The position of an object moving along a line is given by #p(t) = sin(3t- pi /4) +2 #. What is the speed of the object at #t = (3pi) /4 #?

1 Answer
Jun 4, 2016

Answer:

Velocity of an object is the time derivative of it's position coordinate(s). If the position is given as a function of time, first we must find the time derivative to find the velocity function.

Explanation:

We have #p(t) = Sin (3t - pi/4) + 2#

Differentiating the expression,

#(dp)/dt = d/dt [Sin (3t - pi/4) + 2] #

#p(t)# denotes position and not momentum of the object. I clarified this because #vec p# symbolically denotes the momentum in most cases.

Now, by definition, #(dp)/dt = v(t)# which is the velocity. [or in this case the speed because the vector components are not given].

Thus, #v(t) = Cos (3t - pi/4).d/dt(3t - pi/4)#
#implies v(t) = 3Cos (3t - pi/4)#

At #t = (3pi)/4#

#v((3pi)/4) = 3Cos (3.(3pi)/4 - pi/4)#

#implies# Speed #= 3Cos 2pi = 3# units.