To determine which falling object will result in the least kinetic energy when it collides with the ground, we can use the formula for kinetic energy (KE):
\[ KE = \frac{1}{2}mv^2 \]
where:
- \(m\) is the mass of the object (in kilograms),
- \(v\) is the velocity of the object (in meters per second).
We will calculate the kinetic energy for each option:
-
100 grams of dirt falling at 15 m/s:
- Mass \(m = 100 , \text{grams} = 0.1 , \text{kg}\)
- Velocity \(v = 15 , \text{m/s}\)
- \(KE = \frac{1}{2} \times 0.1 \times (15)^2 = 0.5 \times 0.1 \times 225 = 11.25 , \text{J}\)
-
100 grams of water falling at 10 m/s:
- Mass \(m = 100 , \text{grams} = 0.1 , \text{kg}\)
- Velocity \(v = 10 , \text{m/s}\)
- \(KE = \frac{1}{2} \times 0.1 \times (10)^2 = 0.5 \times 0.1 \times 100 = 5.0 , \text{J}\)
-
10 grams of dirt falling at 2 m/s:
- Mass \(m = 10 , \text{grams} = 0.01 , \text{kg}\)
- Velocity \(v = 2 , \text{m/s}\)
- \(KE = \frac{1}{2} \times 0.01 \times (2)^2 = 0.5 \times 0.01 \times 4 = 0.02 , \text{J}\)
-
10 grams of water falling at 1 m/s:
- Mass \(m = 10 , \text{grams} = 0.01 , \text{kg}\)
- Velocity \(v = 1 , \text{m/s}\)
- \(KE = \frac{1}{2} \times 0.01 \times (1)^2 = 0.5 \times 0.01 \times 1 = 0.005 , \text{J}\)
Now we can compare the kinetic energies:
- 100 grams of dirt at 15 m/s: 11.25 J
- 100 grams of water at 10 m/s: 5.0 J
- 10 grams of dirt at 2 m/s: 0.02 J
- 10 grams of water at 1 m/s: 0.005 J
The object with the least kinetic energy is the 10 grams of water falling at 1 m/s with 0.005 J of kinetic energy.