# An object with a mass of 3 kg is acted on by two forces. The first is F_1= < 1 N , 4 N> and the second is F_2 = < 2 N, -3 N>. What is the object's rate and direction of acceleration?

Aug 6, 2016

Rate of acceleration
$\left\mid \vec{a} \right\mid = \sqrt{{1}^{2} + {\left(\frac{1}{3}\right)}^{2}} = \frac{\sqrt{10}}{3} \setminus \frac{m}{s} ^ 2$

Direction
$\arctan \left(\frac{1}{3}\right)$ measured CCW from x-axis

#### Explanation:

$\Sigma \vec{F} = m \vec{a}$

here

$\left(\begin{matrix}1 \\ 4\end{matrix}\right) + \left(\begin{matrix}2 \\ - 3\end{matrix}\right) = 3 \left(\begin{matrix}{a}_{x} \\ {a}_{y}\end{matrix}\right)$

$\left(\begin{matrix}3 \\ 1\end{matrix}\right) = 3 \left(\begin{matrix}{a}_{x} \\ {a}_{y}\end{matrix}\right)$

$\left(\begin{matrix}{a}_{x} \\ {a}_{y}\end{matrix}\right) = \left(\begin{matrix}1 \\ \frac{1}{3}\end{matrix}\right)$

Rate of acceleration
$\left\mid \vec{a} \right\mid = \sqrt{{1}^{2} + {\left(\frac{1}{3}\right)}^{2}} = \frac{\sqrt{10}}{3} \setminus \frac{m}{s} ^ 2$

direction is: $\arctan \left(\frac{1}{3}\right)$ measured CCW from x-axis