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

Lily Adams · Accepted Answer

#2.12s#

We can solve this just a like a one-dimensional constant acceleration problem; the distance covered is

#Deltax = sqrt (3^2 + 1^2 + 4^2) - sqrt(1^2 + 7^2 + 2^2)= -2.249 m#

And since the acceleration #a_x = 1m/(s^2)#, the time #t# it takes the object to travel #2.249m# (in the reverse direction) is

#t = sqrt((2Deltax)/(a_x)) = sqrt((2(2.249m))/(1m/(s^2))) = 2.12 s#

This is derived from the equation #x = x_0 + v_(0x)t + 1/2a_xt^2#, where #v_(0x)# is #0# (initially at rest), and I made #x# the distance #2.249m# (sign is arbitrary, but needs to be positive to yield a real answer) and #x_0# #0#.

Nora Arzate · Answer

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

$$ d = v_i t + \frac{1}{2} a t^2 $$

Where:
- $d$ is the displacement
- $v_i$ is the initial velocity (which is 0 m/s since the object is at rest)
- $a$ is the acceleration (1 m/s^2)
- $t$ is the time

We rearrange the equation to solve for $t$:

$$ t = \sqrt{\frac{2d}{a}} $$

Given that the initial position is (1, 7, 2) and the final position is (3, 1, 4), we calculate the displacement $d$ using the distance formula:

$$ d = \sqrt{(3 - 1)^2 + (1 - 7)^2 + (4 - 2)^2} $$

$$ d = \sqrt{2^2 + (-6)^2 + 2^2} $$

$$ d = \sqrt{4 + 36 + 4} $$

$$ d = \sqrt{44} $$

$$ d = 2\sqrt{11} $$

Substitute $d$ into the equation for $t$:

$$ t = \sqrt{\frac{2 \times 2\sqrt{11}}{1}} $$

$$ t = \sqrt{4\sqrt{11}} $$

$$ t \approx 2.65 \, \text{seconds} $$

Therefore, it will take approximately 2.65 seconds for the object to reach point B.