Final answer:
To find the rock's speed at the bottom of the hole, we use the kinematic equation, plugging in the initial velocity, distance fallen, and acceleration due to gravity, resulting in a final speed of -16.4 m/s.
Step-by-step explanation:
To determine the rock's speed as it hits the bottom of the hole, we must consider its initial velocity, the distance it falls, and the acceleration due to gravity. Given parameters include the initial vertical velocity (vo = -13.0 m/s), the distance fallen (y1 = -5.10 m), and the acceleration due to gravity (a = -9.80 m/s²).
Using the kinematic equation v² = vo² + 2a(y - yo), where v is the final velocity and y is the final position, we find:
v² = (-13.0 m/s)² + 2(-9.80 m/s²) (-5.10 m - 0 m) = 268.96 m²/s²
The final step is to take the square root of both sides and choose the negative root to indicate that the rock is still heading down, so:
v = -16.4 m/s
This negative value signifies that the rock is moving downwards when it hits the bottom of the hole.