Question

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

Answer 1

`vecV = 2hati+2/3hatj`

Explanation:

The object is performing 2 dimensional motion. Its position vector can be resolved into the following:
`vec R = ( 2sin(2t + pi)hat i+ t-sec(t/2) hatj)`
where `hati , hatj` are unit vectors in the x and y direction.

Now, the velocity is the first derivative of its position with respect to time. Therefore,

`d/dtvec R = vec V` where `vec V` is the velocity vector.

`d/dtvecR = d/dt( 2sin(2t + pi)hat i+ t-sec(t/2) hatj)`
`=> vec V = (-4cos(2t))hati + (1 - 2 sin^3(t/2) csc^2(t))hatj`

plugging in `t = pi/3`

`vec V = 2hati+2/3hatj`