A toy cannon uses a spring to project a 5.28-g soft rubber ball. The spring is o

Question

A toy cannon uses a spring to project a 5.28-g soft rubber ball. The spring is originally compressed by 4.93 cm and has a force constant of 8.10 N/m. When the cannon is fired, the ball moves 14.4 cm through the horizontal barrel of the cannon, and the barrel exerts a constant friction force of 0.031 7 N on the ball. As a side note: In question 8b, the ball has maximum velocity when its acceleration has dropped to zero, because at that point it has had positive acceleration the maximum amount of time.

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 postion)

c. What is this maximum speed? _________ m/s

Explanation / Answer

a.

the speed is calculated as follows:

v = sqrt[2*dE/m]

  = sqrt[2*[0.5*kx2 - fd]/m]

=  sqrt[2*[0.5*8.10*0.04932 - 0.0317*0.144]/0.00528]

= 1.414 m/s

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b.

the required location for maximum speed is,

x' = x - [F/k] = 0.0493 - [0.0317 / 8.1] = 0.04538 m = 0.0454 m = 4.54 cm

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c.

the maximum speed is,

v = sqrt[2*dE/m]

  = sqrt{[2*[{0.5*kx2 - 0.5*k[x-x']2} - fx']/m}

= sqrt{[2*[{0.5*8.10*0.04932 - 0.5*8.1*[0.0493- 0.0454']2} - 0.0317* 0.0454]/0.00528}

= 1.77767 m/s

= 1.78 m/s

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