Respuesta :
Answer:
The particle will travel 6 feet in first 2 seconds.
Step-by-step explanation:
We have been given that a particle moves according to the velocity equation [tex]v(t)= 6t^2-18t+12[/tex]. We are asked to find the distance that the particle will travel in its first 2 seconds.
[tex]s(t)=\int |v(t)|dt[/tex]
[tex]s(t)=\int\limits^2_0 |6t^2-18t+12|dt[/tex]
Now, we will eliminate the absolute value sign as:
[tex]s(t)=\int\limits^1_0 6t^2-18t+12dt+\int\limits^2_1 -6t^2+18t-12dt[/tex]
[tex]s(t)=[\frac{6t^3}{3}-\frac{18t^2}{2}+12t]^1_0 +[\frac{-6t^3}{3}+\frac{18t^2}{2}-12t]^2_1[/tex]
[tex]s(t)=[2t^3-9t^2+12t]^1_0 +[-2t^3+9t^2-12t]^2_1[/tex]
[tex]s(2)=2(1)^3-9(1)^2+12(1)-(2(0)^3-9(0)^2+12(0))-2(2)^3+9(2)^2-12(2)-(-2(1)^3+9(1)^2-12(1))[/tex]
[tex]s(2)=2-9+12-(0)-16+36-24-(-2+9-12)[/tex]
[tex]s(2)=5-(0)-4-(-5)[/tex]
[tex]s(2)=5-4+5[/tex]
[tex]s(2)=6[/tex]
Therefore, the particle will travel 6 feet in first 2 seconds.
The distance is 6 feet.
To understand more, check below explanation.
Velocity :
The velocity equation is given as,
[tex]v=6t^{2}-18t+12[/tex]
As we know that, distance is rate of change of velocity with respect to time.
[tex]d=\frac{dv}{dt}\\\\d=12t-18[/tex]
We have to find distance, when t = 2 seconds.
[tex]d(2)=(12*2)-18\\\\d(2)=24-18=6feet[/tex]
Hence, the distance is 6 feet.
Learn more about the distance here:
https://brainly.com/question/17273444
