Question

Suppose \(s(t)\) is the position of an object moving along a line at time \(t \geq 0 .\) What is the average velocity between the times \(t=a\) and \(t=b ?\)

Short answer

Question: Given the position function \(s(t)\) for an object moving along a line for time \(t \ge 0\), find the average velocity between the times \(t=a\) and \(t=b\). Answer: The average velocity between the times \(t=a\) and \(t=b\) is given by the formula \(V_{avg} = \dfrac{s(b) - s(a)}{b - a}\).

Step-by-step solution

Determine the position of the object at time t=a

Write down the position function for the object at time t=a as \(s(a)\), which represents the position at time t=a.

Determine the position of the object at time t=b

Write down the position function for the object at time t=b as \(s(b)\), which represents the position at time t=b.

Calculate the difference in position

Subtract the position at time t=a from the position at time t=b to find the difference in position: \(\Delta s = s(b) - s(a)\)

Calculate the time interval

Subtract the time t=a from the time t=b to find the time interval: \(\Delta t = b - a\)

Compute the average velocity

Divide the difference in position by the time interval to find the average velocity: \(V_{avg} = \dfrac{\Delta s}{\Delta t} = \dfrac{s(b) - s(a)}{b - a}\)

Key concepts

Understanding the Position Function

The position function is a mathematical representation used to describe the location of an object at any given time when it is moving along a path. It is typically denoted by \(s(t)\), where \(t\) represents time. This function helps us track the movement of the object over time, making it straightforward to determine where the object is at specific moments.
The position function is like a time travel tool in mathematics; it gives us the ability to "see" where the object is at any snapshot in time without actually watching it move.
In practical applications, the position function can be derived from observations or differential equations. It serves as a foundation for analyzing further aspects of motion, such as velocity and acceleration.

Calculating the Difference in Position

The difference in position is a crucial step in determining average velocity. It measures the change in the object's location between two specific times. Mathematically speaking, this can be expressed as \(\Delta s = s(b) - s(a)\), where \(s(a)\) is the position at time \(a\) and \(s(b)\) is the position at time \(b\).
This difference tells us how far, and in which direction, the object has moved over the time interval. It is vital because, without it, we wouldn't know how much total distance the object has covered.
By understanding the difference in position, we can start piecing together the story of the object's journey from one point to another.

Defining the Time Interval

The time interval refers to the duration between two distinct points in time, over which we observe an object's motion. We calculate it by subtracting the initial time \(t=a\) from the final time \(t=b\), resulting in \(\Delta t = b - a\).
This interval is important as it establishes the period over which calculations, such as average velocity, are made.
By clarifying the time interval, you ensure that all subsequent calculations reflect the precise time frame over which the object's movement is assessed. Without a clear time interval, it would be challenging to draw meaningful conclusions from the data on the object's movement.

Steps to Calculate Average Velocity

Average velocity is a key concept in motion analysis. It describes the overall rate of change in position over a specified time period. To find average velocity, we use the formula \(V_{avg} = \dfrac{\Delta s}{\Delta t} = \dfrac{s(b) - s(a)}{b - a}\).
This formula divides the change in position by the time interval, providing a measure of how fast and in which direction the object is moving on average over the period in question.
It's important to remember that average velocity can be positive or negative, depending on whether the object moves forward or backward. Additionally, average velocity differs from instantaneous velocity, which measures speed at a specific instant. Understanding these differences is crucial for analyzing and interpreting objects' motion in more detail.