To find the maximum height that the stone can reach, we can use the principles of energy conservation. The stone's initial kinetic energy will convert entirely to gravitational potential energy at the highest point of its trajectory (where its velocity becomes zero).

Step-by-Step Solution

1. Calculate Initial Kinetic Energy (KE): $\text{KE} = \frac{1}{2} m v^2$ where:

   • $m = 3 \, \text{kg}$
   • $v = 30 \, \text{m/s}$

   Substituting the values: $\text{KE} = \frac{1}{2} \times 3 \times 30^2 = \frac{1}{2} \times 3 \times 900 = 1350 \, \text{J}$

2. Calculate Gravitational Potential Energy (PE) at Maximum Height: At maximum height, all the kinetic energy will have converted to gravitational potential energy: $\text{PE} = m \times g \times h$ where:

   • $g = 9.8 \, \text{m/s}^2$
   • $h$ is the maximum height

   Since KE at the start equals PE at the maximum height: $1350 = 3 \times 9.8 \times h$

3. Solve for $h$: $h = \frac{1350}{3 \times 9.8} = \frac{1350}{29.4} \approx 45.9 \, \text{m}$

Answer

The maximum height that the stone can reach is approximately: $\boxed{45.9 \, \text{m}}$

Would you like further details on any part of this calculation, or do you have additional questions?

Here are some related questions to explore:

1. What would the maximum height be if the stone's mass were doubled?
2. How would air resistance affect the maximum height?
3. What would happen to the maximum height if the initial velocity increased by 50%?
4. How long would it take for the stone to reach this maximum height?
5. What would be the velocity of the stone halfway to its maximum height?

Tip: In problems involving vertical motion and energy conservation, equating kinetic and potential energy can often simplify calculations without needing to solve for time.