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Part A

1. Question: What is the current state represented by $x_A$?

   • Given: $x_A = 15$

2. Question: Write down the equation of motion for $x_A$ in terms of $v_A$ and time $t$.

   • Equation: $v_A = \frac{dx_A}{dt}$

3. Question: If the initial condition is $x_A(0) = 0$, solve the equation of motion to find $x_A(t)$.

   • Solution: $x_A(t) = \int_0^t v_A \, dt$

Part B

1. Question: What is the state represented by $x_B$?

   • Given: $x_B = 30$

2. Question: Write down the equation of motion for $x_B$ in terms of $v_B$ and time $t$.

   • Equation: $v_B = \frac{dx_B}{dt}$

3. Question: With the same initial condition $x_B(0) = 0$, solve the equation of motion to find $x_B(t)$.

   • Solution: $x_B(t) = \int_0^t v_B \, dt$

4. Question: What is the distance between the two states represented by $x_A(t)$ and $x_B(t)$?

   • Distance Formula: $\text{Distance} = |x_B(t) - x_A(t)|$

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This is a breakdown of the questions and the approach to solving them. The exact solutions require additional context, such as the function for $v_A$ and $v_B$, or additional given values, which might be part of your assignment.

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Related Questions:

1. How do you integrate a function of time?
2. What are the general steps for solving a first-order differential equation?
3. How do you interpret the distance between two states in a physical system?
4. What does the graph of $x_A(t)$ vs. $t$ typically look like?
5. How would you find $v_A(t)$ if $x_A(t)$ was given?
6. What initial conditions are necessary to solve a differential equation?
7. Can $x_A(t)$ and $x_B(t)$ ever be the same? Under what conditions?
8. How can the relative motion of $x_A$ and $x_B$ be analyzed?

Tip:

When solving differential equations, always ensure that your initial conditions are applied correctly to find the particular solution.