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.

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Answer 1 · 2016-07-31T08:55:35

#2.447" "# seconds to 3 decimal places

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#