# An object is at rest at (1 ,2 ,1 ) and constantly accelerates at a rate of 2/3 m/s as it moves to point B. If point B is at (4 ,9 ,5 ), how long will it take for the object to reach point B? Assume that all coordinates are in meters.

Sep 29, 2017

5.08 seconds

#### Explanation:

$A \left(1 , 2 , 1\right)$ and $B \left(4 , 9 , 5\right)$
Displacement $A B = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2} + {\left({z}_{2} - {z}_{1}\right)}^{2}}$
$A B = \sqrt{{\left(4 - 1\right)}^{2} + {\left(9 - 2\right)}^{2} + {\left(5 - 1\right)}^{2}} = \sqrt{{3}^{2} + {7}^{2} + {4}^{2}}$
$A B = \sqrt{9 + 49 + 16} = \sqrt{74} m$

Acceleration $a = \frac{2}{3} \frac{m}{s} ^ 2$
Initial Velocity $u = 0$

Using Newton's Second Equation
$S = u t + \frac{1}{2} a {t}^{2}$
S is displacement
$\sqrt{74} = 0 + \frac{1}{2} \left(\frac{2}{3}\right) {t}^{2}$
$\frac{2}{3} {t}^{2} = 2 \sqrt{74}$
${t}^{2} = 3 \sqrt{74}$
$t = \sqrt{3 \sqrt{74}}$
$t = 5.08 \text{seconds}$