Final Answer:
The formula for the displacement of the particle at time t seconds is given by s(t) = ∫v(t) dt, where v(t) is the velocity function. The total distance traveled by the particle at time t seconds is given by ∫|v(t)| dt.
Step-by-step explanation:
To find the displacement of the particle, we use the formula s(t) = ∫v(t) dt, where v(t) represents the velocity function. This formula calculates the area under the velocity-time graph, which gives us the displacement of the particle at time t seconds. If the velocity function v(t) is given, we can integrate it with respect to time to obtain the displacement function s(t).
For the total distance traveled by the particle at time t seconds, we use the formula ∫|v(t)| dt. This formula accounts for both positive and negative values of velocity, ensuring that we consider the entire path traveled by the particle without regard to direction. By integrating the absolute value of the velocity function with respect to time, we obtain the total distance traveled by the particle up to time t seconds.
In summary, to find the displacement of a particle and its total distance traveled at time t seconds, we use integration to calculate s(t) = ∫v(t) dt for displacement and ∫|v(t)| dt for total distance traveled.