In the figure here, two children are playing a game in which they try to hit a s

Question

In the figure here, two children are playing a game in which they try to hit a small box on the floor with a marble fired from a spring-loaded gun that is mounted on a table. The target box is horizontal distance D = 2.17 m from the edge of the table. Bobby compresses the spring 1.15 cm, but the center of the marble falls 29.5 cm short of the center of the box. How far should Rhoda compress the spring to score a direct hit? Assume that neither the spring nor the ball encounters friction in the gun.

Explanation / Answer

1/2 g t^2 = h    where h is height of table

t = (2 h / g)^1/2

s1 = 2.17 - .295 = 1.875 = v1 t      since the marble is fired horizontally

s1 = 1.875 (2 h / g)^1/2 t

s2 = 2.17    direct hit

s2 = v2 t    by the same reasoning

s2 / s1 = 1.157 = v2 / v1     since t is the same in both cases

1/2 m v1^2 = 1/2 k x1^2

1/2 m v2^2 = 1/2 k x2^2

v2 / v1 = x2 / x1 = 1.157

x2 = 1.157 x1 = 1.157 * 1.15 = 1.33 cm

Agrees with above answer.

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