3.5 Linear motion under a variable force

When force depends on position or time, Newton’s law still gives ma=F, but constant-acceleration formulae may fail. Work–energy is often the cleanest route when F is a function of x.

Use v dv/dx=a to convert a position-dependent acceleration into an integrable equation, or integrate F dx for work. State the interval and initial condition before applying limits.

If F=kx on a frictionless line, work from 0 to x is ½kx², so ½mv²=½mu²+½kx²; speed grows with x rather than time uniformly.

A variable force does not justify using v²=u²+2as with an average force unless that average has been derived from work.