Question

An object of mass$m$ is dropped from the top of a tall building of height $h$. Two forces act on it during its fall: gravity and air resistance. The later is always directed against the motion and its magnitude is proportional to the speed of the object. The acceleration due to gravity is $g$. Which of the following statements is true about the motion of the object?

• A. The object will fall throughout with uniform speed.
• B. The object will fall throughout with uniform acceleration.
• C. The sum of the kinetic energy and gravitational potential energy remains constant throughout the fall.
• D. The maximum kinetic energy of the object will be less than $mgh$.

Answer 1

The correct option is D

The maximum kinetic energy of the object will be less than $mgh$.

Explanation for correct option: (d) The maximum kinetic energy of the object will be less than $mgh$.

1. The gravitational potential energy of an object having mass $m$ and at a height of $h$ from the earth is $mgh$.
2. The kinetic energy of an object of mass $m$ and velocity $v$ is $\frac{1}{2}m{v}^2$.
3. At height $h$ the kinetic energy of the object is zero as it's velocity is zero.
4. As the object starts falling it's potential energy is converted into kinetic energy and heat energy due to air resistance.
5. At the bottom of the building the kinetic energy is maximum because the velocity of the object is maximum.
6. When the object reaches the bottom of the building all its potential energy gets converted into kinetic energy and heat energy.
7. So, $mgh=Kineticenergy+Heatenergy$
8. Therefore, $Kineticenergy=mgh-Heatenergy$
9. We know that heat energy will have a non-zero value. So, The maximum kinectic energy will be less than $mgh$.

Explanation for incorrect options:

(a) The object will fall throughout with uniform speed.

The net force on the object is non-zero and in downward direction. So, the object has an acceleration. Therefore the speed can not constant.

(b) The object will fall throughout with uniform acceleration.

Upward acceleration due to air resistance is changing due to change in velocity of the object. So, the net acceleration of the object will not be constant.

(c) The sum of the kinetic energy and gravitational potential energy remains constant throughout the fall.

The potential energy of the object gets converted into kinetic energy and heat energy. So, the sum of the kinetic energy, gravitational potential energy and heat energy remains constant.

Hence, the maximum kinetic energy of the object will be less than $mgh$.

Thus, option (d) is the correct option.