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William Tell shoots an apple from his son's head. The speed of the 102-g arrow just before it strikes the apple is 26.7 m/s, and at the time of impact it is traveling horizontally. If the arrow sticks in the apple and the arrow/apple combination strikes the ground 6.90 m behind the son's feet, how massive was the apple? Assume the son is 1.85 m tall.

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To develop the problem, we require the values concerning the conservation of momentum, specifically as given for collisions.

By definition the conservation of momentum tells us that,
m_1V_1+m_2V_2 = (m1+m2)V_f

To find the speed at which the arrow impacts the apple we turn to the equation of time, in which,


t= \sqrt{(2h)/(g)}

The linear velocity of an object is given by


V=(X)/(t)

Replacing the equation of time we have to,


V_f = (X)/(t)\\V_f =\frac{X}{\sqrt{(2h)/(g)}}\\V_f = \frac{6.9}{\sqrt{(2(1.85))/(9.8)}}\\V_f = 11.23m/s

Velocity two is neglected since there is no velocity of said target before the collision, thus,


m_1V_1 = (m1+m2)V_f

Clearing for m_2


m_2 = (m_1V_1)/(V_f)-m_1\\m_2 = ((0.102)(26.7))/(11.23)-0.102\\m_2 = 0.1405KG= 140.5g

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