Q. Why does a helium balloon go forward in an accelerating car?

A.
"The problem is I have a Helium balloon tied to the back seat of my car. The Balloon is not touching anything but the string that it is tied to. When I slam on the gas pedal of the car and start accelerating the balloon drifts to the front of the car. The faster I accelerate the more the balloon leans to the front of the car, until I reach cruising speed."

When you accelerate, there is a “pseudo-force” that pushes most things to the back of the car – this is the feeling of being pushed back in your seat, when you accelerate and also throws you against the seatbelts when you brake suddenly. This is not a real force – it is an effect of your inertia (your tendency to remain in the same state of motion) hence the name pseudo-force. Pseudo-forces are present anytime your frame of reference accelerates; "centrifugal forces", due to the constant acceleration of circular motion are another famous, familiar example. They even affects the air in your car – so when you accelerate, the air tends to move backwards a little – not so much that people on the front seat suffocate, but enough that the air gets a little more dense (and higher pressure) the further back you go.

As a helium balloon floats due to Archimedes' Principle it rises in the direction reduces the pressure of the air fastest. In normal air, the pressure drops as you go directly upwards (only slightly, but the balloon feels this slight force) but in your car, when the pressure builds up briefly at the back, the balloon “rises” forwards. The balloon will stand vertically if you have constant speed and to go backwards when you brake except the pseudo force on it when you brake sharply may overwhelm the pressure gradient and make it go forwards anyway.