Question

The figure shows the velocity function for motion along a line. Assume the motion begins with an initial position of ​s(0). Determine the following. a. The displacement between t=0 and t=10. b. The distance traveled between t=0 and t=10. c. The position at t=10.

Answer 1

The displacement between t=0 and t=10 is 8 m. The distance traveled between t=0 and t=10 is also 8 m. The position at t=10 is 468 m.

Step-by-step explanation:

The displacement between t=0 and t=10 can be found by integrating the velocity function over the interval [0, 10]. The velocity function is given as v(t) = 10t – 12t² m/s.

The displacement is the antiderivative of the velocity function, which is the integral of v(t) with respect to t. Integrating v(t), we get s(t) = 5t² - 4t³ + C, where C is the constant of integration. To find the displacement between t=0 and t=10, we evaluate s(10) - s(0). After evaluating this expression, we find that the displacement is 8 m.

The distance traveled between t=0 and t=10 can be found by integrating the absolute value of the velocity function over the interval [0, 10]. We need to consider the absolute value of the velocity function because distance is a scalar quantity and does not depend on direction.

Taking the absolute value of v(t), we get |v(t)| = |10t – 12t²|. Integrating |v(t)|, we get d(t) = 5t² - 4t³ + D, where D is the constant of integration. To find the distance traveled between t=0 and t=10, we evaluate d(10) - d(0). After evaluating this expression, we find that the distance traveled is 8 m.

The position at t=10 can be found by evaluating the position function at t=10. The position function is given as s(t) = 5t² - 4t³ + C, where C is the constant of integration. Evaluating s(10), we find that the position at t=10 is 468 m.