A ball is thrown upward from the top of a 27.0-m-tall building. The ball's initial speed is 12 m/s. At the same instant, a person is running on the ground at a distance of 31.2 m from the building. What must be the average speed of the person if he is to catch the ball at the bottom of the building?

Calculate the time t it will take the ball to reach the ground.

27 = -12 t + (g/2)t^2
where g = 9.8 m/s^2
Solve the quadratic for t and take the postive root. (There will be two roots)
Once you have t, to catch the ball, require that
31.2 meters = V t
Solve for the velocity V

calulating the quadratic= 3.872 sec. 31.2 m/ 3.872sec= 8.057m/s. Thank you so much, the answer is correct!

You're welcome! I'm glad my calculations were correct and helped you find the right answer. Remember, when it comes to catching balls, a little speed goes a long way! Keep up the good work!

You're welcome! I'm glad I could help. If you have any more questions, feel free to ask.

To calculate the time it will take for the ball to reach the ground, we use the equation:

27 = -12t + (g/2)t^2

where g = 9.8 m/s^2 (acceleration due to gravity).

To solve this equation, we can rearrange it to get:

(g/2)t^2 - 12t + 27 = 0

This is a quadratic equation, so we can use the quadratic formula to find the values of t. The quadratic formula is:

t = (-b ± √(b^2 - 4ac)) / (2a)

For this equation, a = g/2, b = -12, and c = 27. Plugging in these values, we get:

t = (-(-12) ± √((-12)^2 - 4(g/2)(27))) / (2(g/2))

Simplifying further:

t = (12 ± √(144 - 4(g/2)(27))) / g

Now we can calculate the value of t. Taking the positive root (since time cannot be negative in this context):

t = (12 + √(144 - 4(9.8/2)(27))) / 9.8

t ≈ 3.872 seconds

Now, to catch the ball, the person running on the ground must cover a distance of 31.2 meters (the distance from the building to the person's starting point) in the same time it takes the ball to reach the ground.

The average speed of the person can be calculated using the formula:

average speed = total distance / total time

In this case, the total distance is 31.2 meters, and the total time is 3.872 seconds. Plugging in these values, we get:

average speed = 31.2 / 3.872

average speed ≈ 8.057 m/s

Therefore, the person must have an average speed of approximately 8.057 m/s in order to catch the ball at the bottom of the building.