m

Initial velocity, u=0

`F_x=kx`

Substitute the values

`6500=1300x`

`x=(6500)/(1300)=5`m

Work done due to friction force

`W_f=fscos\theta`

We have
`\theta=180^(\circ)`

Substitute the values

`W_f=50* 5cos180^(\circ)`

`W_f=-250J`

Initial kinetic energy, Ki=0

Initial gravitational energy,
`U_(grav,1)=0`\

Initial elastic potential energy

`U_(el,1)=(1)/(2)kx^2=(1)/(2)(1300)(5^2)`

`U_(el,1)=16250J`

Final elastic energy,
`U_(el,2)=0`

Final kinetic energy,
`K_f=(1)/(2)(60)v^2=30v^2`

Final gravitational energy,
`U_(grav,2)=mgh=60* 9.8* 2.5`

Final gravitational energy,
`U_(grav,2)=1470J`

Using work-energy theorem

`K_i+U_(grav,1)+U_(el,1)+W_f=K_f+U_(grav,2)+U_(el,2)`

Substitute the values

`0+0+16250-250=30v^2+1470+0`

`16000-1470=30v^2`

`14530=30v^2`

`v^2=(14530)/(30)`

`v=\sqrt{(14530)/(30)}`

`v=22m/s`