An object's two dimensional velocity is given by #v(t) = ( cost , -t^3 +4t )#. What is the object's rate and direction of acceleration at #t=7 #?
We're asked to find an object's magnitude and direction of the acceleration given its velocity equation.
We need to find the object's acceleration as a function of time, which we can do by differentiating the velocity equation.
The derivative of
Velocity equation (component form):
#d/(dt) (cost) = color(red)(-sint#
#d/(dt) (-t^3) = color(darkorange)(-3t^2#
#d/(dt) 4t = color(green)(4#
Now, let's plug in
The magnitude of the acceleration at this time is
The direction of the acceleration is given by