1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Chain rule and Kinematic quantities x,v,a

  1. Sep 28, 2015 #1
    Hello Forum,

    I have a couple of kinematics questions.

    The position of a point object is given by the position vector x(t). Speed is v(t)=dx(t)/dt and the acceleration a(t)= dv(t)/dt. What if we wanted to know the velocity and/or the acceleration as a function of position, i.e v(x) or a(x)?

    For example, given x(t)=3t^2+2, how would we find v(x) and a(x)? I think the chain rule is needed. The chain rule is a a formula for computing the derivative of the composition of two or more functions.

    What if we were given v(x) from the beginning, for example v(x) = 3x +x^2, and wanted to know the position x(t) or a(t) for the object?

    A generic force F can be a function of several independent variables, F(x, t, v):
    • distance x
    • time t
    • speed v
    Are these three independent variables associated to a hypothetical object to which the force F is applied? Since F=ma, then the acceleration a(x, t , v)

    If the acceleration varies with time t, then the acceleration should also automatically be a function of distance x, since the object occupies different positions x at different times t. So does it make sense to write a(x,t) or should we just write a(t) or a(x)? Same goes for v(x,t): if the object occupies different positions at different times it also means its speed is a function of time...

  2. jcsd
  3. Sep 28, 2015 #2


    User Avatar
    Science Advisor

    Obviously, if x(t) fulfills some constraints (for example that you do not drive in circles), you can invert it and find t=f(x). Then v(t) = v(f(x)) etc. But v(t) is still dx(t)/dt, so v(f(x)) = [itex]\frac{1}{\frac{dt}{dx}} [/itex] - which gives you another clue: If you stand still, you cannot create t=f(x).
  4. Sep 28, 2015 #3

    So a function is usually invertible when it has some contraints?
  5. Sep 28, 2015 #4


    User Avatar
    Science Advisor

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook