Final answer:
To find the displacement and distance traveled by a particle, we integrate the velocity function and calculate the difference in position at the initial and final times. The displacement is given by the integral of the velocity function, while the distance traveled is the sum of the absolute values of the displacements in different segments.
Step-by-step explanation:
To find the displacement and distance traveled, we need to integrate the velocity function over the given time interval. Given the velocity function, v(t) = 3t-5 m/s, we can integrate it to find the displacement function, x(t). The displacement is the integral of the velocity function over the time interval, which is x(t) = ∫(3t-5)dt = (3/2)t^2 - 5t + C. To find the displacement during the given time interval, we subtract the value of x(t) at the initial time from the value of x(t) at the final time.
To find the distance traveled, we need to calculate the total path length traveled. If the velocity function is positive, the particle is moving in the positive direction, and if the velocity function is negative, the particle is moving in the negative direction. We need to consider both cases.
Example: If the velocity function changes sign within the given time interval, we need to find the time(s) when the velocity is zero and calculate the path length traveled in each segment. We use the formulas for displacement and distance traveled to find the desired values.
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