How to Calculate Average Velocity

All you need to calculate average velocity is the total distance traveled, or change in position, and the total time. Remember that velocity determines both the direction of motion and speed, so the answer must state the direction of motion, such as “north”, “forward” or “left”. If the problem has constant acceleration, you can learn to take shortcuts to find an easier solution.

Table of Contents

Calculate average velocity from distance traveled and time

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Remember that velocity includes speed and direction of motion. Velocity describes the degree of change in position of an object. It concerns not only the speed but also the direction of motion of the object. The speed “100 m/s south ” is not the same as “100 m/ s east “.

Directed quantities are called vector quantities . ^{[1] X Research Source} They are distinguished from scalar quantities by the arrow mark on the variable. For example, v represents speed, while v ^→ represents velocity (including speed and direction of motion). ^{[2] X Research Source} The v used in this article is velocity.
For science problems, you use meters or another metric unit of distance, but in everyday life you can use any unit you like.

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Find the total distance traveled. Displacement distance is the change in position of an object, or includes the distance and direction of motion between the starting position and the ending position. You don’t care how far the object moves on the path before reaching the final position, only the distance between the starting and ending points is important. For the first example, we assume the object is moving at a constant speed in one direction:

Assume a rocket flies north for 5 minutes at a constant speed of 120 meters/min. To calculate the final position, we use the formula s = vt, or intuitively know that the rocket will be located (5 minutes) (120 meters/minute) = 600 meters north .
For problems with constant acceleration, you can solve using the equation s = vt + ½at ² , or refer to another section for a shorter solution.

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Find the total travel time. In the example problem above, the rocket moves forward for 5 minutes. You can write the average velocity in any unit of time, but the second is the international standard unit. We’ll convert to seconds in this example: (5 minutes) x (60 seconds/minute) = 300 seconds .

Even in a science problem, if the problem uses hours or longer time units, it may be easier to calculate velocity, then convert your answer to meters/second.

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Calculate average velocity over distance traveled and time. If you know how far the object moves, and how long it takes the object to travel that distance, you know how fast it is moving. For the above example, the average velocity of the rocket is (600 meters north) / (300 seconds) = 2 meters per second north .

Remember to write the direction of movement (like “front” or “north”).
As a formula we have v _av = Δs/Δt . The black notation ta Δ indicates “degree of change”, so Δs/Δt is “change of position in degrees of change in time”.
The average velocity is written as v _av , or written as v with the dashed line above it.

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Solve more complex problems. The problem is still not a problem if an object changes direction or changes speed. The average velocity is still only calculated from the total distance traveled and the total time. It doesn’t matter what transition happens between the start and end points. Here are some examples of tracks that have the same travel distance and travel time, so have the same average velocity.

Lan walks west at 1m/s for 2 seconds, then suddenly increases to 3m/s and continues walking west for 2 seconds. Her total teleportation distance is (1m/s west)(2 s) + (3m/s west)(2 s) = 8 meters west. Total travel time is 2s + 2s = 4s. Her average velocity is 8m west / 4s = 2m/s west .
Dung walks west at 5m/s for 3 seconds, then turns back and goes east at 7m/s for 1 second. We can think of eastward motion as “negative motion”, so total displacement = (5m/s west)(3s) + (-7m/s west) = 8 meters. Total time = 4s. Average velocity = 8 m west / 4s = 2 m/s west .
Ngoc goes 1 meter north, then 8 meters west, then 1 meter south. The time it takes her to cover this distance is 4 seconds. Draw a diagram on the page, you will see that she has traveled 8 meters west of the original position, so this is her teleport distance. Total travel time is 4s, so average velocity is still 8m west / 4s = 2m/s west .

Calculate average velocity from constant acceleration

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Note that the initial velocity and acceleration are constant. Suppose we have the problem “The bicycle starts moving to the right with a velocity of 5m/s, with a constant acceleration of 2 m/s ² . If it moves for 5 seconds, what is the average velocity how much?”

If you don’t understand “m/s ² “, write it as “m/s/s” or “meter/second/second”. ^{[3] X Research Source} Acceleration of 2m/s/s means that every second the velocity increases by 2m/s.

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Use the acceleration to find the final velocity. Acceleration, denoted a , is the degree of change in velocity (or speed). Velocity increases at a constant rate. You can tabulate and use the accelerometer to find the speed at different times on this journey. We have to do this calculation for the final time of the problem (at t = 5 seconds), but we will make a longer table to help you understand the concept:

At the initial moment ( t = 0 s), the bicycle is moving to the right with a speed of 5 m/s.
After 1 second ( t = 1), the bicycle is running at 5m/s + at = 5m/s + (2 m/s ² )(1 s) = 7m/s.
At t = 2, the bicycle moves to the right with a speed of 5+(2)(2) = 9m/s.
At t = 3, the bicycle moves to the right with a speed of 5+(2)(3) = 11m/s.
At t = 4, the bicycle moves to the right with a speed of 5+(2)(4) = 13m/s.
At t = 5, the bicycle moves to the right with a speed of 5+(2)(5) = 15m/s.

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Use this formula to find the average velocity. If and only if the acceleration is constant, the average velocity is equal to the average of the final and initial velocities: (v _f + v _i )/2 . For this example, the initial velocity of the bicycle v _i is 5 m/s. As calculated above, the bicycle travels with a final speed v _f of 15 m/s. Substituting these numbers in, we get (15 m/s + 5 m/s) / 2 = (20 m/s) / 2 = 10 m/s to the right .

Remember to write the direction of motion, in this case “right”.
These terms can be alternatively written as v ₀ (velocity at time 0 or initial velocity, and v (final velocity).

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Understand the formula for calculating average velocity intuitively. To find the average velocity, we can calculate the velocities at each instant and take their average. (This is the definition of average.) Since this requires a lot of calculations, let’s build this concept more intuitively. Instead of calculating the velocities at each instant, we just average the velocities at the two points in time and see what we get. One moment is near the start of the ride, when the bike is going slow, and the other is near the end of the ride, when the bike is going fast.

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Test intuition theory. Use the table above to find the velocities at different times. Some pairs of numbers that give equal mean are at (t=0,t=5), (t=1,t=4) or (t=2,t=3). You can test with non-integer t values if you like.

Regardless of which pair of numbers we choose, the average of the 2 velocities at those times will always be the same. For example, ((5+15)/2), ((7+13)/2), or ((9+11)/2), all equal 10m/s to the right.

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Complete an intuitive explanation. If we use this method for a series of times, we will continuously average one velocity in the top half with one velocity in the bottom half of the journey. Each halve has an equal number of times so no velocity will be missed once we’ve done the calculation.

Since any of these pairs of numbers have the same mean, the average of all these velocities will equal that value. With the example above, the average of all velocities is still 10m/s on the right.
We can find this value by averaging any pair of numbers, for example initial and final velocities. That is the velocity at times t=0 and t=5, and can be calculated using the above formula: (5+15)/2 = 10m/s on the right.

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Understand this formula on a mathematical basis. If you’re familiar with the formulas, you can start with the constant acceleration formula, and deduce it from there: ^{[4] X Research Source}

s = v _i t + ½at ² . (In theory it should be written as Δs and Δt, or change of position and change of time, but it is also understood if you use s and t.)
The average velocity v _av is defined as s/t, so write the formula in terms of s/t.
v _av = s/t = v _i + at
Acceleration x time is the total change in velocity, or v _f – vi _i . So we can replace “at” in that formula and get:
v _av = v _i + ½(v _f – v _i ).
Simplifying the equation we have: v _av = v _i + ½v _f – ½v _i = ½v _i + ½v _f = (v _f + v _i )/2 .

Advice

Velocity is different from speed because velocity is a vector quantity, and speed is a scalar quantity. Vector quantities represent both direction and magnitude, while scalar quantities represent magnitude only.
If the object moves in only one direction, such as left-right, you can use a positive value to indicate one direction (such as right) and a negative value to indicate the other direction (left). Write this rule at the top of the page so your reader understands your problem.
To calculate the velocity of an object at a particular point in its path, you need to calculate the instantaneous velocity instead of the average velocity.

Steps

Calculate average velocity from distance traveled and time

Calculate average velocity from constant acceleration

Advice

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