A stone is thrown upward a 20 m/s. It is caught on its way down 5.0 m above where it was thrown. How fast was it going when it was caught?

h max = (V^2-Vo^2)/2g = (0-(20)^2/-19.6 = 20.41 m.

V^2 = Vo^2 + 2g*h = 0 + 19.6*(20.41-5) =
302.
V = 17.4 m/s

To determine the speed at which the stone was caught, we need to use the principles of motion and the equations of projectile motion. Let's break down the problem step by step:

Step 1: Define the relevant variables.
Let's assign variables to the given information in the problem:
Initial velocity (upwards) = 20 m/s (positive because it is upward)
Final displacement (downwards) = -5.0 m (negative because it is downwards)
Acceleration due to gravity (g) = -9.8 m/s² (negative because it acts in the opposite direction to the initial velocity)

Step 2: Determine the time it takes for the stone to reach its highest point.
When the stone reaches its highest point, its vertical velocity becomes zero. We can use the following formula to find the time it takes to reach the highest point:

vf = vi + at

Where vf is the final velocity, vi is the initial velocity, a is the acceleration, and t is the time taken.

Rearranging the equation, we have:
0 = 20 - 9.8t (since vf = 0 at the highest point)

Solving for t:
9.8t = 20
t = 20 / 9.8
t ≈ 2.04 seconds (rounded to two decimal places)

Step 3: Calculate the time it takes for the stone to reach the catching position.
To find this time, we can double the time it took for the stone to reach its highest point. This is because the stone takes the same amount of time to rise and fall:

Total time = 2t
Total time = 2 x 2.04
Total time ≈ 4.08 seconds (rounded to two decimal places)

Step 4: Find the velocity at the catching position.
To find the velocity at the position where the stone was caught, we can use the following formula:

vf = vi + at

Where vf is the final velocity (which we need to find), vi is the initial velocity, a is the acceleration, and t is the total time taken.

Rearranging the equation, we have:
vf = vi + at

Substituting the values we know:
vf = 20 - 9.8(4.08)

Solving for vf:
vf ≈ 20 - 39.84
vf ≈ -19.84 m/s (rounded to two decimal places)

Step 5: Determine the speed.
The speed is the magnitude of the velocity, so we take the absolute value of the velocity:

speed = |vf|
speed = |-19.84|
speed ≈ 19.84 m/s (rounded to two decimal places)

Therefore, the stone was going approximately 19.84 m/s when it was caught.