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 the ratio of

Options

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

Answer

To solve the problem, we need to determine the kinetic energies of the two balls after they have fallen 30 feet. Let's break it down step by step.

### Step 1: Understand the Problem
We have two balls:
- Ball 1: Mass \( m_1 = 2 \, \text{kg} \)
- Ball 2: Mass \( m_2 = 4 \, \text{kg} \)

Both balls are dropped from a height of 60 feet and fall 30 feet. We need to find the ratio of their kinetic energies after falling.

### Step 2: Calculate the Velocity After Falling 30 Feet
Since both balls are dropped from the same height and fall the same distance, they will have the same final velocity when they have fallen 30 feet.

Using the equation of motion:
\[ v^2 = u^2 + 2as \]
where:
- \( v \) = final velocity
- \( u \) = initial velocity (0, since they are dropped)
- \( a \) = acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \))
- \( s \) = distance fallen (30 feet)

First, we need to convert 30 feet to meters:
\[ 30 \, \text{feet} = 30 \times 0.3048 \, \text{m} = 9.144 \, \text{m} \]

Now, substituting the values into the equation:
\[ v^2 = 0 + 2 \times 9.81 \times 9.144 \]
\[ v^2 = 2 \times 9.81 \times 9.144 \approx 179.64 \]
\[ v \approx \sqrt{179.64} \approx 13.38 \, \text{m/s} \]

### Step 3: Calculate Kinetic Energies
The kinetic energy (KE) is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]

For Ball 1 (mass \( m_1 = 2 \, \text{kg} \)):
\[ KE_1 = \frac{1}{2} \times 2 \times (13.38)^2 \]
\[ KE_1 = 1 \times 179.64 \approx 179.64 \, \text{J} \]

For Ball 2 (mass \( m_2 = 4 \, \text{kg} \)):
\[ KE_2 = \frac{1}{2} \times 4 \times (13.38)^2 \]
\[ KE_2 = 2 \times 179.64 \approx 359.28 \, \text{J} \]

### Step 4: Find the Ratio of Kinetic Energies
Now we find the ratio of the kinetic energies:
\[ \text{Ratio} = \frac{KE_1}{KE_2} = \frac{179.64}{359.28} = \frac{1}{2} \]

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

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