Balloon Car

My 12-year-old daughter is proposing an interesting project. She is planning to attach a number of helium balloons to a chair, which in turn would be tethered by means of a rope to a Ferrari. Her 13-year-old friend would then drive the Ferrari around, while she sits in the chair enjoying uninterrupted views of the countryside. Leaving aside the legal and insurance difficulties, my daughter is keen to know the maximum speed that she could expect to attain, and how many helium balloons would be required.

Phil Rodgers, Cambridge, UK

Thanks for getting your dad to send in this question! He said not to worry about the "legal and insurance difficulties," so I think it's safe to assume he's taken care of all that.

Note to police: If you've recently taken into custody two unidentified underage drivers, a stolen Ferrari, and a bunch of helium balloons, the person you're looking for is Phil Rodgers in Cambridge, UK.

Okay, on to your question:

Have you ever run with a balloon? It doesn't point straight up. The air rushing past you pushes it down:

How high the balloon goes depends on which force is stronger--the balloon's buoyancy pulling upward, or the wind dragging the balloon backward. If the drag is too strong, the balloon will stay low to the ground and you won't get a good view.

To figure out how fast you can go, let's first figure out how big our cluster of balloons (or one big balloon, which is probably easier) needs to be to lift you.

People your age weigh an average of 43 kilograms, which means you need a balloon 4 meters wide to lift you—that's about the size of a car. (If you don't weigh 43 kilograms, you can put your weight into this formula.)

A 4-meter balloon will be large enough to cancel out your weight. But that's not enough. It just means you wouldn't fall or float—so you'd be towed along the ground behind the car.

To float upward, you need a bigger balloon. A 5 meter balloon will produce 71 kilograms of lift[1]Usually, physics people will make a big deal about how weight and force are different from mass, but in this case, I'm going to resist the urge, because it's easy to just think of everything in terms of weight.​ (here's the formula!). That's enough to cancel out your 43-kilogram weight, plus a few kilograms for the chair and balloon itself.

The balloon will be dragged backward by the air. The faster your friend drives, the more the air will drag the balloon back. You can use this formula to figure out how much "weight" will pull backward on the balloon for different speeds and sizes. Just change the "20 mph" (driving speed) and "5 meters" (balloon size) in the formula.

If the upward pull from the helium is stronger than the backward pull from the wind, the balloon will float at a high angle. If the backward pull is stronger than the upward pull, the balloon will float at a low angle. If you're using a 5-meter balloon, even if you drive only 10 mph, the balloon will float pretty low behind you.

Fortunately, there's a solution: You can make the balloon bigger. As you make the balloon bigger, the buoyancy starts to win out over the drag.[2]The reason is that the buoyancy equation uses diameter^3 but the drag equation uses diameter^2, so if you make diameter bigger, the buoyancy equation grows more.

If you use a 10 meter balloon, the buoyancy is strong enough that you can drive at 20 or 25 mph and still stay pretty high off the ground. A 15 meter balloon is even better; it would let you go 30 mph while still getting a good view.[3]You could make the cable longer, so that even a low angle still gets you high off the ground. But the cable won't be straight; it makes a curve called a catenary. At a low enough angle, making the cable longer would just mean part of it would drag on the ground.

Unfortunately, there's a problem with using larger and larger balloons.

A 15-meter helium balloon plus a 12-year-old can lift 1,895 kilograms. But a Ferrari 458 (plus a 13-year-old) only weighs 1,532 kilograms.

The solution to all this is to ditch the helium. You don't need a balloon. All you need is a kite or a parachute—a surface to act as a wing and redirect that incoming air to push you upward.

In other words, see if your dad will take you parasailing.