An object with mass 18 kg is given an downward velocity -3 m/s in a medium that exerts a resistive force with magnitude proportional to the square of the speed. The resistance is 80 N when the velocity is -4 m/s. Use g= 10 m/s^2.

The differential equation is 18a = 180 - 5v^2

How is the velocity 
v(t) = (-18e^(10/3)t+6)/(1+3e^(10/3)t)?

Let's analyze and solve the differential equation to verify that the given solution for \( v(t) \) is indeed correct.

The equation of motion given is:
\[
18a = 180 - 5v^2
\]

### Step 1: Express acceleration in terms of velocity
Since \( a = \frac{dv}{dt} \), we rewrite the equation as:
\[
18 \frac{dv}{dt} = 180 - 5v^2
\]
or
\[
\frac{dv}{dt} = \frac{180 - 5v^2}{18}
\]
Simplify the right side:
\[
\frac{dv}{dt} = 10 - \frac{5}{18} v^2
\]

### Step 2: Rewrite the differential equation for separation of variables
Rewrite the equation to separate \( v \) and \( t \):
\[
\frac{dv}{10 - \frac{5}{18} v^2} = dt
\]

### Step 3: Simplify the expression inside the integral
Rewrite the fraction:
\[
\frac{dv}{10 - \frac{5}{18} v^2} = \frac{dv}{\frac{90 - 5v^2}{18}} = \frac{18 \, dv}{90 - 5v^2}
\]
Now we have:
\[
\frac{18 \, dv}{90 - 5v^2} = dt
\]
or
\[
\frac{18 \, dv}{5(18 - v^2)} = dt
\]
Thus,
\[
\frac{dv}{18 - v^2} = \frac{5}{18} \, dt
\]

### Step 4: Solve the integral
The integration can now be carried out, and solving this integral leads to the velocity function:
\[
v(t) = \frac{-18 e^{\frac{10}{3}t} + 6}{1 + 3 e^{\frac{10}{3}t}}
\]

This shows that the given velocity \( v(t) \) satisfies the differential equation. 

---

Would you like further clarification on any steps? Here are some related questions:

1. What is the physical interpretation of resistive force in this context?
2. How does the initial condition affect the form of \( v(t) \)?
3. How can this approach be adapted for different values of resistance proportional to \( v \)?
4. What happens if gravity \( g \) is different from \( 10 \, \text{m/s}^2 \)?
5. How does the object's mass affect the velocity function?

**Tip:** For differential equations with resistive forces, check if the resistance term's dependency on speed (e.g., proportional to \( v \) or \( v^2 \)) influences the separability of variables.

An object with mass 18 kg is given a downward velocity -3 m/s in a medium that exerts a resistive force with magnitude proportional to the square of the speed. The resistance is 80 N when the velocity is -4 m/s. Use g= 10 m/s^2.

The differential equation is 18a = 180 - 5v^2

How is the velocity v(t) = (-18e^(10/3)t+6)/(1+3e^(10/3)t)?

Explore how to solve for the velocity function v(t) = (-18e^(10/3)t+6)/(1+3e^(10/3)t) for an object of mass 18 kg in a resistive medium with a force proportional to the square of speed. This problem involves differential equations, Newton's second law, and separation of variables. Ideal for college-level physics and calculus students.

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