Many people get confused when a word problem includes the word "speed." Fortunately, the math is some of the easiest in all of physics: one division, that's all. The problem is not the math, but knowing which numbers to put in and how to do the more complicated versions of the problem.
The One Formula You Actually Need
All speed problems, regardless of how they're packaged, boil down to this simple core equation:
For example, if a cyclist rides 30 km in 2 hours, the speed is 30 ÷ 2, or 15 km/h. This is the entire calculation. The hard part of most speed problems is not in this formula, but in reading the problem carefully enough to identify the correct distance and time in the wording.
Mental Math Shortcuts for Speed
A calculator isn't always required. There are some habits that make speed math fast enough to be done in your head:
- Round to friendly numbers first: 118 miles in 1.9 hours is close enough to 120 miles in 2 hours to estimate that it's about 60 miles per hour.
- Use known benchmarks: 60 mph equals 1 mile per minute. If you're traveling 90 miles on the highway, where you might be driving a little faster than 60 mph, it will take just under an hour and a half.
- Break big numbers into easy chunks: 236 km in 4 hours is not an easy number to divide at first glance, but 240 km in 4 hours is 4 × 60 km/h, which makes it easy to calculate.
These shortcuts will not be used for actual test math, but they are incredibly useful for sanity checking an answer. If your calculator indicates that a car was traveling at 6 mph on the highway, then something is definitely wrong with the setup.
Finding Speed From Partial Information
Many real problems don't present distance and time directly to you. They must be dug out first.
For instance, if you left your house at 3:00 PM and arrived at a friend's house 45 miles away at 4:15 PM, how long did your trip take? The time elapsed isn't given as a clean number, you have to calculate it. From 3:00 to 4:15 is 1 hour and 15 minutes, or 1.25 hours. Speed is then 45 ÷ 1.25, which comes out to 36 mph.
This pattern occurs repeatedly: a start time and end time rather than a duration, or a distance in some odd units such as feet when the answer should be in miles per hour. The secret is not new math; it's turning what you're given into a clean distance and a clean time before dividing.
When Speed Isn't Constant
Real motion seldom has a constant speed throughout a journey. A car accelerates, decelerates, and accelerates again. The rate of change in velocity with time is called acceleration in physics, which is a different calculation from speed.
In most of the "how fast was it going" questions, however, you're not being asked about the changes along the way; you're being asked for the average. The average speed for a 240-mile trip that took 5 hours with a stop for lunch is 240 miles ÷ 5 hours, or 48 mph, even if the speedometer fluctuated during the trip.
Solving Problems With Two Moving Objects
This is where many students get confused as it seems like a completely different type of problem. It just provides an additional step before the regular formula.
Quick Reference: Converting Between Common Speed Units
Unit mix-ups cause more wrong answers than bad math does. Keep these conversions handy:
- Miles per hour to km/h: multiply by 1.609
- Km/h to m/s: divide by 3.6
- M/s to km/h: multiply by 3.6
- Km/h to mph: divide by 1.609
A speed of 100 km/h, for instance, works out to 100 ÷ 3.6, or about 27.8 m/s. Converting before you calculate, rather than after, saves a lot of second-guessing later in a problem.
A Few Worked Examples
A swimmer covers 200 meters in 2 minutes and 30 seconds.
A delivery van travels 18 km in 24 minutes.
A rocket climbs 12 km straight up in 40 seconds during launch.
Common Mistakes to Avoid
- Forgetting to convert minutes or seconds into hours (or the reverse) before dividing.
- Adding speeds for objects moving in the same direction instead of subtracting them.
- Mixing metric and imperial units in the same calculation without converting one of them first.
Catching these three habits early clears up the vast majority of speed problems, whether they show up in a physics class or in everyday planning.
Frequently Asked Questions
What is the basic formula for finding speed?
Speed is the ratio of distance to time. This is the only formula you'll need for most basic speed calculations: Speed = Distance ÷ Time.
How do I find speed when I only know start and end times?
Calculate the duration by subtracting the start time from the end time, convert that time to hours or seconds as appropriate, and then divide the distance by the duration.
Do I add or subtract speeds when two objects are moving?
If they are moving towards each other, add their speeds. If they are moving in the same direction, subtract their speeds. This is what affects the rate at which the gap narrows between them.
Why do unit conversions matter so much in speed problems?
Units do not combine, so if the division is correct but the units are mixed together, the answer will be wrong. This can be avoided by changing all distances and times to the same units before dividing.