Question

A spherical ball is dropped in a long column of a viscous liquid. The speed of the ball as a function of time may be best represented by the graph

• A. A
• B. B
• C. C
• D. D

Answer 1

The correct option is B B

Initial speed of ball$\left(u\right)=0\text{m/s}$.
$\therefore$ we conclude from that, graph $C$ and $D$ do not represent the motion of the spherical ball in the liquid.
Let us see the forces acting on the ball while it is moving in the liquid of viscosity $\eta$.
FBD of three forces acting on the ball is shown in the figure below.


From this,
Gravitational force, $\left(F_g\right)=mg$ remains constant as weight of the ball remains same during the motion.

Thrust force/Buoyant force, $\left(F_t\right)=\rho gV=$ constant
[ where $V$ gives volume and $\rho \to$ density of liquid]

Viscous force, $\left(F_v\right)=bv$
[$b\to$ proportionality constant and $v\to$ velocity]
As velocity increases, viscous force also increases during the motion.

Now, finding the net force on the ball along vertical direction, we get
$F=F_g-F_t-F_v=ma$
As $F_v$ increases, $F$ decreases.
After sufficiently long time, $F$ become zero i.e. net force on the ball is zero. When the net force becomes zero on the ball while it is in motion, ball attains constant velocity.
As we know, if $F$ decreases with time then acceleration of the ball also decreases with time. $[F=ma]$

Slope of $v-t$ graph gives acceleration.
From graph $B$, we can see that the slope of $v-t$ curve, decreases with time and the curve become parallel to $t-$ axis after sufficiently long time
$\therefore$ Graph $B$ correctly represents the motion of the ball in a viscous liquid.