To determine which falling object will have 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 in kilograms and \(v\) is the velocity in meters per second.
Let's calculate the kinetic energy for each object:
-
100 grams of dirt falling at 15 m/s:
- Mass \(m\) = 100 grams = 0.1 kg
- Velocity \(v\) = 15 m/s \[ KE = \frac{1}{2}(0.1)(15^2) = 0.05 \cdot 225 = 11.25 , \text{J} \]
-
100 grams of water falling at 10 m/s:
- Mass \(m\) = 100 grams = 0.1 kg
- Velocity \(v\) = 10 m/s \[ KE = \frac{1}{2}(0.1)(10^2) = 0.05 \cdot 100 = 5 , \text{J} \]
-
10 grams of water falling at 1 m/s:
- Mass \(m\) = 10 grams = 0.01 kg
- Velocity \(v\) = 1 m/s \[ KE = \frac{1}{2}(0.01)(1^2) = 0.005 \cdot 1 = 0.005 , \text{J} \]
-
10 grams of dirt falling at 2 m/s:
- Mass \(m\) = 10 grams = 0.01 kg
- Velocity \(v\) = 2 m/s \[ KE = \frac{1}{2}(0.01)(2^2) = 0.005 \cdot 4 = 0.02 , \text{J} \]
Now, let's summarize the kinetic energies calculated:
- 100 grams of dirt at 15 m/s: 11.25 J
- 100 grams of water at 10 m/s: 5 J
- 10 grams of water at 1 m/s: 0.005 J
- 10 grams of dirt at 2 m/s: 0.02 J
The object with the least kinetic energy is 10 grams of water falling at 1 m/s, with a kinetic energy of 0.005 J.
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