# Why is the rate of change of velocity with distance not constant?

My physics textbook says, "Galileo concluded that the rate of change of velocity with time is a constant of motion for all objects in free fall. On the other hand, the change in velocity with distance is not constant- it decreases with the increasing distance of fall." How is that possible? Shouldn't the change in velocity of an object increase the greater the distance it falls?

• Two things falling, both are increasing their speed by the same amount per second. But If something were falling slowly, it is speeding more per meter than it speeds up per meter if it is moving really fast. Because the second object goes far per second. Sep 3, 2021 at 10:10

Because the object is moving faster, it spends less time in a given spot, so its velocity has less time to change per unit distance. This can be made precise with the chain rule: $$\frac{dv}{dy} = \frac{dv/dt}{dy/dt} = \frac{g}{v}$$ gets smaller if $$v$$ gets bigger and $$g$$ stays constant.