More Bicycling Uphill

Last time we looked at how weight and speed and grade affect the power requirement for a bicyclist climbing a hill.  Today, we’ll reverse the question and assume that the cyclist has a fixed amount of power to deliver.  How will that affect climbing speed?
This figure shows the speed that can be achieved for two different fixed amounts of power delivered to the bicycle. 

The bluish lines represent 150 W, and the reddish/yellow lines represent 200 W.  

There are a few interesting things to note in this figure.  

First, the lines are slightly non-linear with weight which is particularly noticeable in the 5% grade lines.

Second, the lines get steeper with increasing grade so that the 5% grade at 200 W line crosses the 3% grade at 150 W line right at 50 kg and about 11.5 mph.  In other words, with a total mass of 50 kg, 200 W of power would get one up a 5% grade at the same speed (11.5 mph) that 150 W of power would deliver for a 3% grade. 

Third, for a given grade, the 150 W lines and the 200 W lines are about (with some variation) 2 mph apart.  That gives a sense of what more power can do for you. Eat your Wheaties.

Finally, in this view it is easy to see the effect of total weight on the velocity.  For the 1% grade, velocity goes down by about 1/40 mph per kg and for the 5% grade velocity goes down by about 1/14 mph per kg. Or, to put it differently, on a 5% grade, for every kg you drop, you improve your speed by 1/14 mph without changing your energy output at all.  Go on a diet, or else leave the pressure cooker out of your backpack.