What is the instantaneous velocity of an object moving in accordance to  f(t)= (t^2,tcos(t-(5pi)/4))  at  t=(pi)/3 ?

Jun 2, 2016

$v \left(\frac{\pi}{3}\right) = \frac{1}{3} \sqrt{4 {\pi}^{2} + 9 {\cos}^{2} \left(\frac{\pi}{12}\right) + \pi {\sin}^{2} \left(\frac{\pi}{12}\right) + 6 \pi \cos \left(\frac{\pi}{12}\right) \sin \left(\frac{\pi}{12}\right)}$

Explanation:

The equation f(t)=(t^2;tcos(t-(5pi)/4)) gives you the object's coordinates with respect to time:

$x \left(t\right) = {t}^{2}$
$y \left(t\right) = t \cos \left(t - \frac{5 \pi}{4}\right)$

To find $v \left(t\right)$ you need to find ${v}_{x} \left(t\right)$ and ${v}_{y} \left(t\right)$

${v}_{x} \left(t\right) = \frac{\mathrm{dx} \left(t\right)}{\mathrm{dt}} = \frac{{\mathrm{dt}}^{2}}{\mathrm{dt}} = 2 t$

${v}_{y} \left(t\right) = \frac{d \left(t \cos \left(t - \frac{5 \pi}{4}\right)\right)}{\mathrm{dt}} = \cos \left(t - \frac{5 \pi}{4}\right) - t \sin \left(t - \frac{5 \pi}{4}\right)$

Now you need to replace $t$ with $\frac{\pi}{3}$

${v}_{x} \left(\frac{\pi}{3}\right) = \frac{2 \pi}{3}$

${v}_{y} \left(\frac{\pi}{3}\right) = \cos \left(\frac{\pi}{3} - \frac{5 \pi}{4}\right) - \frac{\pi}{3} \cdot \sin \left(\frac{\pi}{3} - \frac{5 \pi}{4}\right)$
$= \cos \left(\frac{4 \pi - 15 \pi}{12}\right) - \frac{\pi}{3} \cdot \sin \left(\frac{4 \pi - 15 \pi}{12}\right)$
$= \cos \left(\frac{- 11 \pi}{12}\right) - \frac{\pi}{3} \cdot \sin \left(\frac{- 11 \pi}{12}\right)$
$= \cos \left(\frac{\pi}{12}\right) + \frac{\pi}{3} \cdot \sin \left(\frac{\pi}{12}\right)$

Knowing that ${v}^{2} = {v}_{x}^{2} + {v}_{y}^{2}$ you find:

$v \left(\frac{\pi}{3}\right) = \sqrt{{\left(\frac{2 \pi}{3}\right)}^{2} + {\left(\cos \left(\frac{\pi}{12}\right) + \frac{\pi}{3} \cdot \sin \left(\frac{\pi}{12}\right)\right)}^{2}}$
$= \sqrt{\frac{4 {\pi}^{2}}{9} + {\cos}^{2} \left(\frac{\pi}{12}\right) + {\pi}^{2} / 9 \cdot {\sin}^{2} \left(\frac{\pi}{12}\right) + \frac{2 \pi}{3} \cdot \cos \left(\frac{\pi}{12}\right) \sin \left(\frac{\pi}{12}\right)}$
$= \frac{1}{3} \sqrt{4 {\pi}^{2} + 9 {\cos}^{2} \left(\frac{\pi}{12}\right) + \pi {\sin}^{2} \left(\frac{\pi}{12}\right) + 6 \pi \cos \left(\frac{\pi}{12}\right) \sin \left(\frac{\pi}{12}\right)}$