A stone is dropped from the top of a tower. The stone hits the ground after 5 seconds. Find the height of the tower and the velocity with which the stone hits the ground.
The Correct Answer and Explanation is :
To determine the height of the tower and the velocity with which the stone hits the ground, we can use the equations of motion under uniform acceleration due to gravity.
Given:
- Time (t) = 5 seconds
- Acceleration due to gravity (g) ≈ 9.81 m/s² (we can use 10 m/s² for simplicity)
1. Height of the Tower
The formula to calculate the distance (height) an object falls freely from rest is:
[
h = \frac{1}{2} g t^2
]
Substituting the known values:
[
h = \frac{1}{2} \times 10 \, \text{m/s}^2 \times (5 \, \text{s})^2
]
Calculating further:
[
h = \frac{1}{2} \times 10 \times 25 = \frac{250}{2} = 125 \, \text{meters}
]
Thus, the height of the tower is 125 meters.
2. Velocity with Which the Stone Hits the Ground
To find the final velocity (v) of the stone just before it hits the ground, we use the formula:
[
v = g t
]
Substituting the values:
[
v = 10 \, \text{m/s}^2 \times 5 \, \text{s} = 50 \, \text{m/s}
]
Therefore, the velocity with which the stone hits the ground is 50 m/s.
Summary
In summary, after dropping a stone from the top of a tower for 5 seconds, the height of the tower is calculated to be 125 meters, and the final velocity of the stone upon impact with the ground is 50 m/s. This analysis is based on the principles of free fall under gravity, illustrating how time and gravitational acceleration affect the distance fallen and the velocity attained.