Galileo is riding on top of a horse that is gallop- ing at 11 ft/s due east. A minute into his ride, Galileo pulls a key out of his pocket, and holds it over the side of the horse. When he lets go, the key falls 3.0 ft vertically to the ground.
a). How far has the horse traveled when the key hits the ground?a). How far has the horse traveled when the key hits the ground?b). Where is the horse relative to the key at the moment the key hits the ground?c). With what speed does the key hit the ground?
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To answer these questions, we need to analyze the motion of the horse and the key separately.
a) To find the distance the horse has traveled when the key hits the ground, we need to determine the time it takes for the key to fall. We can use the equation for vertical motion:
h = (1/2)gt^2
Where h is the vertical distance (3.0 ft), g is the acceleration due to gravity (32.2 ft/s^2), and t is the time. Rearranging the equation to solve for time:
t = sqrt((2h)/g)
Substituting the values:
t = sqrt((2 * 3.0 ft) / 32.2 ft/s^2) ≈ 0.305 s
Since the key falls for 0.305 s, the horse would have traveled at a constant speed of 11 ft/s for that time, so:
Distance traveled by the horse = speed × time
Distance traveled by the horse = 11 ft/s × 0.305 s ≈ 3.355 ft
Therefore, the horse has traveled approximately 3.355 ft when the key hits the ground.
b) At the moment the key hits the ground, the horse continues to move at its constant speed of 11 ft/s in the same direction. Therefore, the horse is still ahead of the key when it hits the ground.
c) To determine the speed at which the key hits the ground, we can use the equation for vertical motion:
v = gt
Substituting the values:
v = 32.2 ft/s^2 × 0.305 s ≈ 9.831 ft/s
Therefore, the speed at which the key hits the ground is approximately 9.831 ft/s.