Question

An old Biblical story tells of David, a small warrior engaged in man‑to‑man combat with Goliath, a much larger adversary. In the story, David uses a sling to increase the speed of a projectile (in this case a rock), striking Goliath in the forehead and killing him.

The type of sling often used in such combat consists of a long strip of cloth with a pouch halfway down its length. The rock is positioned in the pouch, with the warrior holding both ends of the sling. By whirling the sling in a horizontal circle, a warrior can release one end of the sling, thereby allowing the projectile to fly in a straight line at high speed. Historically, warriors and hunters typically used a vertical circle, but the circle used here is horizontal for simplicity.
In this problem, assume that the sling, when unfolded, was roughly =6.08 ft in total length and that the rock weighed a half a pound.
If David whirled the sling such that the sling was at an angle theta=4.43∘ below the horizontal, with what speed did the rock strike Goliath? Assume there is no air friction and that the drop in height due to gravity is minimal
= ____ mph
Suppose David missed. What distance would the rock travel before striking the ground if David released it from a height of H=4.85 ft? Assume there is no air friction.
= ____ ft

Answer 1

If David missed, the rock would travel approximately 18.83 ft before striking the ground.

To find the speed at which the rock struck Goliath, we can use the principles of projectile motion. We'll first calculate the initial velocity of the rock when released from the sling, and then we'll find its horizontal component of velocity.

1. Calculate the initial velocity (v_initial) of the rock when released from the sling:

- The total length of the sling when unfolded is 6.08 ft.

- The angle θ at which the sling is released is 4.43 degrees below the horizontal.

v_initial = √((g * L) / (2 * sin(θ)))

Where:

- g is the acceleration due to gravity (32.2 ft/s²)

- L is the length of the sling (6.08 ft)

- θ is the angle in radians (convert 4.43 degrees to radians)

θ = 4.43 degrees * (π / 180 degrees) ≈ 0.0774 radians

Now, calculate v_initial:

v_initial = √((32.2 ft/s² * 6.08 ft) / (2 * sin(0.0774 radians))) ≈ 36.95 ft/s

2. Convert the velocity to mph:

- 1 mph = 1.46667 ft/s

v_initial ≈ 36.95 ft/s / 1.46667 ≈ 25.19 mph

So, the speed at which the rock struck Goliath is approximately 25.19 mph.

Now, let's calculate the distance the rock would travel before striking the ground if David missed:

3. Calculate the time of flight (t) to hit the ground:

- We'll use the equation for vertical motion:

H = (1/2) * g * t²

Solve for t:

t = √((2 * H) / g)

Where:

- H is the height from which the rock is released (4.85 ft)

- g is the acceleration due to gravity (32.2 ft/s²)

t = √((2 * 4.85 ft) / 32.2 ft/s²) ≈ 0.51 seconds

4. Calculate the horizontal distance (d) the rock travels before hitting the ground:

- We'll use the horizontal motion equation:

d = v_initial * t

Where:

- v_initial is the initial horizontal velocity (approximately 25.19 mph)

- t is the time of flight (approximately 0.51 seconds)

Convert v_initial to ft/s:

v_initial ≈ 25.19 mph * 1.46667 ft/s/mph ≈ 36.95 ft/s

Now, calculate d:

d ≈ 36.95 ft/s * 0.51 s ≈ 18.83 ft

So, if David missed, the rock would travel approximately 18.83 ft before striking the ground.