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

Jul 2, 2017

$t = 6.51$ $\text{s}$

Explanation:

We're asked to find the time $t$ it takes an object to cover a distance with a given acceleration.

To do this, we can use the equation

$\Delta x = {v}_{0 x} t + \frac{1}{2} {a}_{x} {t}^{2}$

We know it starts from a state of rest, so the initial velocity ${v}_{0 x}$ is zero, so our equation now becomes

$\Delta x = \frac{1}{2} {a}_{x} {t}^{2}$

To find the displacement $\Delta x$, we need to find the distance between the two coordinate points, which is done by the distance formula:

Deltax = sqrt((1-4)^2 + (9-4)^2 + (1-5)^2) = color(red)(7.07 color(red)("m"

And our acceleration is given as $\frac{1}{3}$ ${\text{m/s}}^{2}$

Plugging in known values and solving for $t$, we have

t = sqrt((2Deltax)/(a_x)) = sqrt((2(color(red)(7.07)color(white)(l)color(red)("m")))/(1/3color(white)(l)"m/s"^2)) = color(blue)(6.51 color(blue)("s"