Popcorn

One thing that is most important to realize in this problem is, we have to find the velocity of the "centenr of mass " of popcorn.

For a spherical solid substance, the center of mass lies at its geometric center.

Initially the radius of popcorn was $r_{1} = 1mm = 10^{-3} m$

Hence, initial displacement between the center of mass of popcorn

and the base of the pan on which it was resting is $10^{-3} m$

After expansion, the radius increased to $r_{2} = 1 cm = 10^{-2} m$ in just 0.1 s

Hence, final displacement of the center of mass of popcorn = $10^{-2} m$

Hence, velocity = $v = \frac{\bigtriangleup r}{\bigtriangleup t}$

$= \frac{10^{-2} - 10^{-3}}{0.1}$

$= 9 \times 10^{-2} m s^{-1}$

Speed = distance traveled / time taken

SI unit for distance = meter, SI for time = seconds

So, 1mm = 1/1000 m and 1cm = 1/100m

Thus, distance traveled = (1 / 100) - (1 / 1000) = 9 x (10^-3)m

Then, speed = (9 x (10^-3)) / 0.1 = 0.09 m/s

I am assuming 0.1 seconds is very little time for the popcorn to pop off from the pan. So initially the center of mass of the popcorn is at 1mm from the pan, and finally it is at 1cm or 10mm.

The center of mass moves by 9mm in 0.1 seconds, so its velocity can be averaged to be 9/0.1 mm/sec or 0.09 m/s

dr is the difference of radius, dt is time taken. dr/dt = 0.009/0.1 dr/dt=0.09 m/s