You throw a baseball straight upward. the drag force is proportional to v2. in terms of g, what is the y-component of the ball's acceleration when the ball's speed is half its terminal speed and (a) it is moving up? (b) it is moving back down?

The first thing to do is a free-body diagram in a vertical direction.

We have then:

ma = mg - kv ^ 2

Where,

k: proportionality constant.

For the terminal velocity, we have:

a = 0

Clearing the value of v:

v = root (mg / k)

v ^ 2 = mg / k

The ball's speed is half its terminal speed:

v = root (mg / 4k)

v ^ 2 = mg / 4k

going up

ma = - mg - kv ^ 2

a = - g - gk / 4k

a = - g - g / 4

a = - 5g / 4

a = - 5g / 4

Going down

ma = - mg + kv ^ 2

a = - g + gk / 4k

a = - g + g / 4

a = - 3g / 4

a = - 3g / 4