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This problem involves two airplanes traveling at different speeds, and we are asked to find when the second airplane overtakes the first one.

Let’s define the variables:
- The first airplane leaves at 2:00 PM and flies north at 250 miles per hour.
- The second airplane leaves 30 minutes later, at 2:30 PM, and flies north at 280 miles per hour.

To solve this problem, we need to determine when the distances traveled by the two airplanes are equal.

### Step 1: Express the distance traveled by each airplane

The distance \( d \) traveled by an airplane is given by the formula:
\[
d = \text{speed} \times \text{time}
\]

Let \( t \) be the time (in hours) after 2:00 PM when the second airplane catches up with the first airplane.

- **First airplane**: It has been traveling for \( t \) hours at 250 miles per hour, so the distance it travels is:
\[
d_1 = 250 \times t
\]

- **Second airplane**: It starts 30 minutes later, so it has been traveling for \( t - \frac{1}{2} \) hours at 280 miles per hour. The distance it travels is:
\[
d_2 = 280 \times \left(t - \frac{1}{2}\right)
\]

### Step 2: Set up the equation

Since the second airplane overtakes the first airplane when they have traveled the same distance, we set \( d_1 = d_2 \):
\[
250t = 280 \left(t - \frac{1}{2}\right)
\]

### Step 3: Solve the equation

Let's solve this equation step by step.

\[
250t = 280 \left(t - 0.5\right)
\]
Distribute 280:
\[
250t = 280t - 140
\]
Subtract \( 280t \) from both sides:
\[
250t - 280t = -140
\]
\[
-30t = -140
\]
Solve for \( t \):
\[
t = \frac{-140}{-30} = \frac{14}{3} \approx 4.67 \text{ hours}
\]

### Step 4: Interpret the result

Since \( t = \frac{14}{3} \) hours, this means the second airplane will overtake the first airplane approximately 4 hours and 40 minutes after 2:00 PM. 

Thus, the second airplane overtakes the first at around **6:40 PM**.

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Would you like any further clarification or details on this?

Here are 5 related questions for further practice:
1. If the second airplane had a speed of 300 mph, when would it overtake the first airplane?
2. What if the first airplane started 1 hour earlier instead of 30 minutes?
3. How would the solution change if both airplanes flew at the same speed?
4. What if the second airplane left 45 minutes later and traveled at 270 mph?
5. What is the total distance each airplane travels when the second overtakes the first?

**Tip**: To solve relative motion problems, always set up equations that equate distances and carefully track when each object starts.

An airplane leaves Atlanta at 2 PM and flies north at 250 miles per hour. A second airplane leaves Atlanta 30 minutes later and flies north at 280 miles per hour. At what time will the second airplane overtake the first?

Learn how to solve a relative motion problem where one airplane flying at 280 mph overtakes another flying at 250 mph. Detailed step-by-step solution provided, using the formula Distance = Speed × Time. Suitable for Grades 9-12.

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