Final answer:
To find the distance a particle travels in the first t seconds, integrate the velocity function, which is given here as v(t) = t² e³ t. For example, v(2.0 s) for a simpler velocity function v(t) = 3.0 + 1.5t² m/s would be 9.0 m/s. The actual integration for the provided function would be more complex.
Step-by-step explanation:
To determine how many meters a particle will travel during the first t seconds, you need to integrate the velocity function v(t) which gives you the position function x(t). The velocity function we have is v(t) = t² e³ t meters per second. By integrating this with respect to t, we would obtain the position function, which represents the total distance traveled over time. For instance, if we were given a simple function like v(t) = 3.0 + 1.5t² m/s, integrating from 0 to t provides the position x(t). As an example, at t = 2.0 s, the velocity v(2.0 s) = [3.0 + 1.5(2.0)²] m/s = 9.0 m/s. However, for our given function involving an exponential, the integration is more complex and typically requires advanced integration techniques or numerical methods.