Question 8 of 16 Sapling Learning macmillan learning An old Biblical story tells

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Question 8 of 16 Sapling Learning macmillan learning 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, therefore allowing the projectile to fly in a straight line at high speed In this problem, we will assume that the sling, when unfolded, was roughly 5.63 feet in total length. We will also assume the rock weighed a half a pound If David whirled the sling to the point where the sling subtended an angle of = 4.03° with the horizontal, with what speed did the rock strike Goliath? (Assume no air friction and that the drop in height due to gravity is minimal.) Number

Explanation / Answer

The total length of the ling is 5.63 ft.

The distance from the center of the circular path of the rock to one end of the sling is

r = 5.63 / 2 = 2.815 ft

r = 2.815 * 0.3048 = 0.858 m

use the Newton's second law

Fy = F sin(theta) - mg

0 = F sin(theta) - mg

F = mg / sin(theta) ...................(1)

m is the mass of the projectile

g is the acceleration due to gravity

F is the applied force

the net horizontal force is

Fx = (mv^2/r) - F cos(theta)

0 = (mv^2/r) - F cos(theta)

F = (mv^2 / r*cos(theta)) ..................(2)

Compare the above two equations

mv^2 / r*cos(theta) = mg / sin(theta)

v = sqrt (r*g / tan(theta))

the speed of the rock is

v = sqrt(0.858 * 9.8 / tan(4.03))

= 10.925 m/s

v = 10.925 /0.3048 = 35.842 ft/s

The time of flight is

t = sqrt(2h/g)
  
= sqrt (2 * 4.25 *0.3048 / 9.8)

= 0.5142 s

The required distance is

d = v * t

= 10.925 m/s * 0.5142 s = 5.6176 m

d = 35.842 ft/s * 0.5142 s = 18.430 ft

The impact would likely cause serious injury Suppose David had missed.

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