What is Force-velocity curve?

What is Force-velocity curve?

The force-velocity curve is the graphed relationship between how much force a muscle can produce and how fast it is allowed to shorten. Plot force on one axis and shortening velocity on the other and you get a downward-sloping curve: the two are inversely related. When a muscle contracts against a load so heavy it cannot move (an isometric hold), velocity is zero and force is at its maximum, a value physiologists call F0.

As the load gets lighter the muscle shortens faster, but the force it can generate drops. At the far end, against no load at all, the muscle reaches its highest shortening speed, Vmax, while producing almost no external force. Everything in between sits on the curve. This is why you can grind out a 1RM deadlift at a crawl but throw an empty barbell overhead in a fraction of a second: the same muscles, obeying the same curve, at opposite ends.

For a lifter or athlete the curve is a map. Where an exercise sits on it (heavy and slow versus light and fast) decides what quality you are actually training, whether that is maximal strength, power, or pure speed. Understanding the curve is what separates programming for a powerlifter from programming for a sprinter or a jumper, even when both are in the same weight room.

How it works

The force-velocity curve works because of how muscle generates force at the molecular level. Muscle shortening is driven by myosin heads binding to actin filaments and pulling, the cross-bridge cycle. At slow speeds, more cross-bridges are attached and pulling at any instant, so total force is high. As shortening velocity climbs, each cross-bridge has less time to attach, generate force, and detach before it is dragged out of position, so the fraction of bound heads drops and force falls.

A. V. Hill described this trade-off in 1938 with a rectangular hyperbola, and his equation, (F + a)(v + b) = (F0 + a)b, still models it today. Three things move the whole curve for a trained athlete. First, maximal strength training raises F0, shifting the high-force end up. Second, ballistic and speed work raises Vmax, extending the high-velocity end out.

Third, muscle power, the product of force multiplied by velocity, peaks not at either extreme but in the middle, typically around 30 to 60% of 1RM, because that load is heavy enough to demand real force yet light enough to move fast. Rate of force development, how quickly force rises after a movement begins, governs the early portion of the curve and largely decides real-world explosiveness, since most athletic actions last under 250 milliseconds and never reach maximal force.

The formula

(F + a)(v + b) = (F0 + a)b — Hill's force-velocity equation (1938)

F0 is maximal isometric force (velocity = 0); Vmax is maximal unloaded shortening velocity (force = 0); a and b are constants that set the curvature. Mechanical power = force multiplied by velocity, and peaks at an intermediate load, roughly one third of F0.

How to apply it

• Train maximal strength to raise F0: Heavy work at 85 to 95%+ of 1RM for 1 to 5 reps shifts the high-force end of the curve upward. A bigger F0 lifts the entire curve, so every load below it can be moved with more force and, at the same absolute weight, more speed.
• Train ballistic speed to raise Vmax: Fast, light, projected movements such as jump squats, throws, and sprints push the high-velocity end outward. This trains the nervous system to fire motor units rapidly, so the muscle reaches higher shortening speeds against light or no external load.
• Target peak power in the middle: Load a power exercise at roughly 30 to 60% of 1RM and move it with maximal intent for 3 to 5 reps. This is where force multiplied by velocity is largest, so it develops the output most sports actually express in a jump, sprint start, or throw.
• Move sub-maximal loads with maximal intent: Compensatory acceleration, pushing every rep as fast as possible even when the weight is moderate, trains the high-velocity side of the curve without needing light loads. Intent to move fast is the training variable, not just the weight on the bar.
• Use velocity to place the load: Velocity-based training reads bar speed to confirm which zone you are in. Squat mean velocities near 0.3 m/s signal maximal strength, near 0.75 m/s signal power, and above 1.0 m/s signal speed-strength, so you can autoregulate load to hit the quality you want.
• Sequence qualities across a training block: Many programs build F0 with a strength block first, then convert it toward power and speed in later blocks. Combining strength and power work improves athletic performance more than training either quality alone, per repeated meta-analyses.

Types

Worked example

Take a lifter with a 140 kg back squat 1RM. The same athlete, same muscles, sits at a different point on the force-velocity curve depending only on the load. Here is how load, typical bar speed, and the quality being trained change across the curve. Velocities are representative ranges, not exact prescriptions.

The load never changes what muscles are working, only where on the curve they operate. A complete athlete trains several zones across a year, because raising force at the top and velocity at the bottom both push the whole curve, and with it the power peak in the middle, outward.

Force-velocity curve vs power-velocity curve

The two curves come from the same data. Power is force multiplied by velocity, so the power curve is derived directly from the force-velocity curve. That is why peak power sits in the middle even though both force and velocity are individually higher at the ends.

By goal

• Powerlifters and strength athletes: Live near the top-left of the curve. Train at 85 to 100% of 1RM for low reps to drive F0 as high as possible, since the sport is scored purely on force. Use moderate-load speed work as a tool to keep bar velocity and rate of force development sharp, not as the main focus.
• Field, court, and power athletes: Target the middle and right of the curve. Build a strength base, then spend blocks at 30 to 60% of 1RM moving explosively for peak power, plus sprints and jumps for speed. Most sporting actions are fast and sub-maximal, so power and rate of force development transfer more than a bigger 1RM alone.
• General lifters and beginners: Widen the whole curve. Prioritize building maximal strength first, because a bigger F0 raises everything below it, then add a little fast, intentful movement (light jumps, throws, or fast concentric reps) to keep speed and athleticism. You do not need every zone every week early on.

Common misconceptions

• "Getting stronger automatically makes you faster and more powerful." Raising maximal strength (F0) lifts the high-force end of the curve and helps most beginners, but past a point the high-velocity end and rate of force development must be trained directly. A strong lifter who never moves fast can produce high force slowly yet still be un-explosive in fast, sub-maximal actions.
• "Maximum power is produced when you lift the heaviest weight." Power is force multiplied by velocity, so it peaks at an intermediate load, roughly 30 to 60% of 1RM, not at the top. A maximal lift produces huge force but crawls, so its power output is actually lower than a well-loaded jump or power clean moving far faster.
• "To get faster you should only train with light, fast movements." Speed work extends Vmax, but the whole curve, including its fast end, sits higher when maximal strength is greater. Athletes with a stronger base express more velocity against any given load, which is why strength and speed training combined beats either one alone.
• "The force-velocity curve only matters for elite athletes." The curve governs every contraction for everyone. It explains why your last hard rep grinds to a crawl, why an empty bar flies up, and why a moderate load moved fast feels different from a heavy grind. It shapes how any lifter should distribute strength, power, and speed work.

Related terms

Force-velocity curve FAQ

What is the force-velocity curve in simple terms?

The force-velocity curve shows that the harder a muscle has to push, the slower it moves. Heavy loads move slowly with high force, and light loads move fast with low force. Plotted on a graph, force and velocity trade off along a downward-sloping curve for every contraction.

Why is the force-velocity relationship inverse?

The force-velocity relationship is inverse because of the cross-bridge cycle inside muscle. When shortening is fast, each myosin head has less time to attach to actin and pull before detaching, so fewer are bound at once and total force drops. Slow contractions keep more cross-bridges attached, raising force.

Who discovered the force-velocity curve?

Physiologist A. V. Hill described the force-velocity curve in 1938 using frog muscle, showing the relationship followed a rectangular hyperbola. His equation, (F + a)(v + b) = (F0 + a)b, still models muscle mechanics today and earned foundational status in exercise physiology and biomechanics.

Where on the curve is power greatest?

Power is greatest in the middle of the curve, at an intermediate load, because power equals force multiplied by velocity. For most compound lifts this peak sits around 30 to 60% of 1RM, heavy enough to demand real force but light enough to move fast. Neither extreme maximizes power.

What are the training zones of the force-velocity curve?

The curve is usually split into five zones: maximal strength (above 90% 1RM), strength-speed (about 60 to 85%), peak power (about 30 to 60%), speed-strength (about 30 to 45%), and speed (below 30% or unloaded). Each trains a different blend of force and velocity.

How do I shift the whole force-velocity curve up?

Train both ends. Heavy strength work above 85% of 1RM raises maximal force (F0) and lifts the high-force end, while fast, light, ballistic work such as jumps and sprints raises maximal velocity (Vmax) and extends the high-velocity end. Doing both pushes the power peak in the middle outward.

What is the difference between strength-speed and speed-strength?

Strength-speed is force-dominant power, heavier loads (about 60 to 85% 1RM) moved fast, like a heavy power clean. Speed-strength is velocity-dominant power, lighter loads (about 30 to 45% 1RM) moved even faster, like a light jump squat. Both are power, but they sit on opposite sides of the peak.

Does maximal strength training make you slower?

No. Heavy strength training raises maximal force and, because the whole curve sits higher, lets you move any given sub-maximal load faster. It does not train peak velocity directly, so athletes add fast, light work too, but strength itself does not make you slower when speed is also trained.

How does rate of force development relate to the curve?

Rate of force development is how quickly force rises after a contraction starts, and it governs the early, explosive portion of the curve. Most athletic actions last under 250 milliseconds, too short to reach maximal force, so rate of force development often predicts real-world speed and power better than 1RM alone.

How does velocity-based training use the force-velocity curve?

Velocity-based training measures bar speed to place each set on the curve. Slow squat velocities near 0.3 m/s signal maximal strength, around 0.75 m/s signals power, and above 1.0 m/s signals speed-strength. Coaches use these numbers to pick loads that hit the exact quality they want to train that day.

References

1. Sundberg CW, et al. Hill's equation of muscle performance and its hidden insight on molecular mechanisms. Journal of General Physiology, 2013. PubMed 24277600
2. Alcazar J, et al. On the Shape of the Force-Velocity Relationship in Skeletal Muscles: The Linear, the Hyperbolic, and the Double-Hyperbolic. Frontiers in Physiology, 2019. PMC6593051
3. Cormie P, McGuigan MR, Newton RU. Developing maximal neuromuscular power: Part 1 - biological basis of maximal power production. Sports Medicine, 2011. PubMed 21142282
4. Cormie P, McGuigan MR, Newton RU. Developing maximal neuromuscular power: Part 2 - training considerations for improving maximal power production. Sports Medicine, 2011. PubMed 21615188
5. Maffiuletti NA, et al. Rate of force development: physiological and methodological considerations. European Journal of Applied Physiology, 2016. PMC4995130
6. Force-Velocity Curve. Science for Sport, 2023