https://file.aiquickdraw.com/mathbot/file/1727002988125vzcsyzk2.jpg

The equation given is:

\[
s = \frac{mv_i}{k} \left(1 - e^{-\left(\frac{k}{m}\right)t} \right)
\]

To linearize this exponential equation, we need to express it in a linear form that can be plotted as a straight line, typically in the form \( y = mx + b \), where \( y \) and \( x \) represent variables, and \( m \) is the slope, and \( b \) the intercept.

### Step-by-Step Linearization:
1. Start by rearranging the equation in a more manageable form. Define:
   - \( s \) is the displacement.
   - \( t \) is the time.
   - \( m \) is the mass.
   - \( k \) is the constant.
   - \( v_i \) is the initial velocity.
   
   We focus on the exponential term \( e^{-\frac{k}{m}t} \).

2. Let’s isolate the exponential part:
   
   \[
   1 - \frac{s k}{mv_i} = e^{-\left(\frac{k}{m}\right)t}
   \]

3. Take the natural logarithm (ln) of both sides to remove the exponential:
   
   \[
   \ln\left(1 - \frac{s k}{mv_i}\right) = -\frac{k}{m}t
   \]

4. Now, this equation is linear with respect to \( t \). We can compare it to the linear form \( y = mx + b \):
   - \( y = \ln\left(1 - \frac{s k}{mv_i}\right) \)
   - \( x = t \)
   - The slope \( m = -\frac{k}{m} \)
   
This is the linearized form of the original equation, where you can plot \( \ln(1 - \frac{sk}{mv_i}) \) against \( t \) to obtain a straight line. The slope of this line will be \( -\frac{k}{m} \), which can give you the values of \( k \) and \( m \).

Would you like more details or assistance with any particular aspect of this process? Here are five follow-up questions to explore:

1. How do you interpret the slope \( -\frac{k}{m} \) in terms of the system's physical behavior?
2. What are the conditions under which this equation is valid?
3. How would you experimentally determine the values of \( k \) and \( m \) from this linearized equation?
4. Can you think of other examples where linearizing an exponential relationship is useful in physics?
5. What happens to the linear form if \( t \) becomes very large (i.e., as \( t \rightarrow \infty \))?

### Tip:
When linearizing an equation, always remember that taking logarithms is a common technique to handle exponential terms.

Explore how to linearize the exponential equation s(m) = (mv_i/k) * (1 - exp(-(k/m)t)). This guide covers the process of converting it into a linear form using logarithmic transformation, making it easier to analyze and plot. Suitable for undergraduate students studying physics or mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation