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Force=Mass x Acceleration or Surfing the Force Velocity Curve

March 9, 2007 08:15 AM

"A karate punch is like an iron bar. ' Whack!' A kung fu punch is like an iron chain with an iron ball attached to the end and it go 'WHANG!!!!!' and it hurt inside...." ?Bruce Lee

I was lucky enough to attend the Naked Warrior Bodyweight Training Seminar held recently in Los Angeles. The seminar was held at the Dan Inosanto Martial Arts Academy. (Dan was a student of Bruce Lee ? hence the quote above)

But there's more to the quote than that. It illustrates an interesting training concept.

During the seminar Pavel demonstrated two versions of the bench press. The traditional version where you remain tight throughout your entire body, and throughout the entire repetition, and a version that has become popular in Russia, that is more relaxed. It involves a faster negative, and a ballistic pressing action ? in reality more like a push press than a bench press.

Pavel then went on to explain the difference between the two methods using the physics equation Force = mass x acceleration, and how you can use both methods of tension (a continuous tension and a more relaxed tension ? like the iron bar vs. the iron chain).

The explanation I felt was pretty good ? essentially there are many ways to achieve force. But both Pavel and I agreed that some of the audience perhaps didn't understand the need or the application for two different versions of 'tension' when training.

So Pavel and I discussed it ? and he put me in a choke hold and told me to come up with article for his site. After I had tapped out I agreed.

So ? force equals mass times acceleration. The equation F = ma, shorthand for Newton's second law of motion, is the 1 + 1 = 2 of classical mechanics.

Regardless of training objective we all want to achieve maximal force. But there are many ways to do that.

For example: if we want to achieve a hypothetical force score of 10, we can use any of the following variations.


And we'll end up with the same force. So we can do explosive push ups (low mass, high acceleration) and slower push ups with more tension, or more external load (e.g. weight vest etc).

Now obviously a powerlifter who is bench pressing, isn't concerned with the acceleration component as much ? he or she wants the mass component (the load on the bar) to be as high as possible. So as fast as a powerlifter is trying to move the bar ? it doesn't matter, the event is won by lifting the most weight.

Contrast this with a discus thrower or a shot putter ? where this time the mass is constant (the discus) and the only way to achieve maximal force is to maximize acceleration ? by staying relaxed and only applying muscular force at the right time.

And for the martial artists ? it's the difference between a submission hold ? where you apply maximal tension for as long as it takes for the opponent to tap ? i.e. isometrically with no real acceleration, and a kick or a punch, where the mass is constant and the acceleration is the determining factor in force production.

Surfing the force-velocity curve

The key is to look at the force-velocity curve. In order to be a superior explosive athlete you need to train at both ends of the spectrum.

You need to increase your overall force (i.e. raise the curve higher) and decrease your time taken to APPLY that force ? apply it faster (i.e. move the force-velocity curve to the right). So overall we want to move the curve up and to the right.

In general lifting a heavy weight (a max effort attempt) would be an extremely high force activity but it occurs at a low velocity ? so it would be high up and to the far left. Throwing a wiffleball would be a high velocity activity, but with the mass being so low, we'd have a very low force output
The following table is from Theory and Application of Modern Strength & Power Methods by Christian Thibaudeau and shows the basics of the f=ma equation when applied to training methods.

Ballistic Method Speed-Strength Method Strength-Speed Method Controlled Repetition Method Maximal Method Supra-Maximal Method
Acceleration is very dominant. Mass is low. Acceleration is dominant. Mass is low. Acceleration and Mass are contributing equally. Mass is dominant. Acceleration is low. Mass is very dominant. Acceleration is very low to nil. Mass is very dominant. Acceleration is very low to nil.

*Consider the far left of this chart to be the far right of the curve and vice versa.

So as coaches we need to look at our goal ? where do we want to focus our training to get the most beneficial effect.

And the answer is ? we should do both.

We should definitely be looking at methods to increase our ability to produce force. For aesthetic only clients ? we would focus on the right of the above chart ? where mass is the dominant factor in the equation.

But for athletes we should also look at ways to increase the ability to apply that force (the left side of the chart). We have to realize that force can be increased both by heavy strength work as well as lighter speed work. And for best results, we should be "surfing the curve" and using both methods.

For example, with this information we can answer this question:

You can bench press 400lbs. It takes you 4 seconds to complete the repetition.
I can only bench press 125lbs but it only takes me 0.5s to complete the repetition. Who's the strongest? You. Who would win a powerlifting competition? You.

But in a battle of who can apply their strength (in most sports - speed is king), like a boxing match - I'm 'stronger'. When the referee says "go" I'll be able to hit you with 125lbs of force in about 0.5s. You'd need 4s to apply the 400lbs of force.

In other words ? you can produce a power output of 100lbs per second, while I am producing a power output of 250lbs per second. So in combat sports ? I'd win (more velocity). But in powerlifting ? you'd win (more force). Just because of training at different ends of the Mass x Acceleration equation.
Please see the editor's note at the end of this article.

It also allows us to answer the question as to which is "better"? Powerlifting or Olympic Lifting? Heavy resistance training or plyometrics?

There is no answer ? they just target different ends of the curve. And you can see the forward thinking methods of Louie Simmons and the Westside barbell club, who dedicate a large portion of their training to the dynamic effort method (lifting fast) as well as maximal effort work.

If there is an art or a secret to functional performance training - that's it: Moving the force velocity curve up and to the right. Increasing mass (load) and acceleration, by using and designing programs to improve both ends of the curve.

And to prevent getting into a karate vs. kung fu 'who punches the hardest' argument ? getting hit with a chain or a bar would both suck!

About the Author:
Alwyn Cosgrove is one of the country's most popular fitness coaches, writers, and speakers on the lecture circuit. For more information on Alwyn please visit:

Editor's note: Alwyn and I traded a few emails to clarify the bench press/punching example noted in the article. It is an exaggerated example and basically it comes down to the fact that F=MxA is a flawed and basic equation and you the reader will have to accept that the max attempts given in the example are true max attempts. Bench pressing does not correlate to punching in any sort of exact manner. Bench hits the absolute strength end of the curve and Punching hits the acceleration end of the curve. Athletes need to improve both ends.