Question

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`?

Answer 1

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.