Question

How to calculate the final velocity of an object that falls from a certain height to a lower elevation under the sole influence of gravity, ignoring friction?

Answer 1

To calculate the final velocity of an object falling under the influence of gravity, we can use the principles of kinematics. Assuming there is no air resistance and the only force acting on the object is gravity, we can employ the following kinematic equation:

\[
v^2 = u^2 + 2gh
\]

Where:
- \( v \) = final velocity (m/s)
- \( u \) = initial velocity (m/s) (for an object starting from rest, this is 0)
- \( g \) = acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \) on the surface of the Earth)
- \( h \) = height fallen (m) (this is the vertical distance the object has fallen)

If the object starts from rest, the initial velocity \( u = 0 \). Thus, the equation simplifies to:

\[
v^2 = 0 + 2gh
\]
\[
v^2 = 2gh
\]

To find the final velocity \( v \), take the square root of both sides:

\[
v = \sqrt{2gh}
\]

### Steps to Calculate Final Velocity:

1. **Identify the height** \( h \) from which the object is falling.
2. **Use the value** \( g = 9.81 \, \text{m/s}^2 \) for Earth's gravitational acceleration.
3. **Substitute** the height \( h \) and the value of \( g \) into the equation.
4. **Calculate** \( v \) using the formula \( v = \sqrt{2gh} \).

### Example Calculation:

Suppose an object falls from a height of 20 meters. We want to find the final velocity just before it hits the ground.

1. Height \( h = 20 \, \text{m} \)
2. Gravitational acceleration \( g = 9.81 \, \text{m/s}^2 \)

Using the formula:

\[
v = \sqrt{2 \cdot 9.81 \, \text{m/s}^2 \cdot 20\, \text{m}}
\]
\[
v = \sqrt{392.4}
\]
\[
v \approx 19.8 \, \text{m/s}
\]

Thus, the final velocity of the object just before it hits the ground would be approximately \( 19.8 \, \text{m/s} \).