A 20 g arrow moving at 50 m/s penetrates a tree to a depth of 4.00 cm. a) Use th

Question

A 20 g arrow moving at 50 m/s penetrates a tree to a depth of 4.00 cm.
a) Use the work-kinetic energy theorem to determine the magnitude of the average frictional force that stops the arrow in the tree.
b) Assuming that the frictional force is constant, determine how much time elapsed between the moment the arrow entered the tree and the moment it stopped. Use Newton's Second Law and the kinematic equations.
c) Recalculate the answer part (b) using the definition of Impulse.

Explanation / Answer

a) According to work energy theorem we have f * dx = 0.5 m v^2 ==> frictional force = 0.5 m v^2 / dx = 0.5 * 0.02 * 50 * 50 / 0.04 = 625 N b) acceleration of the arrow is a = - v^2 / 2 dx = - 50 * 50 / 2 * 0.04 = 31250 Therefore v = a t ==> t = v / a = 50 / 31250 = 0.0016 s c) change in momentum = impulse = f * t ==> f * t = m v ==> t = m v / f = 0.02 * 50 / 625 = 0.0016 s

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