To solve the problem, we need to find the ratio of the kinetic energies of two balls of different masses dropped from the same height after falling a certain distance. Here’s a step-by-step solution:
### 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 a distance of 30 feet. We need to find the ratio of their kinetic energies after falling 30 feet.
### Step 2: Calculate the Velocity of Each Ball
Since both balls are dropped from the same height and fall the same distance, we can use the formula for the velocity of an object in free fall:
\[
v = \sqrt{2gh}
\]
where \( g \) is the acceleration due to gravity and \( h \) is the height fallen.
Here, the height fallen \( h = 30 \, \text{feet} \). We need to convert feet to meters for standard calculations:
\[
30 \, \text{feet} = 30 \times 0.3048 \, \text{m} \approx 9.144 \, \text{m}
\]
Now, using \( g \approx 9.81 \, \text{m/s}^2 \):
\[
v = \sqrt{2 \times 9.81 \, \text{m/s}^2 \times 9.144 \, \text{m}} = \sqrt{179.56} \approx 13.39 \, \text{m/s}
\]
### Step 3: Calculate the Kinetic Energy of Each Ball
The kinetic energy (KE) of an object is given by the formula:
\[
KE = \frac{1}{2} mv^2
\]
For Ball 1:
\[
KE_1 = \frac{1}{2} m_1 v^2 = \frac{1}{2} \times 2 \, \text{kg} \times (13.39 \, \text{m/s})^2
\]
Calculating \( v^2 \):
\[
v^2 \approx 179.56 \, \text{m}^2/\text{s}^2
\]
Thus,
\[
KE_1 = \frac{1}{2} \times 2 \times 179.56 \approx 179.56 \, \text{J}
\]
For Ball 2:
\[
KE_2 = \frac{1}{2} m_2 v^2 = \frac{1}{2} \times 4 \, \text{kg} \times (13.39 \, \text{m/s})^2
\]
Thus,
\[
KE_2 = \frac{1}{2} \times 4 \times 179.56 \approx 359.12 \, \text{J}
\]
### Step 4: Find the Ratio of Kinetic Energies
Now, we can find the ratio of the kinetic energies:
\[
\text{Ratio} = \frac{KE_1}{KE_2} = \frac{179.56}{359.12} = \frac{1}{2}
\]
### Conclusion
The ratio of the kinetic energies of the two balls after falling 30 feet is:
\[
\text{Ratio} = 1 : 2
\]
