Textbook Question
{Use of Tech} Decreasing velocity A projectile is fired upward, and its velocity in m/s is given by v(t) = 200e^−t/10, for t≥0.
a. Graph the velocity function, for t≥0.
4. Applications of Derivatives
Motion Analysis
3:01 minutesProblem 3.4.6c
Textbook Question
Motion Along a Coordinate Line
Exercises 1–6 give the positions s = f(t) of a body moving on a coordinate line, with s in meters and t in seconds.
c. When, if ever, during the interval does the body change direction?
s = 25/(t + 5), −4 ≤ t ≤ 0
Verified step by step guidance1
First, understand that the body changes direction when its velocity changes sign. Velocity is the derivative of the position function with respect to time, v(t) = f'(t).
Calculate the derivative of the position function s = \frac{25}{t + 5}. Use the quotient rule for derivatives, which states that if you have a function \frac{u}{v}, its derivative is \frac{u'v - uv'}{v^2}.
Apply the quotient rule: Let u = 25 and v = t + 5. Then, u' = 0 and v' = 1. Substitute these into the quotient rule formula to find v(t).
Simplify the expression obtained from the quotient rule to find the velocity function v(t). This will give you v(t) = \frac{-25}{(t + 5)^2}.
Determine when the velocity changes sign by analyzing the expression \frac{-25}{(t + 5)^2}. Since the numerator is negative and the denominator is always positive for the given interval, the velocity does not change sign, indicating the body does not change direction in the interval −4 ≤ t ≤ 0.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative and Velocity
The derivative of a position function s = f(t) with respect to time t gives the velocity of the body. Velocity indicates the rate of change of position and its sign (positive or negative) shows the direction of motion. To determine when the body changes direction, we need to find when the velocity changes sign, which occurs when the derivative equals zero or is undefined.
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Critical Points
Critical points occur where the derivative of a function is zero or undefined. These points are essential in analyzing the behavior of the function, such as identifying potential changes in direction for motion problems. By evaluating the derivative of s = 25/(t + 5), we can find critical points within the interval −4 ≤ t ≤ 0 to determine when the body might change direction.
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Interval Analysis
Interval analysis involves examining the behavior of a function over a specific range of values. For motion problems, this means checking the sign of the velocity before and after critical points within the given interval. By analyzing the intervals around critical points, we can confirm if and when the body changes direction by observing changes in the sign of the velocity.
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