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. 

Runners use their entire body to generate speed, but a significant portion of that is generated in the lower legs. Sprinters especially push off their toes, using their calf muscles to deliver an extra bit of force with each step. Pistorious' calf muscles are conspicuously absent, meaning all the pushing force his legs generate come from his butt muscles and the back of his thighs. 

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

With every strike of his blades to the ground, there is a slight compression before the blades sufficiently stiffen enough to deliver force to the ground. It is very slight, but there is a time delay between his "foot" hitting the ground, and the ground delivering the force to push him forward.

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

There are two parts of running, putting your foot down and bringing it back up, and the springiness which seems like a disadvantage above, helps him here. Springs do not generate energy, they only store it. But whereas a regular runner would deliver all their muscle's energy into the ground (inevitably pushing them more forward) Pistorious' blades store a portion of that energy when they compress, which we'll see in number 4 is another disadvantage. That stored energy, however, does help him bring his legs back up, and quickly. Combined with the reduced weight of the prosthetics, the blades are what make it possible for him to move his legs much faster than his opponents.

4. But alas, the blades deliver less force. 

Lack of calves aside, any energy the blades store is energy not being used to push him forward. The Olympic committee originally stated that Pistorious is able to run the same distance using less energy. This is only half true. The reduced weight of his legs does reduce the amount of energy he needs, but springs do not create energy, they only store it. Any spring action Pistorious gets out of his legs is energy he already expended compressing them in the first place. The 400 M sprint is such a short race, whatever energy Pistorious saves from the reduced weight of his legs would not be noticeable, though it would perhaps become an issue in a much longer race, such as a marathon. 

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

Because of the shape of his blades, he must run out in an upright position, whereas most runners stay low out of the block. He has consistently performed the worst out of the block over his career for this reason, and it is blamed for his last place finish in the semi-finals.

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.