To solve the problem, we'll break it down step by step.

Step 1: Determine Jim's travel time and distance before Joe started.

• Jim left at 9:30 AM and traveled at 50 km/h.
• Joe started 2 hours later, at 11:30 AM.

So, by the time Joe started, Jim had been driving for 2 hours.

Distance Jim traveled in 2 hours: $\text{Distance} = \text{Speed} \times \text{Time} = 50 \, \text{km/h} \times 2 \, \text{hours} = 100 \, \text{km}$

Step 2: Set up the equations for their travel times.

• Let $t$ be the time (in hours) it took Joe to catch up with Jim after Joe started.
• In that time, Joe traveled a distance of $v \times t$, where $v$ is Joe's speed.

Since Joe caught up with Jim after traveling 225 km, we have: $v \times t = 225 \, \text{km}$ $t = \frac{225}{v}$

Step 3: Express Jim's total travel time and distance.

Jim's total travel time until Joe caught up with him is $t + 2$ hours.

Since Jim was traveling at 50 km/h: $\text{Distance traveled by Jim} = 50 \times (t + 2) \, \text{km}$

Step 4: Set up the equation based on the condition that Joe caught up with Jim.

At the point Joe caught up with Jim, both had traveled the same distance: $\text{Distance Jim traveled} = \text{Distance Joe traveled}$ $50 \times (t + 2) = 225$

Step 5: Solve for $t$.

$50t + 100 = 225$ $50t = 125$ $t = \frac{125}{50} = 2.5 \, \text{hours}$

Step 6: Determine Joe's speed.

Using the earlier equation $t = \frac{225}{v}$: $2.5 = \frac{225}{v}$ $v = \frac{225}{2.5} = 90 \, \text{km/h}$

Final Answer:

Joe was traveling at a speed of 90 km/h.

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Would you like more details or have any questions?

Here are some related questions to further explore the topic:

1. If Joe had started only 1 hour after Jim, what would his required speed be to catch up after 225 km?
2. What would be the time difference if Joe had traveled at 80 km/h instead?
3. How long would it have taken Joe to catch up if he had traveled at 100 km/h?
4. If Jim increased his speed to 60 km/h after the first 2 hours, what would Joe's speed need to be to catch up after 225 km?
5. How far would Jim have traveled by the time Joe caught up if both started at the same time, but Joe at 90 km/h and Jim at 50 km/h?

Tip: Understanding how to set up equations from word problems is key to solving them efficiently. Start by identifying the variables and writing down what you know.