To determine the displacement of the particle when its acceleration is zero, we will need to undertake the following steps:
1. Identify displacement function: We are given the displacement function:
[tex]\[
s(t) = t^3 - 6t^2 + 3t + 4
\][/tex]
2. Find the velocity function: Velocity ([tex]\(v\)[/tex]) is the first derivative of displacement with respect to time [tex]\(t\)[/tex]:
[tex]\[
v(t) = \frac{ds}{dt} = \frac{d}{dt} (t^3 - 6t^2 + 3t + 4)
\][/tex]
Calculate the derivative:
[tex]\[
v(t) = 3t^2 - 12t + 3
\][/tex]
3. Find the acceleration function: Acceleration ([tex]\(a\)[/tex]) is the derivative of velocity with respect to time [tex]\(t\)[/tex]:
[tex]\[
a(t) = \frac{dv}{dt} = \frac{d}{dt}(3t^2 - 12t + 3)
\][/tex]
Calculate the derivative:
[tex]\[
a(t) = 6t - 12
\][/tex]
4. Determine when the acceleration is zero: Set the acceleration function equal to zero and solve for [tex]\(t\)[/tex]:
[tex]\[
6t - 12 = 0
\][/tex]
[tex]\[
6t = 12
\][/tex]
[tex]\[
t = 2
\][/tex]
5. Find the displacement when [tex]\(t = 2\)[/tex]: Substitute [tex]\(t = 2\)[/tex] back into the displacement function:
[tex]\[
s(2) = (2)^3 - 6(2)^2 + 3(2) + 4
\][/tex]
Calculate the value:
[tex]\[
s(2) = 8 - 24 + 6 + 4
\][/tex]
[tex]\[
s(2) = 8 - 24 + 10
\][/tex]
[tex]\[
s(2) = -6
\][/tex]
Therefore, the displacement when the acceleration is zero is [tex]\(-6 \, \text{m}\)[/tex].
The correct answer is:
[tex]\[
\boxed{-6 \, \text{m}}
\][/tex]
