Kinetic energy (KE) can be calculated using the formula:
\[ KE = \frac{1}{2} mv^2 \]
Where:
- \( m \) is the mass of the object,
- \( v \) is the velocity of the object.
Now, let's calculate the kinetic energy for each option:
-
120 kg of sand falling at 12 m/s: \[ KE = \frac{1}{2} \times 120 , \text{kg} \times (12 , \text{m/s})^2 \] \[ KE = 60 \times 144 = 8640 , \text{J} \]
-
12 kg of sand falling at 8 m/s: \[ KE = \frac{1}{2} \times 12 , \text{kg} \times (8 , \text{m/s})^2 \] \[ KE = 6 \times 64 = 384 , \text{J} \]
-
12 kg of water falling at 10 m/s: \[ KE = \frac{1}{2} \times 12 , \text{kg} \times (10 , \text{m/s})^2 \] \[ KE = 6 \times 100 = 600 , \text{J} \]
-
120 kg of water falling at 12 m/s: \[ KE = \frac{1}{2} \times 120 , \text{kg} \times (12 , \text{m/s})^2 \] \[ KE = 60 \times 144 = 8640 , \text{J} \]
Now comparing the kinetic energies calculated:
- 120 kg of sand at 12 m/s: 8640 J
- 12 kg of sand at 8 m/s: 384 J
- 12 kg of water at 10 m/s: 600 J
- 120 kg of water at 12 m/s: 8640 J
The falling object with the least kinetic energy when it collides with the ground is 12 kg of sand falling at 8 m/s, which has 384 J of kinetic energy.