What is the instantaneous velocity of an object moving in accordance to # f(t)= (sin(t+pi),sin(2t-pi/4)) # at # t=(-pi)/3 #?

1 Answer
Rohan A.K
Jan 20, 2016

Given equation can be re-written as a x as a function of time in a cartesian system, that is
#x(t)=sin(t+\pi)\hat{i}+sin(2t-\pi/4)\hat{j}#
Differentiating with respect to time, we get the instantaneous velocity of the object at time #t#,
So, #\frac{d}{dx}(x(t))=\frac{d}{dx}(sin(t+\pi))\hat{i}+\frac{d}{dx}(sin(2t-\pi/4))\hat{j}=v(t)#
So, #v(t)=cos(t+\pi)\hat{i}+2cos(2t-\pi/4)\hat{j}#

At #t=(-\pi)/3#, #v((-\pi)/3)=cos(-\pi/3+\pi)\hat{i}+2cos(-2\pi/3-\pi/4)\hat{j}=cos(2\pi/3)\hat{i}+2cos(\frac{-8\pi-3\pi}{12\pi})\hat{j}=-0.5\hat{i}--1.931\hat{j}#
So, velocity of the object at time #t=(-\pi)/3 " is " -0.5\hat{i}--1.931\hat{j}#
Try to find the magnitude as an exercise.

Related questions
See all questions in Instantaneous Velocity
Impact of this question
1535 views around the world
You can reuse this answer
Creative Commons License