To determine which falling object will result in the least kinetic energy upon collision 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, and \( v \) is the velocity in meters per second.
Let's calculate the kinetic energy for each option:
-
10 grams of dirt falling at 2 m/s:
- Mass: \( m = 10 , \text{grams} = 0.01 , \text{kg} \)
- Velocity: \( v = 2 , \text{m/s} \)
- \( KE = \frac{1}{2} \times 0.01 \times (2)^2 = \frac{1}{2} \times 0.01 \times 4 = 0.02 , \text{J} \)
-
100 grams of water falling at 10 m/s:
- Mass: \( m = 100 , \text{grams} = 0.1 , \text{kg} \)
- Velocity: \( v = 10 , \text{m/s} \)
- \( KE = \frac{1}{2} \times 0.1 \times (10)^2 = \frac{1}{2} \times 0.1 \times 100 = 5.0 , \text{J} \)
-
10 grams of water falling at 1 m/s:
- Mass: \( m = 10 , \text{grams} = 0.01 , \text{kg} \)
- Velocity: \( v = 1 , \text{m/s} \)
- \( KE = \frac{1}{2} \times 0.01 \times (1)^2 = \frac{1}{2} \times 0.01 \times 1 = 0.005 , \text{J} \)
-
100 grams of dirt falling at 15 m/s:
- Mass: \( m = 100 , \text{grams} = 0.1 , \text{kg} \)
- Velocity: \( v = 15 , \text{m/s} \)
- \( KE = \frac{1}{2} \times 0.1 \times (15)^2 = \frac{1}{2} \times 0.1 \times 225 = 11.25 , \text{J} \)
Now, summarizing the kinetic energies:
- 10 grams of dirt at 2 m/s: 0.02 J
- 100 grams of water at 10 m/s: 5.0 J
- 10 grams of water at 1 m/s: 0.005 J
- 100 grams of dirt at 15 m/s: 11.25 J
Based on the calculations, the object that results in the least kinetic energy upon collision with the ground is 10 grams of water falling at 1 m/s (0.005 J).
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