An object is at rest at #(8 ,6 ,9 )# and constantly accelerates at a rate of #1 m/s# as it moves to point B. If point B is at #(6 ,4 ,3 )#, how long will it take for the object to reach point B? Assume that all coordinates are in meters.

Question

An object is at rest at #(8  ,6   ,9    )# and constantly accelerates at a rate of #1  m/s# as it moves to point B.  If point B is at #(6   ,4   ,3   )#, how long will it take for the object to reach point B? Assume that all coordinates are in meters.

Leo Abeyta · Accepted Answer

The time is #=3.64s#

The distance between the points #A=(x_A,y_A,z_A)# and the point #B=(x_B,y_B,z_B)# is

#AB=sqrt((x_B-x_A)^2+(y_B-y_A)^2+(z_B-z_A)^2)#

#d=AB= sqrt((6-8)^2+(4-6)^2+(3-9)^2)#

#=sqrt(2^2+2^2+6^2)#

#=sqrt(4+4+36)#

#=sqrt44#

#=6.63m#

We utilize the equation of motion.

#d=ut+1/2at^2#

#u=0#

thus,

#d=1/2at^2#

#a=5/4ms^-2#

#t^2=(2d)/a=(2*6.63)/(1)#

#=13.27s^2#

#t=sqrt(13.27)=3.64s#

Carson Alexander · Answer

undefined Using the kinematic equation $ \Delta x = v_i t + \frac{1}{2} a t^2 $, where $ \Delta x $ is the displacement, $ v_i $ is the initial velocity, $ a $ is the acceleration, and $ t $ is the time:

For each coordinate direction:

For x-axis:
$ \Delta x = 6 - 8 = -2 $ m
$ v_i = 0 $ m/s
$ a = 1 $ m/s$^2$
$ \Delta x = \frac{1}{2} a t^2 $
$ -2 = \frac{1}{2} t^2 $
$ t^2 = -4 $
$ t = \sqrt{-4} $ (Ignoring the negative root since time cannot be negative)

For y-axis:
$ \Delta y = 4 - 6 = -2 $ m
$ v_i = 0 $ m/s
$ a = 1 $ m/s$^2$
$ \Delta y = \frac{1}{2} a t^2 $
$ -2 = \frac{1}{2} t^2 $
$ t^2 = -4 $
$ t = \sqrt{-4} $ (Ignoring the negative root since time cannot be negative)

For z-axis:
$ \Delta z = 3 - 9 = -6 $ m
$ v_i = 0 $ m/s
$ a = 1 $ m/s$^2$
$ \Delta z = \frac{1}{2} a t^2 $
$ -6 = \frac{1}{2} t^2 $
$ t^2 = -12 $
$ t = \sqrt{-12} $ (Ignoring the negative root since time cannot be negative)

Thus, the object will take the same time to reach point B along all three axes. Taking any one of the axes, $ t = \sqrt{4} = 2 $ seconds. Therefore, it will take 2 seconds for the object to reach point B.