Question

A spherical ball is dropped in a long column of a highly viscous liquid. The curve in the graph shown, which represents the speed of the ball (v) as a function of time (t) is:

• A. A
• B. B
• C. C
• D. D

Answer 1

The Correct Option is B

The problem involves understanding the speed-time relationship for a spherical ball dropped in a highly viscous liquid. This is a classic physics problem involving terminal velocity. To determine the correct curve, we need to consider the forces acting on the ball. Initially, gravity accelerates the ball downwards, increasing its speed. As the ball speeds up, viscous drag force, which is proportional to speed in most fluids, opposes this motion.
Formally, the net force \(F\) on the ball can be expressed as: 

Net Force: \(F=mg-kv\)

where \(m\) is the mass of the ball, \(g\) the acceleration due to gravity, \(k\) the drag coefficient (a constant for a particular fluid-ball system), and \(v\) the velocity.

As time progresses, the viscous force \(kv\) increases until it balances the gravitational force \(mg\), resulting in a net force of zero. At this point, the ball achieves terminal velocity \(v_t\), where the speed remains constant. Hence:

Terminal Velocity: \(v_t=\frac{mg}{k}\)

The graph of velocity \(v\) versus time \(t\) will show an initial increase in velocity steeply, eventually leveling off as the terminal velocity is achieved. This results in an exponential growth curve that asymptotically approaches \(v_t\).

By comparing graph options, the curve representing this behavior is option B.