To find the distance traveled by the particle during the first $t$ seconds, we need to calculate the displacement. The displacement is found by integrating the velocity function $v(t)$ over the interval from $0$ to $t$.

Given: $v(t) = t^2 e^{-3t}$ We calculate the displacement by integrating the velocity: $\text{Displacement} = \int_0^t v(t) \, dt = \int_0^t t^2 e^{-3t} \, dt$

Let me evaluate this integral for you.import sympy as sp

Define the variable and the function

t = sp.symbols('t') v = t**2 * sp.exp(-3*t)

Compute the definite integral of the velocity from 0 to t

displacement = sp.integrate(v, (t, 0, t))

displacement.simplify()The displacement function for the particle during the first $t$ seconds is given by:

$\text{Displacement} = \frac{\left(-9t^2 - 6t + 2e^{3t} - 2\right) e^{-3t}}{27}$

This formula gives the total distance the particle traveled from $t = 0$ to $t$.

Would you like to evaluate this displacement for a specific time $t$ or explore further details?

Here are 5 related questions:

1. How would you calculate the displacement for a specific value of $t$?
2. What is the physical interpretation of negative and positive displacements?
3. How does the particle's velocity affect its displacement?
4. How does integrating velocity give the total distance traveled?
5. How can we find when the particle comes to a stop?

Tip: Always check the units when performing integration for physical quantities; in this case, the displacement will be in meters if the velocity is in meters per second.