Question

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

Answer 1

M = 0.31 kg

Step-by-step 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 =
`(m)/(m+M) \ v_1` (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 +
`v_(oy)` t - ½ g t²

when it reaches the ground, its height is y = 0 and as it comes out horizontally,
`v_(oy) = 0`

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 =
`\sqrt{ ( 2 \ 1)/(9.8) }`

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 \ (v_1)/(v)`

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

we calculate

M = 0.144 + 0.144
`(20.4)/(17.7)`

M = 0.31 kg