A ball of mass`2 kg` and another of mass `4 kg` are dropped together from a `60` feet tall building . After a fall of `30` feet each towards earth , their respective kinetic energies will be in the ratio of

Options

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

Explanation

To solve the problem, we need to find the kinetic energies of two balls of mass 2 kg and 4 kg after they have fallen 30 feet from a height of 60 feet. We will then calculate the ratio of their kinetic energies.

### Step-by-Step Solution:

1. **Identify the masses and initial conditions**:
- Mass of ball 1 (m1) = 2 kg
- Mass of ball 2 (m2) = 4 kg
- Initial velocity (u) = 0 (since they are dropped)
- Height fallen (s) = 30 feet

2. **Convert height from feet to meters** (if needed for calculations):
- 1 foot = 0.3048 meters
- Height fallen in meters = 30 feet × 0.3048 m/foot = 9.144 m

3. **Use the third equation of motion to find final velocity**:
- The third equation of motion states: \( v^2 = u^2 + 2as \)
- Here, \( u = 0 \) (initial velocity), \( a = g \) (acceleration due to gravity, approximately \( 9.81 \, m/s^2 \)), and \( s = 9.144 \, m \).
- Therefore, \( v^2 = 0 + 2 \times 9.81 \times 9.144 \)
- Calculate \( v^2 \):
\[ v^2 = 2 \times 9.81 \times 9.144 \approx 179.57 \]
- Thus, \( v \approx \sqrt{179.57} \approx 13.38 \, m/s \) (final velocity for both balls since they fall together).

4. **Calculate the kinetic energy for each ball**:
- Kinetic energy (KE) is given by the formula: \( KE = \frac{1}{2} mv^2 \)
- For ball 1 (mass = 2 kg):
\[ KE_1 = \frac{1}{2} \times 2 \times (13.38)^2 \approx \frac{1}{2} \times 2 \times 179.57 \approx 179.57 \, J \]
- For ball 2 (mass = 4 kg):
\[ KE_2 = \frac{1}{2} \times 4 \times (13.38)^2 \approx \frac{1}{2} \times 4 \times 179.57 \approx 359.14 \, J \]

5. **Find the ratio of their kinetic energies**:
- The ratio of kinetic energies \( \frac{KE_1}{KE_2} = \frac{179.57}{359.14} = \frac{2}{4} = \frac{1}{2} \).

### Final Answer:
The ratio of the kinetic energies of the two balls after falling 30 feet is \( \frac{1}{2} \).