Solving Derivatives Problems: Understanding dv/dt and dv/ds Differences

[Mathnewbie]

Hello can someone point in the right direction on this one.

A particle moves along a strainght line with displacement s(t), velovity v(t), and acceleration a(t). Show that

a(t) = v(t) dv/ds

Explain the difference between the meanings of the derivatives dv/dt and dv/ds.

Does dv/dt mean difference of velocity over the difference time ?

Does dv/ds mean difference of velocity over the difference displacement ?

Any help would be great? Thanks

 

[HallsofIvy]

Basically yes: how fast the speed changes "per foot" rather than "per second", for example.
To do the first part, use the chain rule:
$$\frac{dv}{dt}= \frac{dv}{ds}\frac{ds}{dt}$$