Final answer:
The motion of the particle can be described by the equation x = t^3 + 2t^2 - 4t + 3. The particle crosses the origin at t = -3 and t = 1. The displacement of the particle between t = 3.0 s and t = 6.0 s is 210.
Step-by-step explanation:
The motion of a particle is defined by the relation x = t^3 + 2t^2 - 4t + 3.
To determine when the particle crosses the origin, we need to find the value of t that satisfies x = 0.
Setting x = 0, we can solve the equation t^3 + 2t^2 - 4t + 3 = 0. By factoring or using the Rational Root Theorem, we find that t = -3 and t = 1 are the solutions.
To find the displacement of the particle between t = 3.0 s and t = 6.0 s, we need to find the values of x at those time points. Substituting t = 3.0 s and t = 6.0 s into the equation x = t^3 + 2t^2 - 4t + 3, we get x = 27 and x = 237 respectively. The displacement is the difference between these two values, which is 237 - 27 = 210.