The position of an object moving vertically along a line is given by the function s(t)=−4.9t2+30t+20s\left(t\right)=$$-4.9t^2+30t+20s(t$$)=−4.9t2+30t+20. Find the average velocity of the object over the following intervals.[0,h]\left\lbrack0,h\right\rbrack[0,h], where h\u003e0h\gt{0}h\u003e0 is a real number

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

Problem 14d

Chapter 2, Problem 14d

The position of an object moving vertically along a line is given by the function $s$\left$(t$\right$)=-4.9t^2+30t+20$ . Find the average velocity of the object over the following intervals.
$\left\[\lbrack\)0,h$\right\]\rbrack$ , where $h\(\gt{0}$$ is a real number

Verified step by step guidance

Identify the formula for average velocity over an interval [a, b], which is given by the change in position divided by the change in time: $ v_{avg} = \frac{s(b) - s(a)}{b - a} $.

In this problem, the interval is [0, h], so we need to find the average velocity over this interval. Set $ a = 0 $ and $ b = h $.

Substitute the values into the average velocity formula: $ v_{avg} = \frac{s(h) - s(0)}{h - 0} $.

Calculate $ s(h) $ by substituting $ t = h $ into the position function: $ s(h) = -4.9h^2 + 30h + 20 $.

Calculate $ s(0) $ by substituting $ t = 0 $ into the position function: $ s(0) = -4.9(0)^2 + 30(0) + 20 = 20 $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Position Function

The position function describes the location of an object at any given time, represented mathematically as s(t). In this case, the function s(t) = -4.9t² + 30t + 20 models the vertical motion of an object under the influence of gravity, where t is time in seconds. Understanding this function is crucial for analyzing the object's motion and calculating its velocity.

Average Velocity

Average velocity is defined as the change in position over the change in time, calculated using the formula (s(b) - s(a)) / (b - a) for an interval [a, b]. In this context, to find the average velocity over the interval [0, h], one would evaluate the position function at the endpoints and apply this formula. This concept is essential for understanding how the object's speed changes over time.

Intervals and Limits

Intervals in calculus refer to the range of values over which a function is analyzed. In this question, the interval [0, h] indicates that we are examining the object's motion from time t = 0 to t = h, where h is a positive real number. Understanding how to work with intervals is important for evaluating functions and determining properties like average velocity over specific time frames.

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