60–63. Equivalent constant velocity Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of ________.


v(t)=2 sin t, for 0≤t≤π

Verified step by step guidance

1. Identify the given velocity function: \(v(t) = 2 \sin t\) over the interval \(0 \leq t \leq \pi\).
2. Recall that the distance traveled over a time interval is the integral of the velocity function over that interval. So, calculate the total distance by evaluating the definite integral \(\int_0^{\pi} 2 \sin t \, dt\).
3. Compute the integral: Use the antiderivative of \(\sin t\), which is \(-\cos t\), to find \(\int 2 \sin t \, dt = -2 \cos t + C\).
4. Evaluate the definite integral from \(0\) to \(\pi\): Calculate \([-2 \cos t]_0^{\pi} = (-2 \cos \pi) - (-2 \cos 0)\).
5. To find the equivalent constant velocity \(v_c\), divide the total distance traveled by the total time interval length \(\pi\): \(v_c = \frac{\int_0^{\pi} 2 \sin t \, dt}{\pi}\).

Key Concepts

Average Velocity

Average velocity over a time interval is defined as the total displacement divided by the total time. It represents a constant velocity that would cover the same distance in the same time period as the varying velocity function.

Definite Integral and Displacement

The definite integral of a velocity function over a time interval gives the total displacement during that period. Calculating the integral of v(t) = 2 sin t from 0 to π provides the total distance traveled.

Relationship Between Velocity and Distance

Velocity is the rate of change of position. To find an equivalent constant velocity, one must use the total distance traveled and divide it by the total time, ensuring the constant velocity covers the same displacement as the original function.