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Step-by-Step Solution
Step 1: Identify the Data
• Mass of ball 1, $m_1 = 2\,\text{kg}$
• Mass of ball 2, $m_2 = 4\,\text{kg}$
• Both are dropped from the same height (60 feet), but we are comparing their kinetic energies after they each fall 30 feet.
Step 2: Understand the Concept of Free Fall
When two objects are dropped from rest in a uniform gravitational field (neglecting air resistance), they both acquire the same velocity after the same vertical distance. This means:
$ v_1 = v_2 $
(because the final velocity in free fall depends only on the distance fallen and acceleration due to gravity, not on the mass).
Step 3: Express the Kinetic Energy
The kinetic energy ($KE$) of a body is given by:
$ KE = \frac{1}{2}\,m\,v^2 $
Since each ball has the same velocity after falling the same distance, we compare their kinetic energies based on their masses alone.
Step 4: Compute the Ratio of Kinetic Energies
Let $KE_1$ be the kinetic energy of the 2 kg ball, and $KE_2$ be the kinetic energy of the 4 kg ball. Because both share the same velocity ($v$):
$ \frac{KE_1}{KE_2}
= \frac{\frac{1}{2} m_1 v^2}{\frac{1}{2} m_2 v^2}
= \frac{m_1}{m_2}
= \frac{2}{4}
= \frac{1}{2}. $
Therefore, the ratio of their kinetic energies is $1:2.$
Step 5: State the Final Answer
After they each fall 30 feet, the ratio of their kinetic energies is $1 : 2.$
