# The velocity of an object with a mass of 8 kg is given by v(t)= sin 5 t + cos 3 t . What is the impulse applied to the object at t= ( 3 pi)/ 4 ?

Jan 13, 2016

$11.3137 k g . m / s$

#### Explanation:

Impulse may be given as change in momentum as follows by $I \left(t\right) = F \mathrm{dt} = m \mathrm{dv}$.

$\therefore I \left(t\right) = m \mathrm{dv}$

$= m \frac{d}{\mathrm{dt}} \left(\sin 5 t + \cos 3 t\right)$

$= 8 \left(5 \cos 5 t - 3 \sin 3 t\right)$

$= 40 \cos 5 t - 24 \sin 3 t$

$\therefore I \left(\frac{3 \pi}{4}\right) = 40 \cos \left(\frac{5 \cdot 3 \pi}{4}\right) - 24 \sin \left(\frac{3 \cdot 3 \pi}{4}\right)$

$= \frac{40}{\sqrt{2}} - \frac{24}{\sqrt{2}}$

$= \frac{16}{\sqrt{2}}$

$11.3137 k g . m / s$