displacement and the velocity of a particle

Question

The displacement of a particle in centimetres is given by y=8 sin (2πft). If f=20 Hz, find the displacement and the velocity of the particle at a time t=0.01 s and 0.07 s.

Answer 1

To find the displacement, we will just plug in our times.  So for t=.01s, we have

                                       y=8sin(2π*(20)[which is our frequency]*(.01))

                                       y= 7.61cm

We would then just plug in t=.07s in order to find the second displacement.

 

To answer this question for velocity, we first need to know the relationship between displacement and velocity; simply, that the derivative with respect to time of the displacement is the velocity. We know that 2πf is a constant with respect to t, so lets just call that C for constant.  Our function then is just of the form

                                       y= 8sin(Ct)

So our derivative is

                                       y'= 8Ccos(Ct)

So we end up with

                                       y'= 8*2π*(20)cos(2π*(20)*t)

Plug in for t=.01 and we see that y'= 310.7 cm/s.

You would just plug in t=.07 for the second velocity.