A ball of mass `12 kg` and another of mass `6 kg` are dropped from a `60` feet tall building after a fall of `30` feet each, towards earth, their kinetic energies will be in the ratio of .

Options

A. `sqrt2:1`
B. `1:4`
C. `2:1`
D. `1:sqrt2`

Explanation

To find the ratio of the kinetic energies of two balls of different masses dropped from the same height, we can use the principle of conservation of energy. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have two balls:
- Ball 1: Mass \( m_1 = 12 \, \text{kg} \)
- Ball 2: Mass \( m_2 = 6 \, \text{kg} \)

Both balls are dropped from a height of 60 feet, and we are interested in their kinetic energies after they have fallen 30 feet.

### Step 2: Calculate the Potential Energy Lost
The potential energy (PE) lost by each ball when it falls a height \( h \) can be calculated using the formula:
\[ \text{PE} = mgh \]
where:
- \( m \) is the mass of the ball,
- \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)),
- \( h \) is the height fallen (30 feet).

First, we need to convert the height from feet to meters (1 foot = 0.3048 meters):
\[ h = 30 \, \text{feet} = 30 \times 0.3048 \, \text{m} = 9.144 \, \text{m} \]

### Step 3: Calculate the Potential Energy for Each Ball
Now we can calculate the potential energy lost for each ball after falling 30 feet.

For Ball 1 (12 kg):
\[ \text{PE}_1 = m_1 g h = 12 \times 9.81 \times 9.144 \]

For Ball 2 (6 kg):
\[ \text{PE}_2 = m_2 g h = 6 \times 9.81 \times 9.144 \]

### Step 4: Calculate the Kinetic Energy Gained
According to the conservation of energy, the potential energy lost is equal to the kinetic energy (KE) gained:
\[ \text{KE} = \text{PE}_{\text{lost}} \]

Thus, the kinetic energy for each ball after falling 30 feet will be:
\[ \text{KE}_1 = \text{PE}_1 \]
\[ \text{KE}_2 = \text{PE}_2 \]

### Step 5: Find the Ratio of Kinetic Energies
The ratio of the kinetic energies of the two balls is given by:
\[ \frac{\text{KE}_1}{\text{KE}_2} = \frac{m_1 g h}{m_2 g h} \]
Notice that \( g \) and \( h \) will cancel out:
\[ \frac{\text{KE}_1}{\text{KE}_2} = \frac{m_1}{m_2} = \frac{12}{6} = 2 \]

### Conclusion
The ratio of the kinetic energies of the two balls after falling 30 feet is \( 2:1 \).