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.

Question

1203 views around the world

You can reuse this answer
Creative Commons License

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#