Question

William Tell shoots an apple from his son's head. The speed of the 118-g arrow just before it strikes the apple is 20 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 7.7 m behind the son's feet, how massive was the apple? Assume the son is 1.85 m tall.

Answer 1

Consider the motion of the combination after the collision

V = Common horizontal speed of the combination

Consider the motion along the vertical direction

V_oy = initial velocity in vertical direction = 0 m/s

a = acceleration = 9.8 m/s²

Y = vertical displacement = height of the son = 1.85 m

t = time of travel

using the equation

Y = V_oy t + (0.5) a t²

1.85 = (0) t + (0.5) (9.8) t²

t = 0.614 sec

Consider the motion along the horizontal direction :

V_ox = velocity of the combination after collision = V

t = time of travel = 0.614 sec

X = horizontal displacement = 7.7 m

Using the equation

X = V_ox t                                             (Since there is no acceleration along x-direction)

7.7 = V (0.614)

V = 12.5 m/s

Consider the collision between arrow and apple

m₁ = mass of arrow = 118 g

v₁ = initial velocity of arrow before collision = 20 m/s

m₂ = mass of apple = m

v₂ = initial velocity of arrow before collision = 0 m/s

V = velocity of the combination after collision = 12.5 m/s

Using conservation of momentum

m₁ v₁ + m₂ v₂ = (m₁ + m₂) V

(118) (20) + (m) (0) = (118 + m) (12.5)

m = 70.8 g

hence mass of apple is 70.8 g