"an object is dropped from a great height. its acceleration due to gravity is 9.8 meters per second per second and its deceleration due to air resistance is proportional to its velocity. after a long period of time, its terminal velocity of 30 meters per second is achieved. what was its velocity after 10 seconds? "

When the object reaches its terminal velocity, its net acceleration is zero:

The net acceleration on the object is given by the difference between the gravitational acceleration g (directed downward) and the acceleration due to the air resistance, :
(1)
the acceleration due to the air resistance is proportional to the velocity, so we can write it as the product between a certain coefficient k and the velocity:
(2)

When the terminal velocity is reached, the net acceleration must be zero. We can rewrite (1) using (2):

from which we find k:


Then we know the object moves by accelerated motion, with net acceleration , so its velocity at time t is given by:

And by re-arranging this equation and using t=10 s and , we find the velocity of the object after 10 seconds: