1. Jan 12, 2009

### jeffro0685

I've tried finding this in the forums elsewhere, but can't seem to find the answer I'm looking for. So here's my dilemma...

There are two objects, object X and object Z, that are the same volume and shape. However, object X has a greater mass than object Z.

Objects X and Z are both dropped out of a stationary helicopter tens of thousands of feet in the air. During free fall, because of the difference in mass, X will eventually reach a terminal velocity of 300 m/s and Z will eventually reach a terminal velocity of 200 m/s.

Do each of the objects fall side by side at a constant rate of gravity (10m/s^2) until Z reaches it's terminal velocity of 200 m/s, at which point object X continues on accelerating at 10 m/s^2 until it comes to it's own terminal velocity of 300 m/s? Or do both objects fall at different rates of acceleration? In other words, because of air resistance, does one object fall at a rate faster than the other?

I understand that in a vacuum all objects, regardless of mass, fall at the exact same rate. However, when objects fall in air they begin at a rate of 10 m/s^2 and eventually reach a terminal velocity, which is a rate of 0m/s^2.

If each object began at a rate of 10 and ended at a rate of 0 would this decrease in rate occur in the form of exponential decay? Or would acceleration remain constant until terminal velocity was reached, at which point acceleration would just drop off to 0?

The place where I'm getting confused is this: I've been told that all objects, regardless of mass, fall at the same rate - even in air. But since objects X and Z are falling in air and are of different masses, they have terminal velocities which are NOT the same.

So as each of these objects approaches it's terminal velocity wouldn't the force caused by air resistance gradually increase and, therefore, gradually and exponentially cause free fall acceleration to decrease from 10 m/s^2 to 0 m/s^2? And if this is the case doesn't this mean, since both X and Z begin at a speed of 0 m/s but end at two different speeds, that they would fall at different rates of acceleration (aka NOT fall side by side)?

Or is terminal velocity a sort of "threshold" when it comes to free fall acceleration in air? Would X and Z continue to accelerate side-by-side at 10 m/s^2 until the threshold of terminal velocity was reached for each respective object, thus dropping the rate of acceleration off to 0?

Perhaps a simpler way to ask it is this: is terminal velocity something that is gradually settled into as the increasing velocity of free fall gradually creates an exponential increase in upwards air resistance? Or does the density constant of the air mean that all objects (given their individual cross-sectional area and mass) have a precise speed limit? And does each object free falling in air travel at the constant rate of gravity as they approach their "speed limit" - halting all acceleration once this speed limit is reached?

I'm clearly missing something here. It's been a while since I sat in a Physics classroom, but even then most of the equations we learned about free fall didn't account for air resistance. So I'm not sure.

Last edited: Jan 12, 2009
2. Jan 12, 2009

### Cyrus

The acceleration will depend on the (a) mass, and the (b) aerodynamic drag.

Because (a) is never the same in both cases, (although (b) could be, if you designed them to have the same aeroynamic drag), the acceleration is not the same.

You could achieve (b) being the same for both masses if they are, for example, two balls of the same size but of different material (lead v plastic).

3. Jan 12, 2009

### gmax137

the usualsolution to this problem is to take the air resistance as proportional to either the velocity or to the velocity squared. Then the total force on the object is the sum of the gravity force (=m*g) plus the friction (=-k*v). then the acceleration is (g-(k/m)*v). (or g-(k/m)*v^2). Integrate this with respect to time to get an equation for velocity vs time. Then you can vary k and m as you wish to see how they affect the velocity.

4. Jan 12, 2009

### DaveC426913

??
He stated this as a given. The two objects are same shape and volume.

5. Jan 12, 2009

### gmax137

search wiki for 'terminal velocity' - it has an integration of the 'v-squared' equation showing the hyperbolic tangent form of the result. You can plot it with different masses to see the variation in shape.

6. Jan 12, 2009

### Cyrus

7. Jan 12, 2009

### jeffro0685

Haha. I apologize for the long thread, but I felt it necessary to do so in order to keep from getting generic answers about the basic differences between free fall in a vacuum and free fall in air

8. Jan 12, 2009

### DaveC426913

I learned a new TLA* for geekdom: tldr (Too long. Didn't read.)

*Three Letter Acronym**

**Actually, this is an ETLA - or Extended TLA