An object is at rest at `(7 ,6 ,4 )` and constantly accelerates at a rate of `5/4 m/s^2` as it moves to point B. If point B is at `(5 ,5 ,7 )`, how long will it take for the object to reach point B? Assume that all coordinates are in meters.

Let distance be `s`
Let acceleration be `a`
Let mean velocity be `v`

Let point 1 be `P_1->(x_1,y_1,z_1)=(7,6,4)`
Let point 2 be `P_2->(x_2,y_2,z_2)=(5,5,7)`

`color(blue)("Determine distance between points")`

`s = sqrt((x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2)`

`s=sqrt( (5-7)^2+(5-6)^2+(7-4)^2)`

`color(blue)(s=sqrt(14))`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Determine time of travel")`

Distance is velocity multiplied by time but we have acceleration. So we need to change the acceleration to mean velocity

Mean velocity is
`v=(0+at)/2" " =" " (at)/2" " =" " 5/4xx1/2xxt" " =" " 5/8t`

`color(red)(s)" "color(green)(=" "vt" ")color(blue)( =" "(5/8 t)xxt)" "color(magenta)(=" "5/8t^2)`

`color(brown)("This is where the "1/2at^2 " in the standard equation")``color(brown)("of distance comes from")`

`s=5/8t^t" " =>" " t^2=8/5xxsqrt(14)`

`color(blue)(=>t=+-sqrt(8/5xxsqrt(14))~~+-2.447" to 3 decimal places")`

The negative time is not logical so we can discount it.

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(magenta)("Foot note about the units for acceleration")`

`color(brown)("You can manipulate the units of measurement in the same way")``color(brown)("you do the counts (values)")`

Acceleration is changing velocity

Let the value of the acceleration is `a`
Let the unit measure of distance be `m`
Let the unit measure of time be `s`
Let the count of time be `t`

Then acceleration is written as `a m/s^2`

We must write the seconds as `s^2` to make what follows work.

The passing of time is `t s`

So the velocity after `t` seconds is

`a m/s^2xxts`

Separating the counts from the units of measurement we have

`(axxt) xx (m/s^2xxs)`

`=>(axxt)xx(m/sxxs/s) = (at)xx(m/sxx1)" "->" "at m/s`