A golf ball is dropped from a height of 9.5m. It hits the pavement and bounces back up, rising 5.7m before falling back down. A boy catches the ball on the way down when it is 1.20m above the pavement. Ignoring air resistance, calculate the total time the ball is in the air, from drop to catch.

To calculate the total time the ball is in the air, we need to consider the time it takes for the ball to fall from the initial drop, the time it takes for the ball to rise up after bouncing, and the time it takes for the ball to fall back down to the boy's hand.

Step 1: Calculate the time it takes for the ball to fall from the initial drop.

We can use the formula for free fall: s = ut + (1/2)gt^2, where s is the distance, u is the initial velocity (which is 0 since the ball is dropped), g is the acceleration due to gravity (which is approximately 9.8 m/s^2), and t is the time taken.

Given: s = 9.5m, u = 0, g = 9.8 m/s^2.

Plugging in these values into the formula, we get:

9.5 = 0 * t + (1/2) * 9.8 * t^2

Simplifying the equation, we have:

4.9t^2 = 9.5

Dividing both sides by 4.9, we get:

t^2 = 1.93877551

Taking the square root of both sides, we have:

t = 1.3923 seconds (rounded to 4 decimal places)

Step 2: Calculate the time it takes for the ball to rise up after bouncing.

When the ball bounces, it rises to a height of 5.7m. We can use the same formula for free fall to determine the time it takes for the ball to rise to this height.

Given: s = 5.7m, u = 0 (since the ball starts from rest after bouncing), g = 9.8 m/s^2.

Plugging in these values into the formula, we get:

5.7 = 0 * t + (1/2) * 9.8 * t^2

Simplifying the equation, we have:

4.9t^2 = 5.7

Dividing both sides by 4.9, we get:

t^2 = 1.16326531

Taking the square root of both sides, we have:

t = 1.0796 seconds (rounded to 4 decimal places)

Step 3: Calculate the time it takes for the ball to fall back down to the boy's hand.

The ball falls from a height of 1.20m to the boy's hand. We can again use the formula for free fall.

Given: s = 1.20m, u = 0, g = 9.8 m/s^2.

Plugging in these values into the formula, we get:

1.20 = 0 * t + (1/2) * 9.8 * t^2

Simplifying the equation, we have:

4.9t^2 = 1.20

Dividing both sides by 4.9, we get:

t^2 = 0.24489796

Taking the square root of both sides, we have:

t = 0.4949 seconds (rounded to 4 decimal places)

To calculate the total time the ball is in the air, we need to add the times from step 1, 2, and 3:

Total time = 1.3923 + 1.0796 + 0.4949 = 2.9668 seconds (rounded to 4 decimal places)

Therefore, the total time the ball is in the air, from drop to catch, is approximately 2.967 seconds.