To find the time it takes for the boulder to reach its maximum height, we need to find the time when the velocity of the boulder becomes 0. This is because at the maximum height, the velocity of the boulder becomes 0 before it starts falling back down due to gravity.
Given that the upward velocity of the boulder is 122 ft/s, we can find the time it takes for the boulder to reach its maximum height using the equation of motion:
v = u + at
where:
v = final velocity (0 ft/s at maximum height)
u = initial velocity (122 ft/s)
a = acceleration due to gravity (-32 ft/s^2, negative because it acts in the opposite direction of the boulder's motion)
0 = 122 - 32t
32t = 122
t = 122/32
t ≈ 3.81 seconds
So, it takes approximately 3.81 seconds for the boulder to reach its maximum height.
To find the boulder's maximum height, we can use the formula for the height of the boulder at any given time:
h(t) = ut + (1/2)at^2
At the maximum height, the velocity of the boulder becomes 0, so we can use the time calculated above to find the maximum height:
h(3.81) = 122 * 3.81 - (1/2) * 32 * 3.81^2
h(3.81) ≈ 464.62 feet
Therefore, the boulder's maximum height is approximately 464.62 feet.
A catapult launches a boulder with an upward velocity of 122 feet per second. The height of the boulder, (h), in feet after t seconds is given by the function . How long does it take the boulder to reach its maximum height? What is the boulder’s maximum height?
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