The position of an object moving along a line is given by #p(t) = cos(t- pi /3) +1 #. What is the speed of the object at #t = (2pi) /4 #?

Question

The position of an object moving along a line is given by #p(t) = cos(t- pi /3) +1  #.  What is the speed of the object at #t = (2pi) /4 #?

Emily Abbate · Accepted Answer

#v((2pi)/4) = -1/2#

We can differentiate the provided equation to find the equation for the object's velocity because we already know the equation for the position:

#v(t) = d/dt p(t) = -sin(t - pi/3)#

plugging in the point at which we want to know speed: #v((2pi)/4) = -sin((2pi)/4 - pi/3) = -sin(pi/6) = -1/2#

Technically, it might be stated that the speed of the object is, in fact, #1/2#, since speed is a directionless magnitude, but I have chosen to leave the sign.

Olivia Askew · Answer

undefined To find the speed of the object at $ t = \frac{2\pi}{4} $, we need to differentiate the position function with respect to time and then evaluate it at the given time. The speed is the absolute value of the derivative of the position function.

The derivative of $ p(t) $ with respect to $ t $ is $ p'(t) = -\sin(t - \frac{\pi}{3}) $.

At $ t = \frac{2\pi}{4} = \frac{\pi}{2} $, $ p'(\frac{\pi}{2}) = -\sin(\frac{\pi}{2} - \frac{\pi}{3}) = -\sin(\frac{\pi}{6}) = -\frac{1}{2} $.

The speed of the object at $ t = \frac{2\pi}{4} $ is $ |\frac{1}{2}| = \frac{1}{2} $.