6. Distance and speed
Relative speed for catch-up and meeting problems. Watch time offsets between travelers.
Core ideas
- Toward each other: speeds add. Same direction: subtract, and gap closes at the speed difference.
- Catch-up time = head start distance / speed difference.
- Different start times: give the early traveler a head start distance, then it is a standard catch-up.
- Average speed = total distance / total time, never the average of the speeds (unless times are equal).
Worked example 1
A truck leaves at noon at 40 mph. A car leaves the same point at 1 pm at 60 mph on the same road. When does the car catch the truck?
Show solution
By 1 pm the truck has a head start of 40 * 1 = 40 miles. The gap closes at 60 - 40 = 20 mph. Catch-up time = 40 / 20 = 2 hours after the car starts, so at 3 pm. At that moment both are 60 * 2 = 120 miles out, and the truck confirms: 40 * 3 = 120.
Worked example 2
Two cyclists start 63 miles apart and ride toward each other, one at 12 mph and one at 9 mph. How far has the faster cyclist ridden when they meet?
Show solution
Closing speed = 12 + 9 = 21 mph, so they meet after 63 / 21 = 3 hours. The faster cyclist covers 12 * 3 = 36 miles. Shortcut: the meeting point splits the distance in the ratio of the speeds, 12 : 9 = 4 : 3, and 4/7 of 63 = 36.
Practice set
Question 1
A driver travels for 2 hours at 50 miles per hour and then for 3 hours at 60 miles per hour. What is the driver's average speed, in miles per hour, for the entire trip?
Average speed is total distance over total time, not the average of the two speeds.
Distance = 2*50 + 3*60 = 100 + 180 = 280 miles in 5 hours. Average speed = 280/5 = 56 mph. Choice B (55) is the unweighted average of 50 and 60.
Question 2
A commuter drives from home to work at 60 miles per hour and returns along the same route at 40 miles per hour. What is the commuter's average speed, in miles per hour, for the round trip?
Equal distances at different speeds means the slower leg takes more time, so pick a convenient distance and compute total distance over total time.
Let each leg be 120 miles. Going takes 120/60 = 2 hours; returning takes 120/40 = 3 hours. Average speed = 240/5 = 48 mph. Choice C (50) is the trap of averaging 60 and 40 directly.
Question 3
A car travels 240 miles at a constant speed of 60 miles per hour. How many hours does the trip take?
Time equals distance divided by speed.
Time = 240/60 = 4 hours. The wrong choices come from arithmetic slips such as 240/80 = 3 or 240/48 = 5.
Question 4
A hiker leaves a trailhead at 10:00 am walking at 4 miles per hour. At 11:30 am, a cyclist leaves the same trailhead along the same trail at 12 miles per hour. How many miles from the trailhead does the cyclist catch the hiker?
Convert the 90 minute head start into a distance gap, then close it at the speed difference.
In 1.5 hours the hiker covers 4*1.5 = 6 miles. The cyclist closes the gap at 12 - 4 = 8 miles per hour, so catching up takes 6/8 = 0.75 hours. The cyclist has then ridden 12*0.75 = 9 miles from the trailhead. Choice A (6) is just the head start gap, and choice E (12) treats the catch-up time as a full hour.