Final answer:
The particle crosses the origin at t = 0 s and t = 6 s. The displacement of the particle between t = 3.0 s and t = 6.0 s is -54 meters, indicating it moved in the negative x-direction.
Step-by-step explanation:
Determining Particle Position and Displacement
The question involves finding the time when a particle crosses the origin and its displacement over a period of time in a one-dimensional motion along the x-axis. The position of a particle is given by x(t) = 12t² - 2t³, where x is in meters and t is in seconds.
Position When Crossing the Origin
To find when the particle crosses the origin, set x(t) to zero and solve for t:
- 0 = 12t² - 2t³
- 0 = t²(12 - 2t)
Thus, t = 0 s or t = 6 s.
Displacement Between Specific Times
The displacement of the particle between t = 3.0 s and t = 6.0 s can be found by subtracting the position at t = 3.0 s from the position at t = 6.0 s.
- x(3.0 s) = 12(3.0 s)² - 2(3.0 s)³ = 108 - 54 = 54 m
- x(6.0 s) = 12(6.0 s)² - 2(6.0 s)³ = 432 - 432 = 0 m
- Displacement = x(6.0 s) - x(3.0 s) = 0 m - 54 m = -54 m
The negative sign indicates that the displacement direction is opposite to the positive x-axis.