Desde un edificio de 15 m se deja caer una piedra. a. ¿Cuánto tiempo tarda en llegar al suelo? b.¿Cuál es su velocidad un instante antes de tocar el suelo?

1. Break Down the Problem

• Part a: Calculate the time taken for the stone to fall to the ground.
• Part b: Calculate the velocity of the stone just before it hits the ground.

2. Relevant Concepts

• For free fall under the influence of gravity, we can use the following kinematic equations:

  • $h = \frac{1}{2} g t^2$ where:
    • $h$ is the height (15 m),
    • $g$ is the acceleration due to gravity (approximately $9.81 \, \text{m/s}^2$),
    • $t$ is the time in seconds.

• To find the velocity just before hitting the ground:

  • $v = g t$ where $v$ is the final velocity.

3. Analysis and Detail

Part a: Time to Fall

1. Rearranging the formula $h = \frac{1}{2} g t^2$: $t^2 = \frac{2h}{g}$ Substituting the known values: $t^2 = \frac{2 \cdot 15 \, \text{m}}{9.81 \, \text{m/s}^2}$ $t^2 = \frac{30}{9.81} \approx 3.06 \, \text{s}^2$
2. Taking the square root to find $t$: $t \approx \sqrt{3.06} \approx 1.75 \, \text{s}$

Part b: Velocity Before Hitting the Ground

1. Using the time calculated: $v = g t = 9.81 \, \text{m/s}^2 \cdot 1.75 \, \text{s}$ $v \approx 17.18 \, \text{m/s}$

4. Verify and Summarize

• Time calculated for the stone to hit the ground is approximately $1.75 \, \text{s}$.
• Velocity just before hitting the ground is approximately $17.18 \, \text{m/s}$.

Final Answer

a. The stone takes approximately 1.75 seconds to reach the ground.
b. Its velocity just before touching the ground is approximately 17.18 m/s.