A toy cannon uses a spring to project a 5.40-g soft rubber ball. The spring is o

Question

A toy cannon uses a spring to project a 5.40-g soft rubber ball. The spring is originally compressed by 5.04 cm and has a force constant of 7.94 N/m. When the cannon is fired, the ball moves 14.8 cm through the horizontal barrel of the cannon, and the barrel exerts a constant friction force of 0.031 5 N on the ball.

(a) With what speed does the projectile leave the barrel of the cannon?
m/s

(b) At what point does the ball have maximum speed?
cm (from its original position)

(c) What is this maximum speed?
m/s

Explanation / Answer

given, mass m = 5.4*10^-3 kg

initial compression of the spring, x = 5.04*10^-2 m

spring constant, k = 7.94 N/m

length of barrel, L = 0.148 m

friction force, F = 0.0315 N

a. energy stored in the spring initially = 0.5kx^2 = 0.5*7.94*(5.04*10^-2)^2 = 0.0100844352 J

energy consumed by friction = F*L = 0.004662 J

so KE of the ball as it leaves the canon = 0.0100844352 - 0.004662 = 0.0054224352 J

let speed of the ball be v

then 0.5*m*v^2 = 0.0054224352

v = 1.417 m/s

b. the ball will have its maximum speed while still being attached to the spring

so let the spring be extended by length p at some time t

energy stored in spring = 0.5k(x - p)^2

energy consumed by friction = F*p

so KE o fthe ball = 0.5k(x - p)^2 + Fp

for max KE, d(KE)/dp = 0

k(x - p) = F

7.94(5.04*10^-2 - p) = 0.0315

p = 0.046432 cm

c. maximum speed = V

0.5mV^2 = 0.5k(5.04*10^-2 - 0.0464)^2 + 0.315*2.332 m/s

V =

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