> And no, you can not get around this problem by doing a multivariate regression of speed on gas pedal and hill. That's because gas pedal and hill will be perfectly colinear.
This is one of the reasons why you check for multicollinearity [1] when performing multivariate linear regressions; aside from introducing significant instability into the model, if your predictors are correlated, then you're essentially measuring the same thing twice. The stated problem is actually a good example as to why you should avoid having highly correlated predictors in a multivariate model; by blindly pursuing the regression despite the inputs being correlated, we miss out on the fact that there is a relationship present (i.e., the dependent variable (speed) is actually a factor of both of the independent variables (pedal height and hill slope)).