Particle Motion

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: where is a constant of proportionality. Solving this differential equation shows that the velocity approaches a constant terminal velocity 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.

Step 1: Understanding the motion equations.
The equations for and describe a circular motion since and , which follow a circular path in the -plane. The -motion is linear, given by , indicating a constant increase in the -direction.

Step 2: Analyzing the options.
- (A) Circular: Incorrect. While and describe circular motion, the -motion indicates that the path is not purely circular.
- (B) Elliptical: Incorrect. The motion in the -plane is circular, not elliptical.
- (C) Helical: Correct. The particle follows a helical path, with circular motion in the -plane and linear motion along the -axis.
- (D) Spiral: Incorrect. A spiral involves decreasing radius in the -plane, which is not the case here.

Step 3: Conclusion.
The correct answer is (C) Helical because the particle's motion is circular in the -plane and linear in the -direction, forming a helical trajectory.