Question

What is the instantaneous velocity of an object moving in accordance to `f(t)= (sin2t-cos^2t,tantsect )` at `t=(-pi)/12`?

Answer 1

Instantaneous velocity is `0.366hati+1.1839hatj`

Explanation:

The function `f(t)=(sin2t-cos^2t,tantsect)`, shows the position of a point at a time `t`. Here in `(sin2t-cos^2t,tantsect)`, first number indicates position along `x`-axis and second number along `y`-axis.

As such instantaneous velocity is given by

`d/(dt)(sin2t-cos^2t)hati+d/(dt)(tantsect)hatj`

or `(2cos2t+2sintcost)hati+(sec^2tsect+tantsect tant)hatj`

or `(2cos2t+sin2t)hati+(sec^3t+tan^2tsect)hatj`

and at `t=-pi/12`

it is `(2cos(-pi/6)+sin(-pi/6))hati+(sec^3(-pi/12)+tan^2(-pi/12)sec(-pi/12))hatj`

or `(sqrt3-1/2)hati+(1.1096+0.0743)hatj`

or `0.366hati+1.1839hatj`