Two Dimensional Kinematics

Question

The position of a particle is given by r = (at²)i + (bt³)j + (ct ^-2)k, where a, b, and c are constants.

What is the velocity as a function of time?

v(t) = ???

What is the acceleration as a function of time?

a(t) = ???

How do I go about doing this? Is it a given formula?

Answer 1

Hi Justin G.!

Velocity is the first derivative of the position equation with respect to time:

v = dr/dt

Our position equation was given as:

r = (at²)i + (bt³)j + (ct ^-2)k

So, we can write the first derivative as follows:

v = dr/dt = d[(at²)i + (bt³)j + (ct ^-2)k]/dt = (2at)i + (3bt²)j + (-2ct^-3)k

Acceleration is the second derivative of the position function with respect to time. Or, we can also say it is the derivative of the velocity function with respect to time:

a = d²r/dt² = dv/dt = d[(2at)i + (3bt²)j + (-2ct^-3)k]/dt = (2a)i + (6bt)j + (6ct^-4)k

I hope this helps!

Cheers