energy problem

[wachaif]

A toy cannon uses a spring tp project a 5.3 g soft rubber ball. The spring is originally compressed by 5 cm and has a force constant 8N/m .When the cannon is fired , the ball moves 15cm through the horizontal barrel of the cannon ,and there is a constant friction force of 0.032 N . between the barrel and the ball.At what point does the ball have maximum speed ? What is this maximum speed?

[HallsofIvy]

The force due to the spring at any instant is the spring constant times the compression at that instant: If we take x= 0 at the point the ball is when the spring is compressed, then the force due to the spring is 8(0.05-x) (as long as x< 0.05 m). The friction force is 0.032 (as long as x< 0.15 cm) so the total force on the ball is:
8(0.05-x)- 0.32= 0.08- 8x for x< 0.05
= -0.032 for 0.05< x< 0.15

Since "force= mass* acceleration", we have the differential equation

0.0053x"= 0.08- 8x for x< 0.05
= -0.032 for 0.05< x< 0.15
with initial conditions x(0)= 0, x'(0)= 0.