An object is at rest at #(4 ,1 ,6 )# and constantly accelerates at a rate of #4/3 m/s^2# as it moves to point B. If point B is at #(3 ,5 ,6 )#, 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 #(4 ,1   ,6    )# and constantly accelerates at a rate of #4/3  m/s^2# as it moves to point B.  If point B is at #(3  ,5   ,6  )#, how long will it take for the object to reach point B? Assume that all coordinates are in meters.

Nathan Ashcroft · Accepted Answer

The time is #=2.49s#

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((3-4)^2+(5-1)^2+(6-6)^2)#

#=sqrt(1^2+4^2+0^2)#

#=sqrt(1+16)#

#=sqrt17#

#=4.123m#

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*4.123)/(4/3)#

#=6.18s^2#

t = sqrt(6.18) = 2.49 s

Emily Abbate · Answer

undefined To find the time it takes for the object to reach point B, we can use the formula for displacement with constant acceleration:

$$ \Delta x = v_0 t + \frac{1}{2} a t^2 $$

Where:
- $ \Delta x $ is the displacement (distance) between the initial position and point B.
- $ v_0 $ is the initial velocity, which is 0 since the object is at rest initially.
- $ a $ is the acceleration rate, given as $ 4/3 $ m/s^2.
- $ t $ is the time taken to reach point B.

Given:
- Initial position: $ (x_0, y_0, z_0) = (4, 1, 6) $
- Final position (point B): $ (x, y, z) = (3, 5, 6) $

First, find the displacement in each dimension:
- $ \Delta x = x - x_0 = 3 - 4 = -1 $
- $ \Delta y = y - y_0 = 5 - 1 = 4 $
- $ \Delta z = z - z_0 = 6 - 6 = 0 $

Substitute into the displacement formula:

$$ -1 = 0 \cdot t + \frac{1}{2} \left(\frac{4}{3}\right) t^2 $$

Solve for $ t $:

$$ \frac{1}{2} \left(\frac{4}{3}\right) t^2 = -1 $$

$$ \frac{2}{3} t^2 = -1 $$

$$ t^2 = -\frac{3}{2} $$

Since time cannot be negative, there is no real solution. This means that the object will never reach point B. There might be a mistake in the provided information or assumptions made.