Question

The velocity function (in meters per second) for a particle moving along a line is given by v(t)=t³-5t². Find the displacement and the distance traveled by the particle during the time interval [-1,6].

Answer 1

The displacement of the particle during the time interval [-1,6] is 28 meters, and the distance traveled is also 28 meters.

Step-by-step explanation:

The velocity function for the particle is given by v(t) = t³ - 5t². To find the displacement, we need to integrate the velocity function from -1 to 6:

∫(t³ - 5t²) dt = (1/4)t⁴ - (5/3)t³

Plugging in the upper and lower limits, we get:

Displacement = [(1/4)(6⁴) - (5/3)(6³)] - [(1/4)(-1⁴) - (5/3)(-1³)] = 28 meters

To find the distance traveled, we need to consider the absolute value of the velocity function. Taking the integral of |v(t)| from -1 to 6:

∫(|t³ - 5t²|) dt = (1/4)t⁴ - (5/3)t³

Since the velocity function is non-negative in the given interval, the distance traveled is the same as the displacement, which is 28 meters.