Superman is so strong, he can do anything, right? Could he punch someone so hard that they ended up in space? Let's do this.

## How High Is Space?

When I say space, you might say "outer space." But how high is that? The Earth's atmosphere doesn't just stop at some height. No, instead the density of air gets lower and lower until you can't even really detect it. But for this problem, we have to pick a height. I am going to pick 420 km above the surface of the Earth as "space." Why? Why not. That is about the height of the International Space Station's orbit, so I think it is a good choice.

## How Fast Would the Person Have to Go?

I am talking about after the punch from Superman. Let's just look at a person moving up at some initial speed *v*_{0}. If this were a problem in an introductory physics course, I would hope you would think of the work-energy principle.

Let's say that Superman is punching a clone of himself (called Superman-b) - just as an example. If I take Superman-b and the Earth as my system, then after the punch from Superman there is no external work done on the system. There will be two types of change in energy - kinetic and gravitational potential.

I know the values of these variables. If I plug in what I know, I get a "launch" speed of 2778 m/s (6214 mph). Yes, that is fast - but actually Superman-b would have to be going even faster than that. Why? Air resistance, that's why.

## Launch Speed With Air Resistance

Here is a diagram of Superman-b shortly after he was hit by Superman.

I will use the two following models for the magnitude of the gravitational force and the air resistance force.

For the gravitational force, the two masses are the mass of the Earth and the mass of Superman-b and *r* is the distance between Superman-b and the center of the Earth. This force will decrease somewhat as Superman-b rises to space.

In the model for air resistance, *A* is the cross sectional area of the object and *C* is some drag coefficient that depends on the shape of the object. The ρ is the density of the air. As you get higher in the atmosphere, this will decrease. So, you see this air resistance force changes with both the speed and the altitude. Actually, the drag coefficient can depend on speed too - but I will pretend like it is constant. So, this isn't such an easy problem.

Let me get some estimates for some of these values. I am going to assume Superman-b is the same size and shape as a normal human. Maybe he has a mass of 70 kg. For the product of *AC*, let me estimate this based on the terminal speed of a sky diver. If a sky diver falls at 120 mph (54 m/s) then the air resistance would be equal to the weight of the sky diver. This means that *AC* would be: