A particle of mass 100 g travelling at an initial speed u collides...

Given mass of each particle, $m = 100 \text{ g} = 0.1 \text{ kg}$.

Let the initial velocity of the first particle be $u$ and the coefficient of restitution be $e$.

The velocity of the center of mass is $v_{cm} = \dfrac{m u + m(0)}{m + m} = \dfrac{u}{2}$.

The final kinetic energy of the system can be written as the sum of the kinetic energy of the center of mass and the kinetic energy relative to the center of mass:

$K_f = \dfrac{1}{2}(2m)v_{cm}^2 + \dfrac{1}{2}\mu v_{rel,f}^2$

where reduced mass $\mu = \dfrac{m \cdot m}{m + m} = \dfrac{m}{2}$ and final relative velocity $v_{rel,f} = e u$.

Substituting these values, we get:

$K_f = m\left(\dfrac{u}{2}\right)^2 + \dfrac{1}{2}\left(\dfrac{m}{2}\right)(eu)^2 = \dfrac{mu^2}{4}(1 + e^2)$

Given $K_f = 0.2 \text{ J}$ and $m = 0.1 \text{ kg}$:

$0.2 = \dfrac{0.1 u^2}{4}(1 + e^2)$

$u^2(1 + e^2) = 8$

$u = \sqrt{\dfrac{8}{1 + e^2}}$

For the minimum value of $u$, the denominator must be maximum, which occurs for a perfectly elastic collision ($e = 1$):

u_\min = \sqrt{\dfrac{8}{1 + 1}} = \sqrt{4} = 2 \text{ m/s}

For the maximum value of $u$, the denominator must be minimum, which occurs for a perfectly inelastic collision ($e = 0$):

u_\max = \sqrt{\dfrac{8}{1 + 0}} = \sqrt{8} = 2\sqrt{2} \text{ m/s}

Thus, the minimum and maximum values of $u$ are $2 \text{ m/s}$ and $2\sqrt{2} \text{ m/s}$.