A golf ball is dropped from rest from a height of 8.40 m. It hits the pavement, then bounces back up, rising just 5.30 m before falling back down again. A boy then catches the ball when it is 1.20 m above the pavement. Ignoring air resistance, calculate the total amount of time that the ball is in the air, from drop to catch.

To calculate the total amount of time that the ball is in the air, we can break down the motion into two parts: the time it takes for the ball to fall from the initial drop to hitting the pavement, and the time it takes for the ball to rise and fall back to the catch height.

First, let's calculate the time it takes for the ball to fall from the initial drop to hitting the pavement. We can use the equation:

h = ½ * g * t^2

where h is the initial height, g is the acceleration due to gravity (approximately -9.8 m/s^2), and t is the time.

For the initial drop:
h = 8.40 m
g = -9.8 m/s^2

Plugging these values into the equation, we get:
8.40 = ½ * -9.8 * t^2

Simplifying the equation, we have:
t^2 = -8.40 / (-4.9)
t^2 = 1.714

To solve for t, we take the square root of both sides:
t ≈ √1.714
t ≈ 1.31 s

So it takes approximately 1.31 seconds for the ball to fall from the initial drop to hitting the pavement.

Next, let's calculate the time it takes for the ball to rise and fall back to the catch height. Since the ball reaches a maximum height of 5.30 m, which is less than the initial height, we can use the same equation as before:

h = ½ * g * t^2

For the rise and fall:
h = 5.30 m
g = -9.8 m/s^2

Plugging in these values, we get:
5.30 = ½ * -9.8 * t^2

Simplifying the equation, we have:
t^2 = -5.30 / (-4.9)
t^2 = 1.082

Taking the square root of both sides, we get:
t ≈ √1.082
t ≈ 1.04 s

So it takes approximately 1.04 seconds for the ball to rise and fall back to the catch height.

Now, to calculate the total amount of time the ball is in the air, we add the time it takes to fall to the pavement and the time it takes to rise and fall back to the catch height:
Total time = 1.31 s + 1.04 s
Total time ≈ 2.35 s

Therefore, the total amount of time that the ball is in the air, from drop to catch, is approximately 2.35 seconds.

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