An object is at rest at #(2 ,1 ,6 )# and constantly accelerates at a rate of #1/4 m/s^2# as it moves to point B. If point B is at #(3 ,2 ,7 )#, 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 #(2 ,1   ,6    )# and constantly accelerates at a rate of #1/4  m/s^2# as it moves to point B.  If point B is at #(3  ,2   ,7  )#, how long will it take for the object to reach point B? Assume that all coordinates are in meters.

Charles Aeschliman · Accepted Answer

#t = sqrt((2s)/a) = sqrt ((2xx1.73)/0.25) = 3.72# #s#

In a 3-dimensional coordinate system, the distance between two points is given by:

#r=sqrt((x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2)#

#r=sqrt((3-2)^2+(2-1)^2+(7-6)^2) =sqrt3 = 1.73# #m#

We can use the equation of motion:

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

Since the initial velocity #u=0#, we can ignore the first term, and rearrange #s= 1/2 at^2# to make #t# the subject.

Note that the term 'r' I was using in the distance calculation above is the same as the distance 's' that the object moves. I don't want the symbol change to be confusing.

#t = sqrt((2s)/a) = sqrt ((2xx1.73)/0.25)#

(I used 0.25 rather than 1/4 for ease of writing, but of course it means the same thing)

Therefore #t = 3.72# #s#.

Amelia Adams · Answer

undefined To find the time it takes for the object to reach point B, we can use the kinematic equation:

$$s = ut + \frac{1}{2}at^2$$

Where:
$s$ = displacement,
$u$ = initial velocity (since the object is at rest, $u = 0$),
$a$ = acceleration, and
$t$ = time.

Given that $s$ = displacement from (2, 1, 6) to (3, 2, 7), we can calculate the displacement using the distance formula:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$

Substituting the coordinates, we find that $d ≈ \sqrt{(1^2 + 1^2 + 1^2)} ≈ \sqrt{3}$.

Now, we plug the values into the kinematic equation:

$$\sqrt{3} = 0 \cdot t + \frac{1}{2} \cdot \frac{1}{4} \cdot t^2$$

Solving for $t$:

$$t^2 = 2 \cdot \sqrt{3}$$

$$t ≈ \sqrt{2 \cdot \sqrt{3}} ≈ 1.732$$ seconds.

Therefore, it will take approximately 1.732 seconds for the object to reach point B.