Car Overtaking Truck: Relative Speed & Catch-Up Problem

Visualizing the Situation

Before diving into the algebra, let's picture what's happening on this highway.

The truck gets a 13-minute head start before the car even enters. Once both vehicles are on the highway, the car needs to make up that initial gap while both vehicles continue moving forward.

What This Problem Teaches

• Converting between time units (minutes to hours) for consistent calculations
• Understanding relative speed as the rate at which one object gains on another
• Setting up distance equations when objects start at different times
• Recognizing that "catch-up" problems involve closing an initial gap
• Interpreting fractional answers in practical time contexts

Solution: Method 1 — The Gap-Closing Approach

The most intuitive way to think about this problem is that the car needs to close the gap created by the truck's 13-minute head start.

Step 1 — Convert the head start time to hours

Since our speeds are in mph, we need consistent time units. Convert 13 minutes to hours:

Step 2 — Calculate the truck's head start distance

While the car is still off the highway, the truck travels for 13 minutes at 60 mph:

Step 3 — Find the relative closing speed

Once the car enters the highway, both vehicles move in the same direction. The car closes the gap at the difference between their speeds:

Step 4 — Calculate time to close the gap

Now we can find how long it takes the car to close the 13-mile gap at a relative speed of 7 mph:

Step 5 — Convert to mixed time format

To express this in a more practical format:

Solution: Method 2 — Equating Distances from Start

An alternative approach is to set up equations based on how far each vehicle has traveled from the original entry point.

Step 1 — Define the variable

Let t = hours the car drives until it catches the truck

Step 2 — Express the truck's total travel time

The truck has been driving for 13 minutes longer than the car:

Step 3 — Set up distance equations

When the car catches the truck, both vehicles are the same distance from the entry point:

Step 4 — Solve for t

Isolate the variable to find the catch-up time:

This confirms our answer from Method 1: the car will catch the truck after driving for 13/7 hours.

Verification

Let's check our answer by calculating where both vehicles are when the car catches the truck.

Car's position after 13/7 hours:

Truck's position after 13/7 hours:
Remember, the truck has been driving for 13/7 + 13/60 hours total.

✓ Both vehicles are at the same position, confirming our answer is correct.

Common Pitfalls

The Underlying Pattern

This problem follows the standard relative motion formula for same-direction pursuit:

This pattern appears whenever a faster object pursues a slower object that had a head start. The key insight is that once both objects are moving, the gap closes at the difference in their speeds, not their individual speeds.

Real Applications

• Military and law enforcement: Calculating intercept times for pursuing vehicles or aircraft
• Logistics and delivery: Determining when a faster truck can overtake a slower convoy
• Sports analytics: Analyzing when a trailing runner can catch the leader in distance races
• Network communications: Computing when faster data packets catch up to slower ones in network traffic analysis

What If?