Question

What is the instantaneous velocity of an object moving in accordance to `f(t)= (sin(2t-pi/2),t)` at `t=(5pi)/4`?

Answer 1

`f'(5pi/4)=(x'(5pi/4),y'(5pi/4))=(1,1).`
speed =magnitude of velocity=`sqrt(1^2+1^2)=sqrt2` unit, & direction at angle `pi/4` with the`+ve` direction of the X-axis.

Explanation:

The instantaneous velocity of the object under question is `f'(5pi/4).`

now, `f(t)=(sin(2t-pi/2),t)=(x(t),y(t)),` say.
`rArr f'(t)=(x'(t),y'(t))`

Now `x'(t)=d/dt(sin(2t-pi/2))=cos(2t-pi/2)*(d/dt(2t-pi/2))=2cos(2t-pi/2)`
`y'(t)=1.`

`x'(5pi/4)`=`2cos{(2)(5pi/4)-pi/2}=2cos(5pi/2-pi/2)=cos4pi/2=cos2pi=1.`

`y'(5pi/4)=1.`

Altogether, `f'(5pi/4)=(x'(5pi/4),y'(5pi/4))=(1,1).`
Recall that velocity, being a vector, we have to express this in magnitude (speed) & direction.

Clearly, speed =`sqrt(1^2+1^2)=sqrt2` unit, direction is at angle`pi/4` with the`+ve` direction of the X-axis.