A rock thrown straight up climbs for 4.24 s, then falls to the ground. Neglecting air resistance, with what velocity did the rock strike the ground?

V = Vo + g*t = 0 @ max ht.

Vo - 9.8*4.24 = 0
Vo = 41.55 m/s. = Initial velocity, up. = Final velocity, down.

To determine the velocity at which the rock strikes the ground, we'll need to consider the motion of the rock in two parts: when it goes up and when it comes down.

First, let's calculate the velocity when the rock reaches its maximum height. We know that the time it takes for the rock to climb is 4.24 seconds, and since the motion of the rock is vertical, the time it takes to reach the maximum height is half of the total time taken to climb.

Therefore, the time taken to reach the maximum height is 4.24 / 2 = 2.12 seconds.

During the upward motion, the rock accelerates due to the force of gravity. The acceleration due to gravity is approximately 9.8 m/s^2 (assuming downward as the positive direction). We can use the kinematic equation:

v = u + at

Where:
v = final velocity
u = initial velocity
a = acceleration
t = time

Since the rock reaches its maximum height, the final velocity is 0 m/s. The acceleration due to gravity is -9.8 m/s^2 (negative because it acts in the opposite direction to the motion). The time taken is 2.12 seconds. So, we can rearrange the equation to solve for the initial velocity:

0 = u - 9.8 * 2.12

Simplifying the equation gives:

u = 9.8 * 2.12

Calculating the value gives:

u ≈ 20.776 m/s

Now, let's calculate the velocity when the rock strikes the ground. When the rock falls back down, it experiences the same acceleration due to gravity but in the opposite direction. The time taken to fall back to the ground is the same as the time taken to climb, which is 4.24 seconds.

Using the same kinematic equation, we can find the final velocity:

v = u + at

Where:
v = final velocity
u = initial velocity
a = acceleration
t = time

Since the rock strikes the ground, the final velocity is what we want to find, and it will be in the positive direction. The initial velocity is (-20.776 m/s) since it is in the opposite direction. The acceleration due to gravity is 9.8 m/s^2 (positive because it acts downward). The time taken is 4.24 seconds. Plugging these values into the equation gives:

v = -20.776 + 9.8 * 4.24

Calculating the value gives:

v ≈ -20.776 + 41.672

v ≈ 20.896 m/s

Therefore, the velocity at which the rock strikes the ground is approximately 20.896 m/s.