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The graph in the image represents the positions (in meters) of both you and your sister as a function of time (in seconds). To determine when you will catch up with your sister, we need to create an equation based on the position versus time graph.

### Step-by-step analysis:
1. **Equation for your sister:**
   - The graph shows that your sister’s line (blue) starts at 5 meters when time is 0 seconds. The slope of this line is constant, meaning her velocity is constant. To determine the slope, observe that from 0 seconds to 4 seconds, the position increases from 5 meters to 15 meters.
   - Slope \( m \) of your sister’s line:
     \[
     m = \frac{\Delta \text{Position}}{\Delta \text{Time}} = \frac{15 - 5}{4 - 0} = \frac{10}{4} = 2.5 \, \text{m/s}
     \]
   - The equation of her position \( P_s(t) \) over time is then:
     \[
     P_s(t) = 2.5t + 5
     \]
     where \( t \) is the time in seconds.

2. **Equation for you:**
   - The graph shows that your line (red) starts at 0 meters when time is 0 seconds. Like your sister, your line is also straight, indicating constant velocity. From 0 seconds to 4 seconds, your position increases from 0 meters to 18 meters.
   - Slope \( m \) of your line:
     \[
     m = \frac{\Delta \text{Position}}{\Delta \text{Time}} = \frac{18 - 0}{4 - 0} = 4.5 \, \text{m/s}
     \]
   - The equation of your position \( P_y(t) \) over time is then:
     \[
     P_y(t) = 4.5t
     \]

3. **Finding when you catch up:**
   To find when you catch up with your sister, set the two position equations equal to each other:
   \[
   4.5t = 2.5t + 5
   \]
   Solving for \( t \):
   \[
   4.5t - 2.5t = 5
   \]
   \[
   2t = 5
   \]
   \[
   t = 2.5 \, \text{seconds}
   \]

### Explanation:
You catch up to your sister at **2.5 seconds**. This result comes from setting your position equation equal to your sister’s position equation and solving for \( t \). At this time, both you and your sister are at the same position, meaning you have caught up.

Would you like more details or have any further questions?

Here are 5 related questions:
1. How would the result change if your velocity were slower than your sister's?
2. What would happen if you started ahead of your sister instead of behind?
3. Can you derive an equation for the distance between you and your sister over time?
4. What is the total distance you traveled when you catch up with your sister?
5. How long would it take if you ran at 6 m/s?

**Tip:** When dealing with motion problems, always check the slope of position vs. time graphs, as it represents velocity!

You run to catch up to your sister. The position vs. time of you and your sister are graphed below. Write out the equation you would solve to find when and where you catch your sister.

Learn how to solve a linear motion problem by analyzing position vs. time graphs. This problem involves finding when two moving objects meet by writing equations based on their velocities. Ideal for Grades 6-9 students studying algebra and motion.

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