My grandmother always encouraged me to try going all around on a swing. At the time I thought this was pretty reckless of her and I was much too scared to try it anyway. But she probably had read this section of Gravity’s Rainbow and knew I would be fine:
But! It actually is possible to go around all the way with the help of rockets! See this video. Yes! There is not only the evil rocket for the world’s suicide but also a good rocket that will take us around the swing.
In the video, they show some blackboard calculations arriving at a required energy of 4000 Joule. But will this really be enough? Let’s double-check, so we can feel confident when we try this at home.
There are three things we need to consider:
1.Getting to the apex
How much energy is needed to get to the top depends on your (and your jetpack’s) weight. If we assume a total weight of 75kg and a swing radius of 2 meters, we can calculate the potential energy at the top:
m*g*h = 75kg * 9.81m/s^2 * 2*2m = 2943 kg*m^2/s^2 = 2943 Joule
(m=mass, g=gravitational force on earth, h = height)
2. Not failing straight down once arriving at the top
It would hurt very badly if you just got to the highest possible point and then crashed on the bar of the swing because you’ve lost all of your speed. So let’s add some extra push to avoid things going awry. We need to make sure that the centripetal force acting at the apex is bigger than the gravitational force:
centripetal force >= gravitational force
=> m* v^2 / r >= m*g
=> v >= sqrt(g*r)
=> v_min = sqrt(g*r)
=> v_min = sqrt(9.81m/s^2 * 2m) = 4.43 m/s = 15.95 km/h
So you need to go at least ~16km/h at the very top for the swing’s chain to stay straight.
To accelerate 75kg to 4.43 m/s you need a kinetic energy of:
0.5 * m * v^2 = 0.5 * 75kg * (4.43m/s)^2 = 735.75 Joule
3. Accounting for air resistance
Air resistance is often neglected in physics thought experiments because it depends on many factors that are often hard to estimate. But in this case, our life might depend on getting this right, so let’s try accounting for it. The drag force can be calculated this way:
To see what numbers need to be plugged into this formula, let’s go through the variables one step at a time:
- is the density of the fluid. The air density depends on temperature, humidity and atmospheric pressure. We will assume a temperature of 20°C, dry air and the average atmospheric pressure on earth, which results in a density of 1.2041 kg/m^3
- is the speed of the object relative to the fluid. We could calculate the speed at every point in time, but that would be a lot of work, so I’ll just assume 5 m/s as top speed.
- is the frontal area of your body. We assume this to be 0.5 m^2
- is the drag coefficient. Lower values mean less friction. I think the drag coefficients of a human on a swing might be similar to that of a ski jumper, which is about 1.2 according to this website.
Now we can put it all together:
= 1.2 kg/m^3 * (5m/s)^2 * 0.5m^2 * 1.2 = 18 Newton
So, there is a force of 18 Newtons slowing us down. How much energy will this cost us? It depends on how far we will travel. We only need to calculate how many meters we travel until we arrive at the top. For the way down air-resistance is mainly irrelevant because we are far away from terminal velocity, so gravity will ensure we get down without slowing down. Half of the swing’s circumference is 2*pi *2m / 2 = 6.28m. So the extra energy we need to overcome air resistance is smaller than 18N * 6.28m = 113.4 Joule.
For our example with 75kg payload and a swing radius of 2 meter, we need a total energy of 2943 + 735.75 + 113.4 = 3792 Joule
This is pretty similar to the result shown in the video. So I’m pretty confident that it will actually be enough. But still: Proceed at your own risk…