How can I describe the motion of an object falling, due to gravity, through a fluid? And when does the object reach terminal velocity?

Lets consider a skydiver jumping from an airplane. Once the skydiver has let go of the plane they are pulled towards the earth due to gravity.

By Newton’s second law:

F = ma,

the sky diver begins to accelerate towards the earth at an acceleration speed dictated by their weight. The earth’s gravitational field acting on an objects mass creates an objects weight.

                                    W = mg,          g =  9.81 ms^-2

However, as the skydiver accelerates towards the earth they will experience an air resistance. Air resistance is an upward force experienced by an object colliding with air molecules as it falls. The amount of upward force a skydiver experiences will vary due to the surface area of the skydiver, the speed the skydiver is falling and the density of the air itself.

As the skydiver accelerates downwards the upward force due to the air resistance will increase. Eventually the downward force due to gravity and the upward force due to air resistance will balance and the skydiver will reach terminal velocity.

                                    W = F_AIR

Terminal velocity is when there are no net forces acting on an object and thus the object will move at a constant velocity.

Now consider what will happen if the skydiver opens their parachute.