Motion Along a Coordinate Line

...

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

Accepted Answer

Welcome back, everyone. A car moves along a straight path in its position as in kilometers at any time C in hours is given by S equals 100 divided by T + 2 for the time interval between 0 and 3 inclusive. When, if ever, does the car change its direction of motion? In this problem, we're given the position function S equals 100 divided by T + 2. If the car changes its direction of motion, this means that velocity must change its sign from positive to negative or from negative to positive. So what we want to do is simply evaluate the velocity function, which is the first derivative of the position function. This means that we want to differentiate 100 divided by T + 2. And what we're going to do is simply differentiate 100 multiplied by T + 2, raise to the power of -1. So from here, we can apply the power rule, which gives us 100 multiplied by T + 2, raise the power of -1 minus 1, which is -2, multiplied by -1, the original exponent. And then according to the chain rule, because we have a composite function, we also have to multiply by the derivative of T + 2, which is basically equal to 1, right? The derivative of T is 1, and the derivative of 2 is 0. So now, if we simplify the result, we get 100 divided by T + 2 squad. Now, let's remember that velocity must change its sign if the car changes its direction of motion. So what we want to do is essentially set velocity equal to 0. To find when there is a potential change in sign from negative to positive or from positive to negative. Setting 1-100 divided by. T + 2 squared equal to 0, we can say that this equation has no solutions. Because this is a rational expression. So if we want to make it equal to 0, then our numerator -100 must be equal to 0, and our denominator T + 22 must not be equal to 0, right? We want to satisfy these conditions, and -100 is never equal to 0, right? It's independent of X and therefore we can say that the car never changes. Direction. That will be our final answer and thank you for watching.

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

Key Concepts

Derivative and Velocity

Critical Points

Interval Analysis

Watch next