Problem 7 - Calculus

Question

At time \(t\) in seconds, a particle's distance \(s(t),\) in \(\mathrm{mi}-\) crometers \((\mu \mathrm{m}),\) from a point is given by \(s(t)=e^{t}-1\) What is the average velocity of the particle from \(t=2\) to \(t=4 ?\)

Step-by-Step Solution

Answer

The average velocity is \(\frac{e^4 - e^2}{2}\).

Step 1: Understand the Average Velocity Formula

The average velocity of a particle over an interval is given by the change in position divided by the change in time. Mathematically, it can be expressed as: \[ v_{avg} = \frac{s(t_2) - s(t_1)}{t_2 - t_1} \]where \(t_1\) and \(t_2\) are the start and end times of the interval, respectively.

Step 2: Determine the Positions at Specific Times

Substitute the given times into the position function to find the positions:- Compute \(s(2) = e^2 - 1\).- Compute \(s(4) = e^4 - 1\).

Step 3: Calculate the Change in Position

Find the change in position by subtracting the initial position from the final position:\[\Delta s = s(4) - s(2) = (e^4 - 1) - (e^2 - 1)\]Simplify the expression to \(e^4 - e^2\).

Step 4: Calculate the Change in Time

Since \(t_2 = 4\) and \(t_1 = 2\), compute the change in time:\[\Delta t = t_2 - t_1 = 4 - 2 = 2\]

Step 5: Compute the Average Velocity

Use the average velocity formula:\[v_{avg} = \frac{e^4 - e^2}{2}\]This gives the average velocity of the particle between \(t=2\) and \(t=4\).

Key Concepts

Change in Position

Change in position is a fundamental concept in physics and mathematics, especially when discussing motion. It refers to the difference in the particle's position at two distinct points in time. In the context of our problem, the position of the particle is defined by the function \(s(t) = e^{t} - 1\). This function provides us with the particle's location at any given time \(t\).
To find how much the particle has moved between two times, \(t_1\) and \(t_2\), we use the formula \(\Delta s = s(t_2) - s(t_1)\). Here, \(s(t_1)\) is the initial position and \(s(t_2)\) is the final position. For our exercise:

• Compute \(s(2) = e^2 - 1\) to find the particle's initial position.
• Compute \(s(4) = e^4 - 1\) to find the particle's final position.

This shows us that the change in position \(\Delta s\) is \(e^4 - e^2\).
Understanding this concept is critical for calculating the average velocity over a time interval, as it represents total displacement.

Change in Time

Change in time is equally important when analyzing motion. It refers to the duration between two events or points in time. Often denoted by \(\Delta t\), this concept is instrumental in determining velocities or speeds. In our exercise, we're looking at the time interval from \(t=2\) to \(t=4\).
To calculate the change in time, subtract the initial time from the final time:

• \(\Delta t = t_2 - t_1\)

Substituting our specific times gives:

• \(\Delta t = 4 - 2 = 2\), seconds.

This value is essential in the average velocity calculation because it tells us over what period the position change occurs. It ensures we compare the position change proportional to the time elapsed.

Exponential Functions

Exponential functions are a key mathematical concept with vast applications, such as modeling growth behaviors. They can be represented as \(f(t) = a^{t}\), with \(a\) as a constant. In our problem, the function \(s(t) = e^{t} - 1\) describes the position of a particle over time.
Exponential functions have unique properties:

• They grow rapidly, which is evident in nature, finance, and other fields.
• The base \(e\), known as Euler's Number, is approximately 2.718 and is foundational in continuous growth calculations.

In this exercise, the function helps us determine the position of the particle precisely at any point in time \(t\).
By inserting the times of interest into the function, we can determine where the particle is located and use these positions to calculate other quantities, such as average velocity. The rapid growth nature of the exponential function lets us see how changes in position can increase more significantly over larger intervals.