The position of an object moving along a line is given by #p(t) = 2t - 2sin(( pi )/8t) + 2 #. What is the speed of the object at #t = 12 #?

1 Answer
May 24, 2017

Answer:

#2.0 "m"/"s"#

Explanation:

We're asked to find the instantaneous #x#-velocity #v_x# at a time #t = 12# given the equation for how its position varies with time.

The equation for instantaneous #x#-velocity can be derived from the position equation; velocity is the derivative of position with respect to time:

#v_x = dx/dt#

The derivative of a constant is #0#, and the derivative of #t^n# is #nt^(n-1)#. Also, the derivative of #sin (at)# is #acos(ax)#. Using these formulas, the differentiation of the position equation is

#v_x(t) = 2 - pi/4 cos(pi/8 t)#

Now, let's plug in the time #t = 12# into the equation to find the velocity at that time:

#v_x(12"s") = 2 - pi/4 cos(pi/8 (12"s")) = color(red)(2.0 "m"/"s"#