The position of an object moving vertically along a line is given by the function $s\left(t\right)=-16t^2+128t$. Find the average velocity of the object over the following intervals.
$\left\lbrack1,4\right\rbrack$

Verified step by step guidance

1. 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: \( v_{avg} = \frac{s(b) - s(a)}{b - a} \).
2. Substitute the given interval [1, 4] into the formula, where \( a = 1 \) and \( b = 4 \).
3. Calculate the position at \( t = 4 \) using the function \( s(t) = -16t^2 + 128t \). Substitute \( t = 4 \) into the equation to find \( s(4) \).
4. Calculate the position at \( t = 1 \) using the same function. Substitute \( t = 1 \) into the equation to find \( s(1) \).
5. Substitute \( s(4) \) and \( s(1) \) into the average velocity formula: \( v_{avg} = \frac{s(4) - s(1)}{4 - 1} \), and simplify the expression to find the average velocity over the interval [1, 4].

Key Concepts

Position Function

The position function describes the location of an object at any given time. In this case, the function s(t) = -16t² + 128t represents the vertical position of an object in motion, where 't' is time. 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. It can be calculated using the formula: Average Velocity = (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 between two points in time.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. The interval [1, 4] indicates that the values include both endpoints, 1 and 4, and all numbers in between. Understanding interval notation is important for correctly interpreting the time intervals over which the average velocity is to be calculated.