Terminal velocity of an object falling through a liquid

[studentoftheg]

I have a question regarding the terminal velocity of an object falling through a liquid. Now the speed will increase up to the terminal veloity in a given time. What I do not understand is in this calculation I'm performing (solving the equation of motion for the speed, when acceleration equals zero), when i increase the overall mass of the object two things happen regarding the terminal velocity:

1. There is no change in the terminal velocity (which i understand, as at the terminal velocity the acceleration is zero so when solving the equation of motion you are left with the same equation for terminal velocity, not affected by mass).

2. The time it takes the object to reach its terminal velocity increases. This is what I do not understand, I'm almost 100% sure the calculation is correct (I'm doing it in mathcad and checking it via a validated Excel sheet) however I cannot figure out why this happens? It seems counterintuitive that inreasing the mass would result in the object taking longer to reach terminal velocity.

Can anyone shed some light on it? Is it something to do with inertia?

 

[stevenb]

1. There is no change in the terminal velocity (which i understand, as at the terminal velocity the acceleration is zero so when solving the equation of motion you are left with the same equation for terminal velocity, not affected by mass).

Can you show your equation? Something may or many not be right here. Greater mass implies greater force causing it to accellerate. This force must be offset by air resistance, so if terminal velocity does not change then it is because in making the object more massive you also made it bigger in surface area. If you simply were to increase the mass by increasing density and maintaining the exact same size and shape, then I'd expect terminal velocity to be greater.

 

[stevenb]

2. The time it takes the object to reach its terminal velocity increases. This is what I do not understand, I'm almost 100% sure the calculation is correct (I'm doing it in mathcad and checking it via a validated Excel sheet) however I cannot figure out why this happens? It seems counterintuitive that inreasing the mass would result in the object taking longer to reach terminal velocity.

Can anyone shed some light on it? Is it something to do with inertia?

So, on this part, the acceleration curve (vs. time) need not be the same. What is your formula for air resistance versus speed. Is it linear, square law, cubic or something more complicated? How does it change with surface area? How does it compare to the force of gravity?

 

[studentoftheg]

Yeah the equation of motion is:

Mtotal dv/dt + Fdrag v(t)^2 - Wtotal = 0

Where M is the total mass, Fdrag is the drag force (proportional to the square of the velocity) and Wtotal is the submerged weight.

The surface area of the object did not increase at all, I only increased Mtotal (initially I had Mtotal as the submerged weight+the added mass. I then realized that it was meant to be the mass in air + the added mass (design code requirement). Thanks.

 

[studentoftheg]

The drag force equation is: 0.5 p * (Cd* A)

where p = density of liquid
Cd = drag coefficient for object
A = drag area

So this doesn't change. Only thing I change is Mtotal.

 

[studentoftheg]

Ok so to clarify...

I only changed Mtotal (the total mass), the submerged weight stayed the same as I didnt actually change the mass, I just increased Mtotal (from submerged weight+added mass, to mass in air+added mass). And to calculate the terminal velocity you set dv/dt=0 so Mtotal is disregarded and Vterminal = sqrt (Wtotal/Fdrag). So it stays the same.

Still not too sure why the time increases though...

 

[jtbell]

It seems counterintuitive that inreasing the mass would result in the object taking longer to reach terminal velocity.

Imagine what happens if you decrease the mass instead of increasing it. How long does it take a soap bubble to reach terminal velocity?

 

[cjameshuff]

The drag force equation is: 0.5 p * (Cd* A)

where p = density of liquid
Cd = drag coefficient for object
A = drag area

So this doesn't change. Only thing I change is Mtotal.

The drag force doesn't change with mass, but the weight (gravitational force + buoyancy force) does, and terminal velocity is where the drag force equals the weight. So terminal velocity certainly does change when you change mass, keeping all else equal.