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Assuming one is biking on flat roads and assuming equal power in both scenarios, how much faster would one go using clipless pedals versus flat pedals?
[Note that this question relates to: https://bicycles.stackexchange.com/questions/5647/how-much-of-a-difference-do-toe-pedal-clips-make-in-comparison-to-platform-pedal. I apologize if someone else has asked my exact question.]
Depends on your personal pedalling style.
Exactly how much faster this makes you depends on how much power you can personally deliver through your pedals. A power meter is the tool to measure this, on your bike. Strava's power estimator is useless at this.
You might be able to use a power-meter equipped spin bike at a Gym, but that's not real on-road riding where things may be different.
Do consider also:
You can play around with http://www.bikecalculator.com/ and get some interesting results. EG if you can produce 5 watts more power then that could be 0.3 km/h faster for everything else being the same.
With flat pedals, the only thing holding your foot to the pedal is friction. Friction only exists if your foot is in contact with the pedal -- what physicists call "normal force" or "contact force". No force, no friction, and your foot is off the pedal, by definition.
Because of this fact, you have to exert some amount of pressure to the pedals all the time, particularly on the back-stroke...unless you have super Jedi warrior reflexes and can move your foot in exactly the same arc as the crank, your foot is either going to come off the pedal, or it's going to have to be pressed against the pedal at least a little bit to maintain traction. This "little bit of retarding force" directly counter-acts the main driving force of the other crank...you are literally working against yourself; your legs are working against each other...and this extra effort that must be expended to maintain traction also won't show up in a power meter, depending on the design of the power meter.
It's a physical fact that flat pedals require you to keep your foot pressed against the pedal, and this backward force generally is non-productive and actually detracts from forward force. The main argument is how significant of an effect this is. There have been studies to measure the difference, but not all of them directly measure the right things. The current consensus is that the effect is small, and efficiency for most practical purposes is not greatly different between flat pedals and clipless/clipped pedals. To directly answer your question, I believe attempts to find a difference through practical testing (the straight road, direct comparison test) have failed to find a reproducable difference.
It may be a difference too small to matter, but other things considered, clipless should be more efficient because the need to apply this retarding force to the cranks is removed.
As cycling power is normally measured at the cranks, i.e. at the input to the drivetrain and not in your muscles, you'll go exactly the same speed as with flats given your "same power" constraint.
Where clipless pedals can help is in improving efficiency by keeping your foot in an optimal position, or improving peak power if you pull up on the pedals (this is rarely sustainable for long). You may also be able to push round more of the stroke, which can help get more power out of your muscles. At their very best, compared to your feet sliding around on the pedals, you should get more power for the same perceived effort, but to actually measure this would be very tricky.
"assuming equal power in both scenarios"
As Chris H said, pedaling power is usually understood as the power provided to the cranks, averaged over a full rotation, regardless of physiological efficiency. Same power = same speed if all else is equal (aero, incline...).
You probably meant "given the same level of exertion, how much faster will I go with clipless pedals?".
From what I read and based on my own mountain bike experience riding both types of pedals, going clipless gives you a power advantage during explosive bursts. This comes in handy on very steep technical climbs, when you need to pass large roots and rocks. Road riders will benefit from it when sprinting, or accelerating in trafic.
This study clearly demonstrates the power advantage over a 30s sprint: The average across test riders is 617W with clipless, Vs 566 W using flats. That is a 9% advantage in power, which translates to a 3% advantage in speed. Using the default parameters with this model, I get a speed of 50,7km/h with clipless Vs 49,1km/h with flats.
However, this extra power is not free, you will tire faster if you're using weak muscles to help stronger ones.
I could not find any study clearly demonstrating a speed advantage on a long flat course. Actually, "semi-scientific" videos like this one by GCN show that there is no power gain except in a sprint.
You might be wondering why everyone rides clipped-in at Tour De France or similar events. Riders spend most of the race in a peloton, shielding each other from the wind, with lesser team members taking turns at the front to help their leader. Whoever manages to save the most energy for the final sprint wins.
