# SS Day 9: Superman and Sidewalks

Thursday, 26 June 2014

Source: Action Comics 1 (June 1938).

Summer school is an odd beast. Sometimes, I’m all crunched for time. Today, I found an extra hour so we talked about my favorite topic, Superman. I asked the kids if Superman’s tall-building-leaping would crack sidewalks. We didn’t think so but how to prove it? Below is our solution, will you check our physics for us? (This was super-scary to solve with the kids because I haven’t solved the problem before.)

How high can Superman leap? According to Action Comics 1 (June 1938), Superman can leap 1/8 of a mile or about 200 meters in the air [1].

What’s his takeoff speed when jumping 200 m in the air? I’m going to assume Superman isn’t experiencing air resistance, so let’s go with my personal favorite of the kinematics equations, $v_{f}^{2}=v_{i}^{2}+2ad$, knowing that $v_{f}=0 m/s$, $a=-9.8 m/s/s$, and $d = 200 m$. You get $v_{i}=63 m/s$.

When Supes crouches down to leap, he has an initial velocity of 0 m/s and must leave the ground at 63 m/s. What acceleration will he give himself in that time? Well, how long do you want to assume that jumping time is? We went with $t=0.1 s$. The definition of acceleration says that $a=\frac{\Delta v}{\Delta t}$. I get an acceleration of 626 m/s/s.

Now, how about that force we promised to find at the start? We know from Newton’s Second Law that $F_{net} = ma$ and from DC Comics that Superman has a mass of 107 kg [2]. I’ve decided on the following free body diagram, with an upward force that his leg muscles generate and Superman’s weight:

Free body diagram of Superman as he’s jumping.

In addition to Newton’s Second Law, I can sum the forces and eventually solve for the jumping force:

$F_{net}=F_{jump}-F_{g}=ma$

$F_{jump}=ma+mg=m(a+g)$

Using the acceleration from above and solving, $F_{jump} = 66,000 N$.

Is that enough force to break concrete? Concrete breaking strength is quoted as a pressure. 20-40 MPa, to be exact [3]. To find pressure, we need the force and area. Assuming Superman’s jumping off the balls of his feet, they occupy a box about 10cm by 20 cm in size. Using $P=\frac{F}{A}$, I get a pressure applied by Superman’s feet to the sidewalk of 3.3 MPa, or about an order of magnitude smaller than concrete’s breaking strength.

You might take issue with some of my assumptions (perhaps his jumping time is not 0.1s or the area of the balls of his feet isn’t right), but I still think we’ll come in under 20 MPa even if you fiddle with them all.

So, good news for the sidewalks of Metropolis — Superman will not break you.

Your homework (in the style of Rhett Allain’s great superhero physics posts): How high would Superman have to jump down from in order to crack a sidewalk?

[1] Action Comics 1, see picture above.