Oscar Pistorius Olympics Controversy: Why Prosthetic Legs Are Not Likely to Lend an Edge

Oscar Pistorious made history this summer at the London 2012 Olympics, by being the first paraplegic runner to compete against able-bodied opponents in the 4x400 relay and the 400 M sprint. 

He has sparked controversy however, by some claiming his carbon-fiber running blades give him an advantage over able-bodied runners. Using my engineering degree for a change, I was lucky enough to catch his startlingly excellent performance in the first round, where he placed second in his race; I did my best to analyze what I saw.

His legs move faster. Maybe it was my imagination, but Pistorious' legs were moving much faster than any of his opponents, yet his speed was not noticeably out of the ordinary. A few things are happening here:

1. He is missing muscles. 

2. His legs apply force over a slower period of time. 

3. The springiness of his legs help him move his legs faster and with less effort. 

4. But alas, the blades deliver less force. 

5. He has a huge disadvantage out of the starting block.

There are two factors that determine a runners speed, the pushing force his legs exert, and the frequency that force is applied. We call these brief impacts of foot on ground impulses, and an impulse is equal to a change in momentum or I = mass (m) x change in velocity (dV). 

The sum of the impulses will equal total change in momentum, which, if we consider initial momentum at the starting position to be zero, will be M = m x Vfinal (or final speed).

At some point a runner hits maximum speed. This does not mean he is not striking the ground anymore, but that forces acting against him are cancelling out the forces he is applying to the ground. This could come in the form of impulses received when his foot first touches the ground in front of him (mostly due to friction), gravity, and wind resistance. Meanwhile, forces applied by the runner lessen overtime as their body tires out. 

Impulses are also equal to force applied over time, or I = F x dT where dT is the physical amount of time the impact takes place.  In this equation, I is constant, so as dT increases, Force of impact decreases. 

Impulse (constant)  = mass x change in speed = Force x duration of force

If you wondered why diving into an empty pool will kill you, but a full one won't, this is why. It is the deceleration of stopping suddenly, where your velocity goes from really fast to zero in a minuscule amount of time, that actually damages you.  

Acceleration = change in speed / time (we can substitute in the equation from above here):

Force of Impact = mass x acceleration = mass x change in speed / duration of force

Because of the springiness of his blades. dT for Pistorious is slightly higher than everyone else. So if he wants to go the same speed (and therefore the same impulse) as another racer, his force is going to decrease, meaning it will take him longer to accelerate:

m x final speed = m x acceleration x dT

m is mass, which is constant unless Pistorious goes on a diet. Final speed is the speed he needs to win, so that is also constant. That leaves just acceleration and dT that change. Since Pistorious' dT is higher, his acceleration is lower. That is crucial seconds wasted getting up to speed.

Runners don't exert force constantly, but in steps, with each impact accelerating them for a brief moment. Pistorius must create more impacts more quickly in order to compensate for the fact that each of his steps accelerates him less than other runners. This is why his legs move so much faster. 

So, to figure out if Oscar Pistorious does indeed have an advantage over other runners one must simply measure him as he runs. What needs to be known is how much faster do his prosthetics let him move his legs (something difficult to find out, since there is no running data for him with natural legs), and just how much less force he can exert per step, between not having calf muscles, and having the springs.

My math is rusty, but without spending too much time deriving equations, I believe if one did take the time to figure it out, they'd see that the relationship between force of step and frequency of step to attain a certain speed is a 1 to 1 ratio. Meaning if he exerts half the force of an able-bodied runner, per step, he must take twice as many steps to run at the same speed as them.

For example: If his legs exert only half the force, but he can move them twice as fast, then he has no advantage nor disadvantage, and just requires a different running technique (we are not considering his problems off the starting block, something he won't deal with in the relay, here).

But if those numbers are not balanced, say he moves his legs twice as fast (keep in mind we are talking about unnaturally fast, not fast because he trained that way), but only exerts 30% less force, then he does, in fact have an advantage.

Oscar Pistorious has broken ground at these Olympics. He has opened the door for other para-Olympians to go head to head against their able-bodied counterparts. Someday someone will do a comprehensive study, if not on Pistorious, on someone like him, where pre-amputee running data is perhaps available, and we'll find out once and for all. For now, South Africa has a lot to be proud of in their star runner.