One model that is used for the interactions between​ animals, including fish in a​ school, is that the fish have an energy of interaction that is given below by a Morse potential. The fish will attract or repel each other until they reach a distance that minimizes the function​ V(r). The coefficients A and a are positive numbers. Complete parts​ (a) through​ (c).
Upper V left parenthesis r right parenthesis equals e Superscript negative r Baseline minus Upper A e Superscript negative ar Baseline comma r greater than 0
Question content area bottom
Part 1
​(a) Assume initially that a​ = one half
 and A​ = 1, what is the behavior of​ V(r) as rright arrow​0?
As rright arrow​0, ​V(r)right arrow
  
0
Part 2
What is the behavior of​ V(r) as rright arrowinfinity​?
As rright arrowinfinity​, ​V(r)right arrow
  
0
Part 3
​(b) Find the value of r that minimizes​ V(r).
requals
  
enter your response here

We are analyzing the given Morse potential function: 

\[
V(r) = e^{-r} - A e^{-ar}, \quad r > 0
\]

with parameters \(A > 0\) and \(a > 0\). Below is the detailed breakdown of the problem. 

---

### Part (a)

#### **Behavior as \(r \to 0\):**

Substitute \(r \to 0\) into the expression for \(V(r)\):  
\[
V(r) = e^{-r} - A e^{-ar}.
\]
At \(r \to 0\), \(e^{-r} \to 1\) and \(e^{-ar} \to 1\), so:  
\[
V(r) \to 1 - A.
\]
For \(A = 1\), this becomes:  
\[
V(r) \to 1 - 1 = 0.
\]

Thus, as \(r \to 0\), \(V(r) \to 0\).

---

#### **Behavior as \(r \to \infty\):**

As \(r \to \infty\), both exponential terms \(e^{-r}\) and \(e^{-ar}\) approach 0 because the exponential decay dominates:  
\[
V(r) \to 0 - 0 = 0.
\]

Thus, as \(r \to \infty\), \(V(r) \to 0\).

---

### Part (b): **Find the value of \(r\) that minimizes \(V(r)\):**

To find the minimum, take the derivative of \(V(r)\) with respect to \(r\) and set it to 0:  
\[
\frac{dV}{dr} = -e^{-r} + aA e^{-ar}.
\]
Set \(\frac{dV}{dr} = 0\):  
\[
-e^{-r} + aA e^{-ar} = 0.
\]
Rearranging gives:  
\[
e^{-r} = aA e^{-ar}.
\]
Divide through by \(e^{-r}\):  
\[
1 = aA e^{-(a-1)r}.
\]
Take the natural logarithm:  
\[
0 = \ln(aA) - (a-1)r.
\]
Solve for \(r\):  
\[
r = \frac{\ln(aA)}{a-1}.
\]

For \(a = \frac{1}{2}\) and \(A = 1\), substitute these values:  
\[
r = \frac{\ln(1 \cdot \frac{1}{2})}{\frac{1}{2} - 1} = \frac{\ln(0.5)}{-\frac{1}{2}}.
\]
\(\ln(0.5) = -\ln(2)\), so:  
\[
r = \frac{-\ln(2)}{-\frac{1}{2}} = 2 \ln(2).
\]

Thus, the minimum value occurs at:  
\[
r = 2 \ln(2).
\]

---

### Summary of Results:

1. As \(r \to 0\), \(V(r) \to 0\).
2. As \(r \to \infty\), \(V(r) \to 0\).
3. The value of \(r\) that minimizes \(V(r)\) is \(r = 2 \ln(2)\).

---

### Follow-Up Questions:

1. What happens to \(V(r)\) if \(A > 1\) instead of \(A = 1\)?
2. How does the minimum \(r\) change if \(a\) increases or decreases?
3. What is the physical interpretation of the minimum distance \(r = 2 \ln(2)\)?
4. How does the Morse potential change qualitatively as \(a \to 1\)?
5. Can you verify the result numerically by plotting \(V(r)\) for given \(a\) and \(A\)?

### Tip:
When dealing with exponential equations like this, using logarithms can simplify the process of isolating variables.

One model that is used for the interactions between animals, including fish in a school, is that the fish have an energy of interaction that is given below by a Morse potential. The fish will attract or repel each other until they reach a distance that minimizes the function V(r). The coefficients A and a are positive numbers. Complete parts (a) through (c). V(r) = e^(-r) - A * e^(-ar), r > 0. Part (a): Assume initially that a = 1/2 and A = 1, what is the behavior of V(r) as r → 0? As r → 0, V(r) → 0. Part (2): What is the behavior of V(r) as r → ∞? As r → ∞, V(r) → 0. Part (b): Find the value of r that minimizes V(r).

Explore the behavior of the Morse potential V(r) = e^(-r) - A * e^(-ar) with detailed analysis as r → 0 and r → ∞. Learn how to find the distance r that minimizes the function using calculus and logarithmic techniques. Suitable for advanced students in physics and mathematics.

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