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

Feb 26, 2016

$I \left(4\right) = 2 \left(4 \cdot {4}^{2} + 3 \cdot {4}^{2}\right) = 7 \cdot {4}^{2} = 112 N s$
Careful however Impulse is not $m v \left(t\right)$ evaluate at $t = 4 s$
it is Force applied for a finite amount of time $F \cdot \Delta t$
in the limit $I = {\int}_{{t}_{1}}^{{t}_{2}} F \mathrm{dt}$

#### Explanation:

This is a straight implementation of the impulse equation i.e.
$F = m \frac{\mathrm{dv}}{\mathrm{dt}}$ Newton law ==> (1)
$I = F \mathrm{dt} = m \left(\mathrm{dv}\right)$ Impulse equation ==> (2)
While the question is not clear I will make the assumption the Force was applied for 4 seconds causing the change in speed from
$v \left(0\right) = 0 \to v \left(4\right)$ and $I \left(4\right) = 2 \left(4 \cdot {4}^{2} + 3 \cdot {4}^{2}\right) = 7 \cdot {4}^{2} = 112 N s$