Average velocity The velocity in m/s of an object moving along a line over the time interval [0,6] is v (t) = t² + 3t. Find the average velocity of the object over this time interval.

Verified step by step guidance

1. Step 1: Recall the formula for average velocity over a time interval [a, b]. It is given by: $\frac{v_{\mathrm{avg}}}{=\frac{1_{\mathrm{b-a}}}{{\int }_a}}$
2. Step 2: Substitute the given time interval [0, 6] into the formula. The average velocity becomes: $\frac{v_{\mathrm{avg}}}{=\frac{1_{\mathrm{6-0}}}{{\int }_0}}$
3. Step 3: Write the integral expression for the velocity function v(t) = t² + 3t. The integral becomes: ${\int }_0\mathrm{(t²+3t)}\mathrm{dt}$
4. Step 4: Compute the integral of the function t² + 3t. Use the power rule for integration: $\int \mathrm{t²}\mathrm{dt}=\frac{\mathrm{t³}}{3}$ and $\int \mathrm{3t}\mathrm{dt}=\frac{\mathrm{3t²}}{2}$. Combine these results to find the antiderivative.
5. Step 5: Evaluate the definite integral by substituting the limits of integration (0 and 6) into the antiderivative. Then divide the result by the length of the interval (6 - 0) to find the average velocity.

Key Concepts

Average Velocity

Average velocity is defined as the total displacement divided by the total time taken. In calculus, it can be calculated using the formula: average velocity = (s(b) - s(a)) / (b - a), where s(t) is the position function and [a, b] is the time interval. For the given problem, we need to integrate the velocity function over the interval and then divide by the length of the interval.

Velocity Function

The velocity function describes how the velocity of an object changes over time. In this case, the velocity is given by v(t) = t² + 3t, which is a polynomial function. Understanding this function is crucial for determining the object's behavior over the specified time interval, as it provides the rate of change of position with respect to time.

Integration

Integration is a fundamental concept in calculus used to find the area under a curve, which in the context of motion, represents the total displacement. To find the average velocity, we need to integrate the velocity function v(t) over the interval [0, 6] and then divide the result by the length of the interval. This process allows us to calculate the total change in position over the given time period.