Question

A ball of mass m is falling freely under gravity through a viscous medium in which the drag force is proportional to the instantaneous velocity \(v\) of the ball. Neglecting the buoyancy force of the medium, which one of the following figures best describes the variation of \(v\) as a function of time \(t\)? 

• A. A
• B. B
• C. C
• D. D

Hint:

In viscous drag problems, the velocity of a falling object asymptotically approaches a terminal velocity when the drag force balances gravity.

Answer 1

The Correct Option is D

Step 1: Understanding the motion.
In a viscous medium, the drag force is proportional to the velocity of the ball. The net force acting on the ball is the gravitational force minus the drag force, which results in an equation of motion: \[ m \frac{dv}{dt} = mg - kv \] where \(k\) is a constant of proportionality. Solving this differential equation shows that the velocity approaches a constant terminal velocity \(v_t = \frac{mg}{k}\) as time increases.

Step 2: Analyzing the options.
- (A): Incorrect. This curve suggests a linear increase in velocity, which is not the case in this situation.
- (B): Correct. This curve shows an exponential increase in velocity, reaching a terminal velocity asymptotically.
- (C): Incorrect. This curve shows a decrease in velocity, which is not correct for a falling object in a viscous medium.
- (D): Incorrect. This curve suggests an oscillatory motion, which does not match the conditions described.

Step 3: Conclusion.
The correct answer is (B) because the velocity increases exponentially to a terminal value as the drag force balances the gravitational force.