# An object with a mass of 3 kg is acted on by two forces. The first is F_1= < 4 N , 1 N> and the second is F_2 = < 5 N, 9 N>. What is the object's rate and direction of acceleration?

Feb 21, 2017

The rate of acceleration $= 4.48 m {s}^{-} 2$ in the direction $48.0$º

#### Explanation:

The resultant force is

$\vec{F} = {\vec{F}}_{1} + {\vec{F}}_{2}$

$= < 4 , 1 > + < 5 , 9 >$

$= < 9 , 10 >$

we use Newton's second Law

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

$\vec{a} = \frac{1}{m} \cdot \vec{F}$

$= \frac{1}{3} < 9 , 10 > = < 3 , \frac{10}{3} >$

The magnitude of the acceleration is

$| | \vec{a} | | = | | < 3 , \frac{10}{3} > | |$

$= \sqrt{{\left(3\right)}^{2} + {\left(\frac{10}{3}\right)}^{2}}$

$= \sqrt{9 + \frac{100}{9}}$

$= \frac{\sqrt{181}}{3} = 4.48 m {s}^{-} 2$

The direction is $\theta = \arctan \left(\frac{\frac{10}{3}}{3}\right)$

The angle is in the 1st quadrant

$\theta = \arctan \left(\frac{10}{9}\right) = 48.0$º