In the legend of William Tell, Tell is forced to shoot an apple from his son's head for failing to show respect to a high official. In our case, let's say Tell stands 8.7 meters from his son while shooting. The speed of the 144-g arrow just before it strikes the apple is 20.4 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 8 m behind the son's feet, how massive was the apple

M = 0.31 kg

Explanation:

This exercise must be done in parts, let's start by finding the speed of the set arrow plus apple, for this we define a system formed by the arrow and the apple, therefore the forces during the collision are internal and the moment is conserved

let's use m for the mass of the arrow with velocity v₁ = 20.4 m / s and M for the mass of the apple

initial instant. Just before the crash

          p₀ = m v₁ + M 0

instant fianl. Right after the crash

          p_f = (m + M) v

          p₀ = p_f

          m v₁ = (m + M) v

          v =                          (1)

now we can work the arrow plus apple set when it leaves the child's head with horizontal speed and reaches the floor at x = 8 m. We can use kinematics to find the velocity of the set

          x = v t

          y = y₀ + t - ½ g t²

when it reaches the ground, its height is y = 0 and as it comes out horizontally,

          0 = h - ½ g t²

          t² = 2h / g

For the solution of the exercise, the height of the child must be known, suppose that h = 1 m

            t =

            t = 0.452 s

let's find the initial velocity

             v = v / t

             v = 8 / 0.452

             v = 17.7 m / s

From equation 1

              v = m / (m + M) v₁

              m + M =

              M = m + m \  \frac{v_1}{v}

we calculate

              M = 0.144 + 0.144  

              M = 0.31 kg