Re: Why do balls filled with different gases bounce the same height?

The question: "Why do balls filled with different gases bounce the same height? We filled playground-type balls with approximately the same volume of He, O2, CO2, N2, and air. We thought that the lighter molecular weight gas (He) would bounce higher. Why did all the balls bounce the same height and weigh the same?"

First do a search using our search engine to search on "ball bounce pressure" and these previous answers will be listed:
higher with more pressure
increasing bounciness

Then do a search on "buoyancy gases" and you will find an answer in our archives that discusses
buoyancy calculations with different gases.

The one major factor that will not cause differences in ball bounce height is the different gas, if the balls have the same pressure, because the pressure of any gas is mainly due to the number of atoms (or molecules) of the gas, and you will have tried to make the pressure the same in all the balls. I'm assuming that all the balls are the same type and size and mass. I am also assuming that the temperature of all the balls is more or less the same, and that the balls are all dropped on the same surface.

In addition to the previous answers above, see this document about the proper inflation of a basketball. Increased pressure provides the ball with "more bounciness" because the shape of the ball doesn't deform as much with the bounce, so the energy lost in the bounce (due mainly to heating the ball material as it deforms) is less with higher pressure. But if we assume the three factors of the previous paragraph, the balls will all bounce the same.

The buoyant force will cause the balls filled with different gases to have different net forces downward and upward, but the height of the bounces will be the same. The time for the ball to fall and then rebound will not be the same, however. The following paragraphs will explain why these things are so.

The buoyant force is numerically equal to the mass difference between the volume of air displaced by the ball and the mass of the same volume of the filling gas. But keep in mind that the filling gas is at a pressure higher than atmospheric pressure, so the filling gas will be more dense than the same gas at atmospheric pressure. For instance, if we assume that the balls are pressurized to one half atmosphere (7.35 pounds per square inch) then the density of the filling gas will be 1.5 times its density at atmospheric pressure.

You didn't say exactly what kind of balls you were using. But I found online a playground ball that is 8.5 inches (0.216 meters) in diameter, and, although I don't know what its mass is, I'm going to guess that it has a mass of roughly 0.5 kg. The volume displaced is thus the volume of a sphere that is 0.108 m radius, which is about 0.0053 m³ (4*PI*radius³/3). The third reference above indicates that the 1-atmosphere density of air is 1.29 kg / m³ and that of helium is 0.179 kg / m³. Taking the fill density to be 1.5 times these given, the fill density for the air is about 1.9 kg / m³ and that for helium is about 0.27 kg / m³. So the masses of the ball's volume of the various gases are:
air, surrounding the ball: 0.0068 kg
fill air: 0.0101 kg
fill helium: 0.0014 kg
The ball filled with air thus has a mass of 0.5101 kg, compared to the 0.0068 kg of air that it displaces. The helium-filled ball has a mass of 0.5014 kg. So the net weight (gravitational force) of the air-filled ball is (0.5101 - 0.0068 kg)*9.8 m / s² = 4.93 Newtons, while that of the helium-filled ball is (0.5014 - 0.0068 kg)*9.8 m / s² = 4.85 Newtons. (I am making calculations only for air and for helium, but the results can be calculated for all the different gases.)

Newton's 2nd Law tells us that the acceleration of an object is equal to the net force on it divided by its mass. The air-filled ball has a mass of 0.5101 kg, with a net weight of 4.93 Newtons, so it is being accelerated toward the ground at a rate of 9.669 m / s², while the helium-filled ball with a mass of 0.5014 kg is accelerated at 9.667 m / s². (Compare these accelerations to the usual acceleration of gravity at the surface of the Earth: 9.8 m / s²)

Let's say the balls are both dropped from a height "h" of 2 meters. The time required to go that distance for the given acceleration "a" is
t = (2*h/a)^1/2
(See Equations of Motion at Wikipedia.)
For the two different accelerations given above, the air-filled ball takes 0.6432 seconds to fall while the helium-filled ball takes 0.6433 seconds. There is only 0.0001 seconds difference between the two!

The balls will obtain velocities downward of magnitude
v = (2*a*h)^1/2
so the air-filled ball will be going 6.219 m/s when it hits and the helium-filled ball will be going 6.218 m/s when it hits.

When the balls get to the floor they will bounce. If the balls provide a perfectly elastic collision with the floor they will rebound with the same speed with which they hit the floor. How high will they bounce up? For a given speed and acceleration
h = v² / (2*a)
so the air-filled ball will go back up 2.00 meters, and the helium-filled ball will go back up 2.00 meters!!! They both go back up the same height, even though their accelerations and speeds are slightly different.

In the real world the balls won't rebound with the same speed with which they hit the floor, but they should all rebound with about the same fraction of the original speed, so even though they won't actually go back to 2 meters height they will all go back to pretty much the same height. Also in the real world there is some air resistance (drag), but within the limits of the precision with which you can make measurements it is safe to ignore the drag over the small distances and speeds that you are encountering.

John Link, MadSci Physicist