A ball is thrown upward from the top of a 40.0 m tall building. The ball's initial speed is 12.0 m/s. At the same instant, a person is running on the ground at a distance of 31.0 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?
hf=0=40 + 12t-4.9t^2
solve for t, the time when the ball is at the bottom.
avg speed= 31/time
To find the average speed of the person to catch the ball at the bottom of the building, we need to determine the time it takes for the ball to reach the ground.
Let's start by finding the time it takes for the ball to reach its highest point (when it's at its maximum height above the ground).
We can use the equation of motion for vertical motion:
vf = vi + at
Here:
- vf is the final velocity at the highest point (0 m/s because the ball momentarily stops at the highest point)
- vi is the initial velocity of the ball (12.0 m/s upward)
- a is the acceleration due to gravity (-9.8 m/s², negative because it acts downward)
Using these values, we can calculate the time it takes for the ball to reach its highest point:
0 m/s = 12.0 m/s + (-9.8 m/s²) * t
Rearranging the equation, we have:
-12.0 m/s = -9.8 m/s² * t
Dividing both sides by -9.8 m/s², we get:
t = -12.0 m/s / -9.8 m/s² ≈ 1.22 s
Now, to find the total time it takes for the ball to reach the ground, we double the time it took to reach the highest point:
Total time = 2 * t ≈ 2 * 1.22 s ≈ 2.44 s
Since the person is running on the ground while the ball is in the air, we need to find the distance the person covers during this time.
The horizontal distance covered by the person can be calculated using the formula:
distance = speed * time
Here:
- distance is the distance between the person and the building (31.0 m)
- time is the total time it takes for the ball to reach the ground (2.44 s)
Substituting these values, we can find the average speed of the person:
31.0 m = speed * 2.44 s
Solving for speed:
speed = 31.0 m / 2.44 s ≈ 12.70 m/s
Therefore, the person must have an average speed of approximately 12.70 m/s to catch the ball at the bottom of the building.