However, my experience remains that going up hills doesn’t lower my overall average speed as much as I expect it to. This could perhaps be because the momentum of the downhill has a greater affect than any additional recovery required once I’ve reached the crest after an uphill.

For example, Consider a course that is one mile uphill followed one mile downhill and then one mile of flat. For some distance on the flats I’ll be able to continue to put the downhill momentum to work as my speed tapers back town a “normal” speed for flat ground.

In the rolling hills of Indiana, that also means that some amount of the downhill momentum is going into making climbing the raise immediately following it easier.

Of course, I’m using intuition again and you are using physics equations, so I’m prepared to be wrong about the reality again. ]]>

For example, if you climb a hill for X miles at a pace of 3 MPH, then it is impossible to achieve an average speed for both climbing and descending of 6 MPH or greater. This is because even if you descend at the speed of light — not recommended on most bikes! — you merely double your total distance traveled to 2X, and doubling the distance merely doubles the average speed.

In the example in the post, climbing a hill for X miles at 3 MPH and descending at 21 MPH yields the following:

T = D / V (velocity)

D-up = X miles (given)

V-up = 3 MPH (given)

T-up = X miles / 3 MPH

D-down = X miles (given)

V-down = 21 MPH (given)

T-down = X miles / 21 MPH

T-total = (X miles / 3 MPH) + (X miles / 21 MPH)

D-total = 2X miles (given)

V-total = D-total / T-total

V-total = 2X miles / ((X miles / 3 MPH) + (X miles / 21 MPH))

V-total = 5.25 MPH

So in the above example, going up a hill at 3 MPH and down at 21 MPH gives an average speed of 5.25 MPH. This is not as encouraging of a result — sorry!

Two comments:

(1) Try this for yourself. Go up a hill at 3 MPH and down at 21 MPH. Measure how far you went in total up and down the hill as well as how much time elapsed. Then find a flat stretch the same distance and go 5.25 MPH. You should find that it takes the same time.

(2) An intuitive way to understand the math is that you spend a lot of time going up the hill slowly but zip down the hill quickly. Therefore, the uphill portion penalizes your average speed much more than the downhill. ]]>