# The velocity of an object with a mass of 8 kg is given by v(t)= sin 3 t+ cos 2 t . What is the impulse applied to the object at t= ( 3 pi)/ 4 ?

Feb 15, 2016

#### Explanation:

This is an ill-posed problem. I see a whole lot of questions asking What is the impulse applied to an object at a given instant. You can talk about force applied at a given instant. But when we talk about Impulse, it is always defined for a time interval and not for an instant of time.

By Newton's Second Law,
Force: $\setminus \vec{F} = \setminus \frac{d \setminus \vec{p}}{\mathrm{dt}} = \setminus \frac{d}{\mathrm{dt}} \left(m . \setminus \vec{v}\right) = m \setminus \frac{d \setminus \vec{v}}{\mathrm{dt}}$

Magnitude of the force : $F \left(t\right) = m \setminus \frac{\mathrm{dv}}{\mathrm{dt}} = m . \setminus \frac{d}{\mathrm{dt}} \left(\sin 3 t + \cos 2 t\right)$,
$F \left(t\right) = m . \left(3 \cos 3 t - 2 \sin 2 t\right)$

$F \left(t = \frac{3 \setminus \pi}{4}\right) = \left(8 k g\right) \setminus \times \left(3 \cos \left(\frac{9 \setminus \pi}{4}\right) - 2 \sin \left(\frac{3 \setminus \pi}{2}\right)\right) m {s}^{- 2} = 32.97 N$

Impulse : $J = \setminus {\int}_{{t}_{i}}^{{t}_{f}} F \left(t\right) . \mathrm{dt}$ is defined for the time interval $\setminus \Delta t = {t}_{f} - {t}_{i}$. So it makes no sense to talk about impulse at an instant.