Two-Train Overtaking Problem: Solve for Speed

What This Problem Teaches

• Setting up equations when objects start at different times but travel the same distance
• Converting time differences into hours for distance calculations
• Understanding that "overtaking" means both objects have traveled identical distances from the starting point
• Recognizing how speed differences compensate for time disadvantages in catching up scenarios
• Building confidence with multi-step distance-rate-time problems involving relative motion

Solution: Method 1 — Equal Distance at Meeting Point

Step 1 — Set up the variable

Let r = speed of the first train (mph). Then the speed of the second train = r + 84 mph.

Step 2 — Calculate travel times until overtaking

Both trains are overtaken at 10:00 PM, so we need to find how long each train travels:

• First train: leaves at 4:00 PM, overtaken at 10:00 PM → travels for 6 hours
• Second train: leaves at 8:00 PM, overtaken at 10:00 PM → travels for 2 hours

Step 3 — Set up the distance equation

When the second train overtakes the first, they have traveled the same total distance from Boston. Using Distance = Speed × Time:

Step 4 — Solve for r

Expand and solve the equation:

Step 5 — Find both speeds

• First train speed: r = 42 mph
• Second train speed: r + 84 = 42 + 84 = 126 mph

Solution: Method 2 — Relative Speed and Gap Closing

Step 1 — Calculate the head start distance

The first train gets a 4-hour head start (4:00 PM to 8:00 PM). In this time, it travels 4r miles ahead of the second train.

Step 2 — Find the relative speed

The second train moves (r + 84) - r = 84 mph faster than the first. This is how quickly it closes the gap.

Step 3 — Set up the catch-up equation

The second train needs 2 hours to close a 4r mile gap at a relative speed of 84 mph:

This gives us the same result: first train at 42 mph, second train at 126 mph.

Verification

Let's check that both trains travel the same distance by 10:00 PM:

✓ Both trains travel exactly 252 miles from Boston when they meet at 10:00 PM.

Common Pitfalls

Does This Seem Reasonable?

Let's check if our answer makes sense in the real world:

• Speed range: 42 mph is reasonable for a local train, while 126 mph falls within high-speed rail territory
• Time advantage: The second train gets only 2 hours vs. 6 hours for the first — a 3:1 disadvantage. The speed advantage is 126:42 = 3:1, which exactly compensates
• Distance check: 252 miles in 6 hours is a sensible journey length for intercity rail

The mathematical relationship makes physical sense: when one train has triple the time, the other needs triple the speed to cover the same ground.

Recognizing This Problem in the Wild

Look for these key phrases that signal an overtaking problem:

• "leaves at different times" + "same direction" + "overtakes" or "catches up"
• "faster by [amount]" combined with a time delay
• Two speeds given as a difference rather than individual values
• A specific time when the faster object catches the slower one

Problem structure: Any scenario where a delayed but faster entity catches up to an earlier but slower one — whether trains, cars, runners, or even business growth rates.

The Underlying Pattern

All overtaking problems follow this general structure:

The key insight is that equal distance ≠ equal time. The object that starts later must have proportionally higher speed to compensate for less time.

Special case: If the speed difference and time difference are given, you can often solve using relative motion — the faster object closes the gap at a rate equal to the speed difference.

What If?