You throw a 20-N rock vertically into the air from ground level. You observe that when it is a height 14.8m above the ground, it is traveling at a speed of 25.0m/s upward.

A) Use the work-energy theorem to find its speed just as it left the ground. What is it?
B) Use the work-energy theorem to find its maximum height. What is it?

m*g = 20 N.

m = 20/g = 20/9.8 = 2.041 kg

A. Work = 0.5m*Vo^2 - 0.5m*V^2 = mg*h
1.02*Vo^2 - 1.02*25^2 = 20*14.8
1.02Vo^2 - 637.5 = 290
1.02Vo^2 = 290 + 637.5 = 927.5
Vo^2 = 909.31
Vo = 30.15 m/s.

B. h = (V^2-Vo^2)/2g =
(0-30.15^2)/-19.6 = 46.38 m.

To solve this problem using the work-energy theorem, we need to consider the work done on the rock and the change in its kinetic energy.

A) Let's first determine the speed of the rock just as it left the ground.

Work done on the rock is given by the formula:

Work = Change in kinetic energy

The work done on the rock can be calculated as the product of the force exerted on it and the distance over which the force is applied:

Work = Force × Distance

Here, the force being exerted on the rock is its weight, which is given by the formula:

Force = mass × acceleration due to gravity

Acceleration due to gravity is approximately 9.8 m/s², and the mass of the rock is not given, so we need to solve for it.

Using Newton's second law of motion:

Force = mass × acceleration

Since the rock is in free fall, the only force acting on it is its weight. Therefore, we can write:

mass × acceleration due to gravity = mass × acceleration

Rearranging the equation, we get:

mass × (acceleration - acceleration due to gravity) = 0

Since acceleration - acceleration due to gravity = 0, the mass of the rock does not affect its speed. Thus, we can consider it canceled out when calculating the work done.

Now, we know that the work done is equal to the change in kinetic energy. At the ground level, the rock's speed is zero, so its initial kinetic energy is zero. At a height of 14.8 m, the rock's speed is 25.0 m/s upward, so its final kinetic energy is given by:

Kinetic energy = (1/2) × mass × speed²

The work done on the rock can be calculated as:

Work = Final kinetic energy - Initial kinetic energy
= (1/2) × mass × speed² - 0

Since the mass cancels out, we can rewrite the equation as:

Work = (1/2) × speed²

Substituting the known values, we get:

20 N × 14.8 m = (1/2) × speed²

Simplifying the equation, we find:

speed = sqrt((20 N × 14.8 m × 2) / 1)
= sqrt(592 m²/s²)
≈ 24.3 m/s

Therefore, the speed just as the rock left the ground is approximately 24.3 m/s.

B) To find the maximum height reached by the rock, we can use the conservation of mechanical energy. At the maximum height, the rock's velocity is zero, so its kinetic energy is zero. The only form of energy present is gravitational potential energy.

The change in potential energy can be calculated as the difference between the initial and final potential energies:

Change in potential energy = Final potential energy - Initial potential energy

At the ground level, the initial potential energy is zero. At the maximum height, the final potential energy is given by:

Potential energy = mass × acceleration due to gravity × height

Since the mass cancels out, we can rewrite the equation as:

Potential energy = 20 N × 9.8 m/s² × 14.8 m

Simplifying the equation, we find:

Potential energy = 20 N × 14.8 m × 9.8

Therefore, the maximum height reached by the rock is approximately 2,744 m.

Hence, the answer to part A is approximately 24.3 m/s, and the answer to part B is approximately 2,744 m.