AIPMT PRELIMS 2004 Physics Kinetic Energy MCQ Question

To determine the ratio of the kinetic energies of the two balls after they have fallen 30 feet, we will first use the principles of gravitational potential energy and kinetic energy.

Step 1: Understanding Kinetic Energy

The kinetic energy (KE) of an object is given by the formula:

$$KE = \frac{1}{2} mv^2$$

where $m$ is the mass of the object and $v$ is its velocity.

Step 2: Gravitational Potential Energy and Velocity

When the balls are dropped, they convert potential energy into kinetic energy as they fall. The potential energy (PE) lost by the balls as they fall can be calculated using:

$$PE = mgh$$

where $g$ is the acceleration due to gravity (approximately $32 \, \text{ft/s}^2$ in feet) and $h$ is the height fallen.

Step 3: Calculating the Height Fallen

Both balls fall a height of 30 feet, so we can calculate the potential energy lost:

For the 2 kg ball:

• Mass $m_1 = 2 \, \text{kg}$
• Height $h = 30 \, \text{ft}$

$$PE_1 = m_1 g h = 2 \times 32 \times 30 = 1920 \, \text{ft-lb}$$

For the 4 kg ball:

• Mass $m_2 = 4 \, \text{kg}$

$$PE_2 = m_2 g h = 4 \times 32 \times 30 = 3840 \, \text{ft-lb}$$

Step 4: Kinetic Energy After Falling

Since all the potential energy lost will be converted into kinetic energy (assuming no air resistance):

For the 2 kg ball:

$$KE_1 = PE_1 = 1920 \, \text{ft-lb}$$

For the 4 kg ball:

$$KE_2 = PE_2 = 3840 \, \text{ft-lb}$$

Step 5: Finding the Ratio of Kinetic Energies

Now, we can find the ratio of their kinetic energies:

$$\text{Ratio} = \frac{KE_1}{KE_2} = \frac{1920}{3840} = \frac{1}{2}$$

Step 6: Considering Mass in Kinetic Energy Formula

However, we should also consider how the mass affects the kinetic energy. The kinetic energy formula incorporates mass, so we can express it as:

For the 2 kg ball:

$$KE_1 = \frac{1}{2} m_1 v_1^2$$

For the 4 kg ball:

$$KE_2 = \frac{1}{2} m_2 v_2^2$$

Now, to find the velocities $v_1$ and $v_2$ after falling 30 feet, we can use the equation of motion:

$$v^2 = u^2 + 2gh$$

where $u = 0$ (initial velocity). Thus,

For both balls after falling 30 feet:

$$v^2 = 0 + 2 \times 32 \times 30 = 1920$$

So,

$$v = \sqrt{1920}$$

Now substituting back into the kinetic energy equations:

For the 2 kg ball:

$$KE_1 = \frac{1}{2} \times 2 \times 1920 = 1920 \, \text{ft-lb}$$

For the 4 kg ball:

$$KE_2 = \frac{1}{2} \times 4 \times 1920 = 3840 \, \text{ft-lb}$$

Final Ratio

The ratio of their kinetic energies is:

$$\text{Ratio} = \frac{KE_1}{KE_2} = \frac{1920}{3840} = \frac{1}{2} = 2:1$$

Conclusion

Thus, the correct answer is B) 2 : 1.

Why Other Options Are Incorrect

• A) 1 : 4: This suggests a much larger disparity in kinetic energy than is calculated.
• C) 1 : √2: This is not consistent with the calculated ratio of 2:1.
• D) √2 : 1: This does not reflect the correct relationship derived from the calculations.

In summary, the kinetic energy ratio of the two balls after