Acceleration is always the time rate of change of velocity. But a problem may give position, velocity, or acceleration as a function of time or position. Select what the problem gives you and practice the correct derivative or integral.

v = ds/dt
and
a = dv/dt

1. What does the problem give you?

CASE: POSITION AS A FUNCTION OF TIME

Differentiate with respect to time

    2. Try the numerical example

    The complete decision map

    GivenOperationRelationship
    s(t)Differentiate with respect to timev = ds/dt; a = d²s/dt²
    v(t)Differentiate with respect to timea = dv/dt
    a(t)Integrate with respect to timeΔv = ∫a dt
    v(s)Chain rule; differentiate with respect to positiona = v(dv/ds)
    a(s)Separate and integratev dv = a ds
    FE exam check: Look at the independent variable—the symbol inside the parentheses or on the horizontal axis. If it is t, work with respect to time. If it is s or x, connect position to time using ds/dt = v.

    Engineering Record

    Explain how the independent variable determines which derivative or integral you should use.