AIIMS 2006 Physics Free Fall MCQ Question

To solve the problem, we need to analyze the motion of the two spheres as they fall under the influence of gravity. Both spheres are dropped from the same height and are subject to the same gravitational acceleration.

Correct Answer: D) Acceleration

Explanation: When both spheres are in free fall, they experience the same acceleration due to gravity, regardless of their masses. The acceleration due to gravity (denoted as $g$) is approximately $9.81 \, \text{m/s}^2$ near the Earth's surface. This acceleration acts downwards and affects all objects equally, irrespective of their mass.

Therefore, both spheres, when they are 1 m above the ground, have the same acceleration of $g$.

Why Other Options Are Incorrect:

A) Momentum: Momentum ($p$) is defined as the product of mass and velocity:

$p = m \cdot v$

At the moment they are both 1 m above the ground, they will have different velocities because they were released at the same time from the same height. The velocity can be calculated using the equation of motion for free fall:

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

where $u$ is the initial velocity (0 in this case since they are dropped), $g$ is the acceleration due to gravity, and $h$ is the distance fallen. The distance fallen when they are 1 m above the ground is:

$h = 72 \, \text{m} - 1 \, \text{m} = 71 \, \text{m}$

Plugging in the values, we find:

$v^2 = 0 + 2 \cdot 9.81 \cdot 71$ $v^2 = 1396.62$ $v = \sqrt{1396.62} \approx 37.43 \, \text{m/s}$

Now, calculating momentum for both spheres:

• For the 2 kg sphere: $p_1 = 2 \, \text{kg} \cdot 37.43 \, \text{m/s} \approx 74.86 \, \text{kg m/s}$
• For the 4 kg sphere: $p_2 = 4 \, \text{kg} \cdot 37.43 \, \text{m/s} \approx 149.72 \, \text{kg m/s}$

Since the momenta are different, this option is incorrect.

B) Kinetic Energy: Kinetic energy ($KE$) is given by the formula:

$KE = \frac{1}{2} m v^2$

Since the velocities of both spheres are the same when they reach 1 m above the ground (as calculated previously), the kinetic energies will differ because their masses are different:

• For the 2 kg sphere: $KE_1 = \frac{1}{2} \cdot 2 \cdot (37.43)^2 \approx 1395.73 \, \text{J}$
• For the 4 kg sphere: $KE_2 = \frac{1}{2} \cdot 4 \cdot (37.43)^2 \approx 2791.46 \, \text{J}$

Thus, the kinetic energies are different, making this option incorrect.

C) Potential Energy: Potential energy ($PE$) is given by:

$PE = mgh$

When both spheres are 1 m above the ground, their potential energy will also differ due to their different masses:

• For the 2 kg sphere: $PE_1 = 2 \cdot 9.81 \cdot 1 \approx 19.62 \, \text{J}$
• For the 4 kg sphere: $PE_2 = 4 \cdot 9.81 \cdot 1 \approx 39.24 \, \text{J}$

Since the potential energies are not the same, this option is also incorrect.

Conclusion:

The only quantity that remains the same for both spheres during free fall, despite their differing masses, is the acceleration due to gravity, which is $9.81 \, \text{m/s}^2$. Therefore, the correct answer is D) acceleration.