How To Find A Distance From Velocity & Time

Velocity is the change in position (x), or distance, over time. If you know the change in position and the amount of time taken to complete the journey, you can determine velocity. Similarly, if you have any two of these variables, you can always solve for the third.

The relationship between these three variables is as follows:

\(V = \dfrac{\bigtriangleup x}{t}\)

A car drives from Baltimore to Washington, D.C. in 1.5 hours. If you know that it's 38 miles between the two cities, what was the car's average velocity during the trip? Since, this is a trip that goes in one direction, the change in position is the same as distance. Since you know time and the distance, you can solve for velocity by plugging in the distance formula in physics:

\(V = \dfrac{\bigtriangleup x}{t}\)

\(V = \dfrac{38}{1.5} = 25.3\)

So you know your answer is 25.3, but this isn't quite complete: 25.3 what? Units are just as important as the numerical answer when it comes to physics problems, so don't lose track of what you're using to measure distance and time. Since you're measuring distance in miles and time in hours, your final answer is miles divided by hours, or miles per hour.

Try another example:

A bicyclist completes a 550 meter race in 1.5 hours. What is the bike's velocity in meters per second? Here, since you need to determine the velocity in meters per second, first convert time to seconds:

\((1.5 hours)(60 minutes)(60 seconds) = 5,400 seconds\)

Then, plug your known variables into the velocity formula:

\(V = \dfrac{\bigtriangleup x}{t}\)

\(V = \dfrac{550}{5400} = 0.1m/s\)

If you know how fast and how long something was traveling, you can solve for the distance traveled. You just need to rearrange the velocity formula above to get the distance formula in physics:

\({\bigtriangleup x} =(velocity)(time)\)

A plane travels 150 miles per hour on it's way from Atlanta to San Diego. How far has the plane traveled in 3.5 hours?

Since the plane appears to be going in one direction (toward San Diego) in a straight line, you can assume that the change in position equals distance. Plug your known variables into the distance formula:

\({\bigtriangleup x} =(150mph)(3.5h) = 525 miles\)

If you need to solve for time, you just rearrange the formula one more time:

\(time = \dfrac{\bigtriangleup x}{velocity}\)

Say a turtle crawls at 3 mph. How long will it take the turtle to finish a 5-mile race?

\(time = \dfrac{5}{3} = 1.67 h\)

People tend to use "speed" for "velocity" and vice versa, but they are slightly different concepts. Speed doesn't take into account direction, while velocity does. If you look at the formula, velocity is the change in position over time, while speed is distance over time. Let's look at an example to illustrate:

Say you drive 20 miles from your house to your college campus and then head back. It took you an hour round trip. What is your average speed?

You know your total distance and the time taken, so plug into the formula for speed:

\(speed= \dfrac{distance}{time}\)

\(speed= \dfrac{40}{1} = 40mph\)

Now, what is your average velocity? Keep in mind that you use change in location or displacement to determine velocity because direction matters:

\(velocity= \dfrac{\bigtriangleup distance}{time}\)

Since you end at your beginning location, your change in position or distance is actually 0, which means your velocity is also 0. Velocity is equal to the formula for speed only if you're traveling in a straight line.