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Step-by-Step Solution
Step 1: Identify the Physical Situation
A uniformly charged spherical shell of total charge Q is initially at rest with radius $R_0$. Because of mutual electrostatic repulsion, it expands and its radius $R(t)$ increases over time.
Step 2: Apply Energy Conservation
Since no external forces act on the system, the total mechanical (kinetic + electrostatic potential) energy remains constant. At any instant, let the mass of the shell be m and its radius be $R$. Then:
• Kinetic Energy (K.E.) = $ \tfrac{1}{2} m V^2 $
• Electrostatic Potential Energy (P.E.) of a uniformly charged spherical shell of charge Q and radius R is $ \tfrac{K Q^2}{2 R} $, where $K = \tfrac{1}{4 \pi \epsilon_0}$ in SI units.
Initially (when $R = R_0$):
• K.E. $= 0$ (since it is at rest)
• P.E. $= \dfrac{K Q^2}{2 R_0}$.
Hence, the total initial energy is
$$
E_{\text{initial}}
= 0
+ \dfrac{K Q^2}{2 R_0}
= \dfrac{K Q^2}{2 R_0}.
$$
Step 3: Write the Energy Conservation Equation
At any later time, with radius $R(t) = R$ and speed $V(t) = V$, the total energy is
$$
\text{K.E.} + \text{P.E.}
= \dfrac{1}{2} m V^2 + \dfrac{K Q^2}{2 R}.
$$
Since total energy is conserved,
$$
\dfrac{1}{2} m V^2 + \dfrac{K Q^2}{2 R}
= \dfrac{K Q^2}{2 R_0}.
$$
Step 4: Solve for the Speed $V$
Rearranging for $V^2$:
$$
\dfrac{1}{2} m V^2
= \dfrac{K Q^2}{2 R_0}
- \dfrac{K Q^2}{2 R}
\quad \Longrightarrow \quad
V^2
= \dfrac{K Q^2}{m}
\Bigl(\dfrac{1}{R_0} - \dfrac{1}{R}\Bigr).
$$
Therefore,
$$
V
= \sqrt{
\dfrac{K Q^2}{m}
\Bigl(\dfrac{1}{R_0} - \dfrac{1}{R}\Bigr)
}.
$$
Step 5: Interpret the Velocity vs. Radius Graph
As $R$ begins at $R_0$ (and $ \tfrac{1}{R_0} - \tfrac{1}{R_0} = 0$), the velocity starts from zero. As $R$ increases, $V$ increases accordingly but with a decreasing slope because the term
$ \bigl(\tfrac{1}{R_0} - \tfrac{1}{R}\bigr) $ approaches a constant value at large $R$ (it never grows unbounded). Hence, the graph of $V$ versus $R$ starts at zero (when $R=R_0$) and increases, but its slope keeps decreasing.
Correct Answer Figure
The correct representative curve that shows this behavior (starting from zero velocity at $R_0$, then gradually increasing with a diminishing slope) is:
