Physics and a Balloon in a Car

Date: 06/09/2005 at 06:01:49
From: Phil
Subject: Applied mathematics, forces on a helium balloon.

Consider a balloon filled with helium, hanging from the floor of a 
car.  The car is sealed, and is full of air at ambient surface 
pressure.  If the car undergoes constant acceleration forwards only, 
which way, relative to the car, will the balloon move?

Obviously there is a rearwards force on the balloon, the most obvious 
force in opposition to the acceleration of the car.  But there must 
also be a forwards force on the balloon due to the fact that the air 
in the car is also accelerated rearwards.  Hence there is a higher 
air pressure at the bottom or rear end of the vehicle than there is at 
the forward or top end.

My thinking is that initially, due to the movement of the air within 
the car as the car initially accelerates, as well as the opposite 
reaction, the balloon will move rearwards with the airflow.  Then, as 
the air inside reaches equilibrium, the balloon will move forwards 
again.  I just don't know by how much--will the balloon end up with a 
slightly forwards lean due to it being lighter than air, or will the 
"opposite" reaction overwhelm this force, and keep the balloon with a 
rearwards lean?

Date: 06/09/2005 at 12:17:16
From: Doctor Douglas
Subject: Re: Applied mathematics, forces on a helium balloon.

Hi Phil.

Your reasoning is excellent.  The balloon and surrounding air ought to
move backwards at first, but the duration of this is so short
(approximately equal to some length within the car divided by the
speed of sound in air, probably a few milliseconds or so) that this
motion is barely visible. 

At this point, the pressure inside the car equilibrates, but the
acceleration of the car continues.  What happens now?  You already 
have an intuitive feel for what is happening, I think; the heavier air 
flows to the back and pushes the balloon forward.  You can also think 
of the accelerated situation as being equivalent to a small 
gravitational force pointing backwards in the car (this is actually a 
theory first put forth by Einstein).  So the heavier air will flow to 
the back and the lighter balloon will therefore lean forward.  Or 
equivalently, what would a helium balloon do if you were in a parked 
car facing uphill?  Of course it will lean towards the front of the 
car.  As long as the car is accelerating, the balloon will lean 
forward.  If the car's speed levels off at a constant cruising speed 
(acceleration goes to zero), the balloon will return to an upright 
position of equilibrium (perhaps oscillating back and forth around 
this position as it gets there).

I encourage you to actually perform the experiment:  tie a helium 
balloon to your stick shift, and go for a test drive.  Watch what the 
balloon does when you accelerate forward (try to find a place where 
you can accelerate gradually for several seconds), when you decelerate 
(acceleration points backward), and when you go around a curve.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/