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.
$\left\lbrack0,3\right\rbrack$

Verified step by step guidance

1. To find the average velocity of the object over the interval [0, 3], we need to use the formula for average velocity: \( v_{avg} = \frac{s(t_2) - s(t_1)}{t_2 - t_1} \), where \( t_1 = 0 \) and \( t_2 = 3 \).
2. First, calculate \( s(t_1) = s(0) \) by substituting \( t = 0 \) into the position function \( s(t) = -4.9t^2 + 30t + 20 \). This will give us the initial position of the object.
3. Next, calculate \( s(t_2) = s(3) \) by substituting \( t = 3 \) into the position function. This will give us the position of the object at \( t = 3 \).
4. Subtract \( s(0) \) from \( s(3) \) to find the change in position, \( \Delta s = s(3) - s(0) \).
5. Finally, divide \( \Delta s \) by the change in time, \( \Delta t = 3 - 0 \), to find the average velocity: \( v_{avg} = \frac{\Delta s}{\Delta t} \).

Key Concepts

Position Function

The position function describes the location of an object at any given time. In this case, the function s(t) = -4.9t² + 30t + 20 represents the vertical position of an object in motion, where 't' is time in seconds. Understanding this function is crucial for analyzing the object's movement and calculating other related quantities like velocity.

Average Velocity

Average velocity is defined as the change in position divided by the change in time over a specified interval. Mathematically, it is calculated as (s(b) - s(a)) / (b - a), where [a, b] is the interval of interest. This concept is essential for determining how fast the object is moving on average during the given time period.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. The notation [0, 3] indicates that the interval includes both endpoints, meaning time 't' can take any value from 0 to 3 seconds, inclusive. Understanding interval notation is important for correctly interpreting the time frame over which the average velocity is calculated.