Two balls of equal size are dropped from the same height from the roof of a building . One ball has twice the mass of the other. When the balls the ground, how do the kinetic energies of the two balls compare ?

Options

A. The lighter one has one fourt has much kinetic energy as the other does.
B. The lighter one has one half as much kinetic energy as the other does.
C. The lighter one has the same kinetic energy as the other does.
D. The lighter one has twice as much kinetic energy as the other does.

Answer

To solve the problem, we will analyze the kinetic energies of the two balls when they reach the ground after being dropped from the same height.

### Step-by-Step Solution:

1. **Identify the Masses of the Balls:**
- Let the mass of the lighter ball be \( m \).
- The mass of the heavier ball is given as twice that of the lighter ball, so it is \( 2m \).

2. **Understand the Initial Conditions:**
- Both balls are dropped from the same height \( h \).
- Initially, both balls have zero kinetic energy since they are at rest.

3. **Calculate Potential Energy at the Height:**
- The potential energy (PE) of the lighter ball at height \( h \) is given by:
\[ PE_1 = mgh \]
- The potential energy of the heavier ball at height \( h \) is:
\[ PE_2 = 2mgh \]

4. **Use Conservation of Energy:**
- According to the conservation of energy, the total mechanical energy (potential + kinetic) remains constant if we ignore air resistance.
- When the balls reach the ground, all potential energy is converted into kinetic energy (KE).

5. **Calculate Kinetic Energy at the Ground:**
- For the lighter ball:
\[ KE_1 = PE_1 = mgh \]
- For the heavier ball:
\[ KE_2 = PE_2 = 2mgh \]

6. **Compare the Kinetic Energies:**
- Now we compare the kinetic energies:
\[ KE_1 = mgh \quad \text{and} \quad KE_2 = 2mgh \]
- From this, we can see that:
\[ KE_2 = 2 \times KE_1 \]
- This means the kinetic energy of the heavier ball is twice that of the lighter ball.

7. **Conclusion:**
- The lighter ball has half the kinetic energy of the heavier ball when they reach the ground.

### Final Answer:
The lighter ball has half as much kinetic energy as the heavier ball.