How to Calculate Speed, Distance & Time

1. 1
   Isolate the different variables to solve for distance and time. Once you know the basics of the speed equation, you can use it to find more than just speed. For example, if you start out knowing speed and one of the other variables, you can rearrange the equation to find the missing piece of information.

   • For example, let's say that we know that a train traveled at 20 kilometers per hour for four hours, but we need to know how far it went. In this case, we can rearrange the equation and solve like this:

     speed = distance/time

     speed × time = (distance/time) × time

     speed × time = distance

     20 km/h × 4 h = distance = 80 kilometers
2. 2
   Convert your units as needed. Sometimes, you may calculate speed in one set of units but need it in another. In this case, you'll need to use conversion factors to get your answer into the correct units. To do this, just write the relationships between your units as a fraction and multiply. When you multiply, flip your fraction as needed to get rid of the units you don't want. This is a lot easier than it sounds!

   • For example, let's say that in the example problem above, we need our answer in miles instead of kilometers . There are about 1.6 kilometers in a mile, so we can convert like this:

     80 kilometers × 1 mile/1.6 kilometers = 50 miles

   • Note that because kilometers appears on the bottom of the fraction, it cancels with the kilometers in the original answer, leaving the answer in miles.
   • This site contains conversions for most common units.
3. 3
   Replace the "distance" variable with distance formulas as needed. Objects don't always travel in nice, convenient straight lines. In cases when they don't, you may not be able to simply plug a numerical value for distance into the standard speed equation. Instead, you may need to replace the d in s = d/t with a formula that models the distance the object traveled.

   • For example, let's say that an airplane flies in a circle that is 20 miles wide five times. The plane completes this journey in half an hour. In this example, we still need to find exactly how far the plane has traveled before we can find its speed. We can use the equation for the distance around a circle (its circumference) in place of d in our equation. This equation is circumference = 2πr where r = the circle's radius. ^[1] We would solve like this:

     s = (2 × π × r)/t

     s = (2 × π × 10)/0.5

     s = 62.83/0.5 = 125.66 miles/hour
4. 4
   Understand that s = d/t gives an average speed. The simple, convenient equation we have been using to find speed has one significant flaw. The value it gives you is technically an average speed. This means that it assumes the object you're measuring went the same speed for the whole trip . As we'll see below, finding an object's speed at a single given moment can be more difficult.

   • To illustrate this difference, imagine the last trip you took in a car. It's highly unlikely that you traveled the same speed for the whole trip. Instead, you started out slow and gradually reached your cruising speed, slowing down at stoplights, traffic jams, and so on. If you use the standard speed equation to figure out your speed for the trip, these changes in speed won't be reflected. Instead, you'll get an answer that's somewhere in the middle of all the different speeds you traveled at. ^[2]
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Note: This section uses techniques that may be unfamiliar to those who have not studied calculus. See our calculus articles for help.

1. 1
   Understand that speed is defined as the magnitude of velocity. Higher-level speed calculations can get confusing because mathematicians and scientists use different definitions for "speed" and "velocity". A velocity has two components: a magnitude and a direction . The magnitude is equal to the object's speed. A change in the direction will cause a change in the velocity, but not in the speed.

   • For example, let's say that there are two cars moving in opposite directions. Both cars' speedometers read 50 km/hr, so they both have the same speed. However, since they are moving apart from each other, we say that one car has a velocity of -50 km/hr and one has a velocity of 50 km/hr.
   • Just as you can calculate instantaneous speed, you can also calculate instantaneous velocity .
2. 2
   Use absolute values for negative velocities. Objects can have velocities with a negative magnitude (if they are moving in a negative direction relative to something else). However, there's no such thing as a negative speed, so in these cases the absolute value of the magnitude gives the object's speed.

   • For this reason, in the example problem above, both cars have a speed of 50 km/hr .
3. 3
   Take the derivative of a position function. If you have a function s(t) that gives you the position of an object with regards to time, the derivative of s(t) will give you its velocity with regards to time. Just plug a time value into this equation for the variable t (or whatever the time value is) to get the velocity at this given time. From here, finding the speed is easy.

   • For example, let's say that an object's position in meters is given with the equation 3t ² + t - 4 where t = time in seconds. We want to know what the speed of the object is at t = 4 seconds. In this case, we can solve like this:

     3t ² + t - 4

     s'(t) = 2 × 3t + 1

     s'(t) = 6t + 1

   • Now, we plug in t = 4:

     s'(t) = 6(4) + 1 = 24 + 1 = 25 meters/second . This is technically a velocity measurement, but since it's positive and direction is not mentioned in the problem, we can essentially use it for speed.
4. 4
   Take the integral of an acceleration function. Acceleration is a way of measuring the change in an object's velocity over time . This topic is a little too complex to explain fully in this article. However, it's useful to note that when you have a function a(t) that gives acceleration with regards to time, the integral of a(t) will give you velocity with regards to time. Note that it's helpful to know the object's initial velocity so that you can define the constant that results from an indefinite integral.

   • For example, let's say that an object has a constant acceleration (in m/s ² given by a(t) = -30. Let's also say that it has an initial velocity of 10 m/s. We need to find its speed at t = 12 s. In this case, we can solve like this:

     a(t) = -30

     v(t)= ∫ a(t)dt = ∫ -30dt = -30t + C

   • To find C, we'll solve v(t) for t = 0. Remember that the object's initial velocity is 10 m/s.

     v(0) = 10 = -30(0) + C

     10 = C, so v(t) = -30t + 10

   • Now, we can just plug in t = 12 seconds.

     v(12) = -30(12) + 10 = -360 + 10 = -350. Since speed is the absolute value of velocity, the object's speed is 350 meters/second .
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