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 a position function, \(s(t)=4t^2+5t-8\), and a time interval, \(t=[1,3]\), calculate the average velocity of the object over that time period. Solution: Step 1: Calculate change in position. \(\Delta s = s(3) - s(1) = (4(3)^2 + 5(3) - 8) - (4(1)^2 + 5(1) - 8)\) \(\Delta s = (36 + 15 - 8) - (4 + 5 - 8)\) \(\Delta s = 43 - 1 = 42\) Step 2: Calculate change in time. \(\Delta t = 3 - 1 = 2\) Step 3: Calculate the average velocity. Average velocity \(= \frac{\Delta s}{\Delta t} = \frac{42}{2} = 21\) Thus, the average velocity of the object over the given time interval is 21 units per second.

Step-by-step solution

Calculate change in position

Determine the change in position by evaluating the position function, \(s(t)\), at both \(t=a\) and \(t=b\). Subtract the initial position, \(s(a)\), from the final position, \(s(b)\), to calculate the change in position. This can be represented as: \(\Delta s = s(b) - s(a)\).

Calculate change in time

Determine the change in time by subtracting the initial time, \(t=a\), from the final time, \(t=b\). This can be represented as: \(\Delta t = b - a\).

Calculate the average velocity

To find the average velocity, divide the change in position, \(\Delta s\), by the change in time, \(\Delta t\). Use the values found in Steps 1 and 2. The formula for average velocity is: Average velocity \(= \frac{\Delta s}{\Delta t}\). Plugging the values found in Steps 1 and 2, we get: Average velocity \(= \frac{s(b) - s(a)}{b - a}\).

Key concepts

Position Function

A position function helps us understand where an object is located on a line at any given time. It is usually denoted as \( s(t) \), where \( s \) represents position and \( t \) represents time.
The position function provides a clear picture of how an object's location changes over time. For example:

• When you plug a specific time into the function \( s(t) \), it outputs the exact position of the object at that time.
• Mathematically, the position function is crucial for determining various other concepts, like velocity and acceleration.

Understanding the notion of a position function allows us to track motion and analyze movement patterns, which is vital in both everyday scenarios and complex computations.

Change in Position

Change in position reflects the difference in an object's location between two points in time. This change can be expressed mathematically using the position function \( s(t) \).
The formula to calculate change in position is:

• \( \Delta s = s(b) - s(a) \)

This represents the position at time \( b \) minus the position at time \( a \).
It's important to remember:

• Change in position tells us how far the object has moved, but not specifically the path it took to get there.
• It gives insight into whether the object is moving forward, backward, or staying stationary.

Accurately calculating this change is crucial for further analysis of motion, such as determining average velocity.

Calculus Concepts

Calculus provides us tools to explore dynamic processes like velocity and acceleration through concepts like derivatives. When working with position functions, calculus allows us to derive insightful information about an object's velocity.
To find average velocity, we use:

• The formula \( \frac{s(b) - s(a)}{b - a} \), which divides the change in position by the change in time.

Calculus helps us understand:

• How the position changes over time (via the derivative of the position function) to find instantaneous velocity at precise moments.
• How these instant changes accumulate over intervals to find average results.

It provides a deeper understanding of motion by linking the mathematical representations with physical phenomena, making it an essential tool in physics and engineering.