Five balls move through the air as shown in FIGURE Q6.17 All five have the same size and shape. Air resistance is not negligible. Rank in order, from largest to smallest, the magnitudes of the accelerations aa to ae. Some may be equal. Give your answer in the form $a7b=c7d7e$and explain your ranking.

FIGURE Q6.17

This force's magnitude is proportional to the magnitude of the acceleration.

The ball has mass here.$m=50\mathrm{g}$

If the ball is simply released, there will be no air drag.

Then comes the acceleration.

$a_a=g$.

we get mass of ball is$m=100\mathrm{g}$

After that, the ball is simply released.

There will be no opposition from the air.

Then comes the acceleration$a_b=g$.

The net acceleration of the ball when it is travelling downward is

$a=-\left(g-\frac{D}{m}\right)$

The force of drag is$D=\frac{1}{4}A{v}^2$

The ball's mass is$m=50\mathrm{g}$

$=0.50\mathrm{kg}$

we have$v=20\mathrm{m}/\mathrm{s}$

$a_c=-\left(g-\left(\frac{\frac{1}{4}A(20\mathrm{m}/\mathrm{s}{)}^2}{0.05\mathrm{kg}}\right)\right)$

$\text{Magnitude of acceleration is}\left|a_c\right|=g-\left(\frac{\frac{1}{4}A(20\mathrm{m}/\mathrm{s}{)}^2}{0.05\mathrm{kg}}\right)$.

The ball's mass is now$m=100\mathrm{g}$$=0.1\mathrm{kg}$

The ball is currently on its way down.

Then$a_d=-\left(g-\frac{\frac{1}{4}A(20\mathrm{m}/\mathrm{s}{)}^2}{0.1\mathrm{kg}}\right)$

$\left|a_d\right|=g-\left[\frac{\frac{1}{4}A(20\mathrm{m}/\mathrm{s}{)}^2}{0.1\mathrm{kg}}\right]$

e) The ball's mass is now$m=50\mathrm{g}$$=0.05\mathrm{kg}$

The ball is now going upward.

The ball's acceleration is then increased$a_e=\left(g+\frac{\frac{1}{4}A{v}^2}{m}\right)$

$=-\left(g+\frac{\frac{1}{4}A(20\mathrm{m}/\mathrm{s}{)}^2}{0.05\mathrm{kg}}\right)$

The magnitude of the ball's acceleration is

$\left|a_e\right|=\left(g+\frac{\frac{1}{4}A(20\mathrm{m}/\mathrm{s}{)}^2}{0.05\mathrm{kg}}\right)$

Now, based on the information presented above,$a_e>a_a=a_b>a_d>a_c$.