109. Average velocity Find the average velocity of a projectile whose velocity over the interval 0 ≤ t ≤ π is given by
v(t) = 10 * sin(3t).

Verified step by step guidance

1. Recall that the average velocity over the interval \([a, b]\) is given by the formula: \(\text{Average velocity} = \frac{1}{b - a} \int_a^b v(t) \, dt\).
2. Identify the interval and the velocity function: here, \(a = 0\), \(b = \pi\), and \(v(t) = 10 \sin(3t)\).
3. Set up the integral for the average velocity: \(\frac{1}{\pi - 0} \int_0^{\pi} 10 \sin(3t) \, dt = \frac{1}{\pi} \int_0^{\pi} 10 \sin(3t) \, dt\).
4. Evaluate the integral \(\int_0^{\pi} 10 \sin(3t) \, dt\) by using the substitution method or by recalling the integral of sine: \(\int \sin(k t) \, dt = -\frac{1}{k} \cos(k t) + C\).
5. After finding the definite integral value, divide it by \(\pi\) to find the average velocity over the interval \([0, \pi]\).

Key Concepts

Average Velocity

Average velocity over a time interval is defined as the total displacement divided by the total time elapsed. It can be calculated by integrating the velocity function over the interval to find displacement, then dividing by the length of the interval.

Definite Integral of Velocity

The definite integral of a velocity function v(t) from time a to b gives the net displacement of the object during that interval. This integral is essential to find total change in position when velocity varies with time.

Trigonometric Integration

Integrating functions involving sine or cosine requires knowledge of basic trigonometric integrals. For example, the integral of sin(k t) with respect to t is -cos(k t)/k, which is used to evaluate the displacement from the given velocity function.