A golf ball is dropped from rest from a height of 8.20 m. It hits the pavement, then bounces back up, rising just 5.40 m before falling back down again. A boy then catches the ball when it is 1.40 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 the motion of the ball into two parts: the time it takes to fall from the initial height to the pavement, and the time it takes to rise from the pavement to the final height where the boy catches it.
First, let's calculate the time it takes for the ball to fall from the initial height to the pavement. We can use the equation of motion:
π = π£βπ‘ + Β½ππ‘Β²
Where:
- π is the distance
- π£β is the initial velocity (in this case, since the ball is dropped, π£β = 0)
- π‘ is the time
- π is the acceleration (acceleration due to gravity, π = -9.8 m/sΒ²)
For the first part of the motion, the distance is 8.20 m, and the initial velocity is 0. We can rearrange the equation to solve for time:
8.20 = 0 + Β½(-9.8)π‘Β²
16.4 = -4.9π‘Β²
π‘Β² = -16.4 / -4.9
π‘Β² = 3.3469
π‘ β β3.3469
π‘ β 1.8311 s
Therefore, it takes approximately 1.8311 seconds for the ball to fall from the initial height to the pavement.
Next, let's calculate the time it takes for the ball to rise from the pavement to the final height where the boy catches it. Again, we can use the equation of motion:
π = π£βπ‘ + Β½ππ‘Β²
For the second part of the motion, the distance is the difference between the final height and the height of the pavement, which is (5.40 - 0) = 5.40 m. The initial velocity is the final velocity when the ball hits the pavement, and the acceleration is still the acceleration due to gravity (-9.8 m/sΒ²).
Using the equation, we have:
5.40 = π£β + Β½(-9.8)π‘
5.40 = π£β - 4.9π‘
Since the ball reaches its maximum height when its velocity is 0, π£β = 0. Substituting π£β = 0 gives us:
5.40 = 0 - 4.9π‘
5.40 = -4.9π‘
π‘ β -5.40 / -4.9
π‘ β 1.102 seconds
Therefore, it takes approximately 1.102 seconds for the ball to rise from the pavement to the final height.
To calculate the total time, we add the time it takes to fall to the pavement and the time it takes to rise from the pavement to the final height:
Total time β 1.8311 + 1.102
Total time β 2.9331 seconds
Therefore, the total amount of time that the ball is in the air, from drop to catch, is approximately 2.9331 seconds.