Superman and Conservation of Energy

In the pantheon of superheroes, Superman is unquestionably king. The Flash has his speed, Spiderman has his web slinging, Wolverine has his claws, Batman has his, um, grappling hooks...but Superman. Superman has it all. He can lift mountains, survive nuclear explosions, fly through space, shoot lasers out of his eyes—power, it seems, that's limitless.

Turns out old Supes might need to spend a little more time sunbathing and a little less time fighting crime to truly be "faster than a speeding bullet, more powerful than a locomotive, and able to leap tall buildings in a single bound.'' This looks like a job for science.

Superman gets his mighty powers from Earth's sun, and on a generously sunny day in Metropolis, the sun is going to be radiating about 1400 W/m². We'll assume Superman is able to convert 100% of that energy to useful work (another superpower—the best solar cells on the market currently are hard-pressed to beat 30%).

Superman's around 2 m tall and, depending on the exact comic book you're reading, looks to be about 1 m wide at his barrel of a chest, so we'll give him the benefit of the doubt and say he's got a good 2 square meters of surface area to soak up some rays with. That puts him at 2800 J of energy from the sun per second when he whips out the tanning lotion.

Faster Than a Speeding Bullet

Every kind of gun is going to fire bullets at different speeds, but a good typical value for small rifles is right around the speed of sound in air, or 340 m/s. Superman's a burly guy, and probably tips the scales at 100 kg. So in order to run at speeds exceeding a bullet, he's going to need  of energy—about 35 minutes of exposure to the sun. We'll assume another superpower is the ability to ignore friction and air resistance, a power we employ often in physics exercises.

More Powerful Than a Locomotive

A modern diesel-electric locomotive, such as the JT56ACe used in China, weighs 150,000 kg and can pull another 5,000,000 kg at speeds of 120 kilometers per hour (75 mph, or 33 m/s). That's a kinetic energy of , or almost twelve whole days of constant sunbathing. (Superman can fly fast enough to stay in the sun for 24 hours, obviously.)

Able to Leap Tall Buildings in a Single Bound

The tallest building in the world is the Burj Khalifa in Dubai, shooting up over 800 m from the sand below. We'll assume some really tall buildings in Metropolis are a more reasonable 400 m, somewhere between the Empire State Building's roof and its spire's tip.

The gravitational potential energy Superman has at the top of the building is U = mgh = (100kg)(9.81m/s²)(400 m) = 392,400 J—a quick two and a half minute power-tan. Five minutes, and he can make it over the Burj Khalifa faster than you can say "Tom Cruise in a catsuit."

So for each battle against Lex Luthor, where Superman is running around and punching buildings to rubble to his heart's content, he's going to need a lengthy Caribbean vacation to recover his strength. After all, even superpowers are no use against the might of the law of conservation of energy.