Problem 52 - Calculus Early Transcendentals

Question

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 _____. $$v(t)=2 \sin t, \text { for } 0 \leq t \leq \pi$$

Step-by-Step Solution

Answer

Answer: The constant velocity is $\frac{4}{\pi}$.

Step 1: Find the absolute value of velocity function

Since the maximum value of sin(x) is 1 and minimum value is -1, the range of the given velocity function is from -2 to 2. Considering the interval 0 to π, sin(t) is always positive. So, |2sin(t)| is equal to 2sin(t). Therefore, we can consider the velocity function as it is.

Step 2: Integrate the velocity function

Now, integrate the given velocity function over the interval [0, π] to find the distance traveled. $$\int_0^{\pi} 2\sin(t) dt = 2\int_0^{\pi} \sin(t) dt$$

Step 3: Evaluate the integral

To find the distance traveled, evaluate the integral from step 2: $$2\int_0^{\pi} \sin(t) dt = 2[-\cos(t)]_0^{\pi} = 2[-\cos(\pi)+\cos(0)] = 2[-(-1)+1] = 4$$ The total distance traveled during the interval is 4 units.

Step 4: Find the constant velocity

To find the constant velocity that would result in the same distance traveled, divide the distance by the total time (π): $$\text{Constant velocity } = \frac{\text{Total distance}}{\text{Total time}} = \frac{4}{\pi}$$

Step 5: Write the conclusion

The same distance could have been traveled over the given time period at a constant velocity of $$\frac{4}{\pi}$$.