How to Find Velocity: Formulas & Units for Average, Initial, & More

Velocity, sometimes called celerity, is an object's speed in a particular direction. Mathematically, velocity is often described as the change in position over the change in time. This fundamental concept shows up in many basic physics problems, including finding instantaneous and final velocities. We spoke to physics tutor and founder of Alexander Tutoring Sean Alexander to get the best tips on how to tackle your math problems. Keep reading to learn which equation is right for your question.

Velocity Formulas

Average velocity = $v_{{av}}={\frac {x_{f}-x_{i}}{t_{f}-t_{i}}}$ or $v_{av}={\frac {D}{t}}$
Average velocity if acceleration is constant = $v_{{av}}={\frac {v_{i}+v_{f}}{2}}$
Average velocity if acceleration is zero and constant = $v_{{av}}={\frac {x}{t}}$
Final velocity = $v_{f}=v_{i}+at$
Circular velocity = $v_{c}={\frac {2\pi r}{T}}$
Velocity from kinetic energy = $v^{2}={\frac {2KE}{m}}$

Section 1 of 8:

Calculating Average Velocity Using Position and Time

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Find the distance between the start and endpoints. Subtract the starting point value from the endpoint to find displacement. Use the value in the equation $v_{{av}}={\frac {x_{f}-x_{i}}{t_{f}-t_{i}}}$ , or "final position - initial position divided by final time - initial time.” Distance isn’t always the same as displacement, so check that you’re using the correct value. ^{[1]
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- This equation is often written as $V={\frac {D}{t}}$ , or “velocity equals displacement over time.” If you’re not given the time period and displacement, you’ll have to find those values from your problem.
- Example 1: A car traveling due east starts at position x = 5 meters. After 8 seconds, the car is at position x = 41 meters. What was the car's displacement?
  - The car was displaced by (41m - 5m) = 36 meters east.
- Example 2: A ball is thrown 3 meters straight into the air, then falls downward for 5 meters before hitting the ground. What is the diver's displacement?
  - The ball ended up 2 meters below the starting point, so the displacement is 2 meters downward, or -2 meters. (0 + 5 - 3 = -2). Even though the ball traveled 8 meters (3 up, then 5 down), what matters is that the endpoint is two meters below the start point.
2
Calculate the change in time. How long did the object take to reach the endpoint? Many problems will tell you this directly. If it does not, subtract the start time from the end time to find out. ^{[2]
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- Example 1 (cont.): The problem tells us that the car took 8 seconds to go from the start point to the endpoint, so this is the change in time.
- Example 2 (cont.): If the ball was thrown at t = 8 seconds and hit the ground at t = 10 seconds, the change in time = 10s - 8s = 2 seconds.
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3
Divide the total displacement by the total time. In order to find the velocity of the moving object, you will need to divide the change in position by the change in time. Specify the direction moved, and you have the average velocity. ^{[3]
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- Example 1 (cont.): The car changed its position by 36 meters over 8 seconds. $v_{{av}}={\frac {36m}{8s}}=$ 4.5 m/s east.
- Example 2 (cont): The ball changed its position by -2 meters over 2 seconds. $v_{av}={\frac {-2m}{2s}}=$ -1 m/s . (In one dimension, negative numbers are usually used to mean "down" or "left." You could say "1 m/s downward" instead.)
4
Solve problems in two dimensions. Not all word problems involve movement back along one line. If the object turns at some point, you may need to draw a diagram and solve a geometry problem to find the distance .
- Example 3: A man jogs for 3 meters east, then makes a 90º turn and travels 4 meters north. What is his displacement?
  - Draw a diagram and connect the start and end points with a straight line. This is the hypotenuse of a triangle, so solve for the length of this line using properties of right triangles . In this case, the displacement is 5 meters northeast.
  - At some point, your math teacher may require you to find the exact direction traveled (the angle above the horizontal). You can do this by using geometry or by adding vectors.
5
Find average velocity when acceleration is constant. If an object is accelerating at a constant rate, calculate average velocity with: $v_{{av}}={\frac {v_{i}+v_{f}}{2}}$ . ^{[4]
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Research source} In this equation $v_{i}$ is the initial velocity , and $v_{f}$ is the final velocity. Remember, you can only use this equation if there is no change in acceleration.
- As a quick example, let's say a train accelerates at a constant rate from 30 m/s to 80 m/s. The average velocity of the train during this time is ${\frac {30+80}{2}}=55m/s$ .

Section 2 of 8:

Final Velocity

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1
Multiply the acceleration by the change in time. Alexander says to just think of acceleration as the change in velocity, so if the acceleration is constant, the velocity continues to change at the same rate. ^{[5]
X
Expert Source

Sean Alexander, MS
Academic Tutor

Expert Interview. 14 May 2020.} Use the equation $v_{f}=v_{i}+at$ to find final velocity. By multiplying acceleration by the change in time, you can see how much the velocity increased (or decreased) over this period.
- $v_{f}=v_{i}+at$ , or "final velocity = initial velocity + (acceleration * time)"
- Initial velocity $v_{i}$ is sometimes written as $v_{0}$ ("velocity at time 0").
- Example : A ship sailing north at 2 m/s accelerates north at a rate of 10 m/s ² . How much did the ship's velocity increase in the next 5 seconds?
  - a = 10 m/s ²
  - t = 5 s
  - (a * t) = (10 m/s ² * 5 s) = 50 m/s increase in velocity.
2
Add the initial velocity. Once you know the total change in the velocity, add it to the initial velocity of the object, and you have your answer. ^{[6]
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- Example (cont) : In this example, how fast is the ship traveling after 5 seconds?
  - $v_{f}=v_{i}+at$
  - $v_{i}=2m/s$
  - $at=50m/s$
  - $v_{f}=2m/s+50m/s=52m/s$
3
Specify the direction of movement. Unlike speed, velocity always includes the direction of movement. Make sure to include this in your answer. ^{[7]
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- In our example, since the ship started going north and did not change direction, its final velocity is 52 m/s north.
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Solve related problems. As long as you know the acceleration , and the velocity at any one point in time, you can use this formula to find the velocity at any other time. ^{[8]
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Research source} Here's an example solving for the initial velocity:
- "A train accelerates at 7 m/s ² for 4 seconds and ends up traveling forward at a velocity of 35 m/s. What was its initial velocity?"
  - $v_{f}=v_{i}+at$
    $35m/s=v_{i}+(7m/s^{2})(4s)$
    $35m/s=v_{i}+28m/s$
    $v_{i}=35m/s-28m/s=7m/s$

Section 3 of 8:

Circular Velocity

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1
Learn the formula for circular velocity. Circular velocity refers to the velocity that one object must travel to maintain its circular orbit around another object, usually a planet or other gravitating mass. ^{[9]
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- The circular velocity of an object is calculated by dividing the circumference of the circular path by the time period over which the object travels.
- When written as a formula, the equation is:
  - v = ^(2πr) / _T
- Note that 2πr equals the circumference of the circular path.
- r stands for "radius"
- T stands for "time period"
2
Multiply the circular radius by 2π. The first stage of the problem is calculating the circumference . To do this, multiply the radius by 2π. If you are calculating this by hand, you can use 3.14 as an approximation for π. ^{[10]
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- Example : Find the circular velocity of an object traveling a circular path with a radius of 8 m over a full-time interval of 45 seconds.
  - r = 8 m
  - T = 45 s
  - Circumference = 2πr = ~ (2)(3.14)(8 m) = 50.24 m
3
Divide this product by the time period. In order to find the circular velocity of the object in question, you need to divide the calculated circumference by the time period over which the object traveled. ^{[11]
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- Example : v = ^(2πr) / _T = ^{50.24 m} / _{45 s} = 1.12 m/s
  - The circular velocity of the object is 1.12 m/s.

Section 4 of 8:

Instantaneous Velocity

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1
Take the derivative of a function for displacement. While the average velocity looks at velocity over a period of time, taking instant velocity requires getting your time period as close to zero as possible. To do this, take the derivative of an equation for displacement, using the formula: If $y=ax^{n}$ , Derivative = $n*ax^{n-1}$ . ^{[12]
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- The derivative of a function gives you a new equation that calculates the slope at a single point.
- Example : For the displacement equation with respect to time $x=t^{3}+3t^{2}+10$ , the derivative is ${\frac {dx}{dt}}=3t^{2}+6t$ .
  - When taking a derivative, constants (numbers without a variable) become 0.
2
Plug in the time when you want to find velocity for “t.” The new equation gives you the velocity at any point in time. ^{[13]
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Research source} When you work out the equation, it gives you the instant velocity.
- Example : To find the instant velocity at 2 seconds for the equation $x=t^{3}+3t^{2}+10$ , substitute “2” for “t” in the equation of the derivative.
  ${\frac {dx}{dt}}=3(2)^{2}+6(2)$
  $=36+12$
  $=48m/s.$

Section 5 of 8:

Velocity from Kinetic Energy

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1
Use algebra to isolate velocity from the formula for kinetic energy. Kinetic energy is directly proportional to the mass of an object and the square of its velocity— $KE={\frac {1}{2}}mv^{2}$ . ^{[14]
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Research source} If you’re given an equation with respect to kinetic energy, rewrite it as $v^{2}={\frac {2KE}{m}}$ .
- If you already have the equation in terms of velocity, skip this step.
- Kinetic energy is often measured in Joules (J) and mass is measured in kilograms. One Joule equals 1 kg ${\frac {m^{2}}{s^{2}}}$ , where m = meters and s = seconds.
- Example : If you have a car of mass = 1000 kg and kinetic energy = 450,000 J, set up the equation as $v^{2}={\frac {2(450,000)}{1000}}$ .
2
Work out the right side of the problem. Multiply the kinetic energy by 2, then divide it by the mass. ^{[15]
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- Continuing the example above,
  $v^{2}={\frac {2(450,000)}{1000}}$
  $={\frac {9,000,000}{1000}}$
  $=9000$ .
3
Take the square root of your answer. Since your current equation solves for the square of velocity, take the square root of both sides. ^{[16]
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Research source} That turns v ² into v, and the right side of the equation into the velocity in meters per second.
- $v^{2}=9000=$
  ${\sqrt {v^{2}}}={\sqrt {9000}}=$
  $v=94.86$

Section 6 of 8:

Velocity vs. Speed vs. Acceleration

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Velocity vs. Speed Velocity and speed are not the same thing. While speed measures distance traveled, velocity also takes direction into account. ^{[17]
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Research source} That is why the equations ask for displacement, and not total distance traveled.
- Velocity is also a vector quantity, and vectors are line segments that have magnitude and direction. ^{[18]
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  Research source} Speed, which is a scalar quantity, does not have direction. ^{[19]
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- In linear algebra, scalars are real numbers by which you can multiply matrices. ^{[20]
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  Research source} A vector is a matrix with only one row and column.
2
Velocity vs. Acceleration Acceleration is the slope of a velocity vs time graph. Velocity vs time graphs show velocity as a function of time, or how the velocity changes as time moves forward. ^{[21]
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Research source} The slope between two points on the graph is the average acceleration since acceleration is the change of velocity over time.
- To find instantaneous acceleration , take the derivative of the function of the velocity vs time graph. ^{[22]
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Section 7 of 8:

How to Add Velocities

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Convert the velocities to matrices. Velocities can be added with linear algebra, but not regular algebra. You can think of two-dimensional velocities as vectors, and write them in a matrix—the top number tells you how far the vector moves on the x-axis, and the top tells you how it moves on the y-axis. ^{[23]
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- For example, if you have a boat going 60 m/s North on a river with a current that moves 30 m/s West and 10 m/s South, you’d make the matrices ${\begin{pmatrix}60\\0\end{pmatrix}}$ for the boat and ${\begin{pmatrix}-30\\-10\end{pmatrix}}$ for the river.
Add the top numbers and bottom numbers together to get a new vector. ^{[24]
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Research source} If you’re using a graph to solve your problem, translate one vector by moving its end to the other’s tip, and draw a third line to make a triangle.

Section 8 of 8:

Velocity Units

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Velocity measures displacement over time, so use the SI units for distance and time, meters, and seconds. ^{[25]
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Research source} Alexander warns that using the wrong units (or not putting them in at all) is one of the biggest mistakes math and physics students make on their tests and homework. ^{[26]
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Expert Source

Sean Alexander, MS
Academic Tutor

Expert Interview. 14 May 2020.}
2
Acceleration is measured in meters per second per second. Acceleration measures how quickly something speeds up, i.e., the change in velocity. For example, if something is moving at 30 m/s at t = 1 and 60 m/s at t = 2, that means the object accelerated 30 m/s over the period of one second, so the acceleration is 30 m/s ² . ^{[27]
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- Alexander notes that many people struggle to visualize what a “meter per second per second” really is. If you think about the units for velocity as a fraction (meters over seconds), and then think of “per second” as a fraction, ¹ /second, you can divide m/s by ¹ /s, and get m/s ² . ^{[28]
  X
  Expert Source
  
  Sean Alexander, MS
  Academic Tutor
  
  Expert Interview. 14 May 2020.}

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When do we have deceleration?

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As the object loses energy passing through a medium. The medium an object is passing through will determine the deceleration.

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How do I calculate the velocity of something given its time traveled and distance covered?

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Divide distance traveled by the time taken to get the average speed. Velocity is the term used for speed when the object travels in a uniform direction (i.e. straight line or circle).

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How does velocity change if the distance decreases and the time increases?

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Velocity decreases. Think about it: It takes a longer time to cover a shorter distance.

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Average velocity measures the average velocity an object travels over the full course of its path. Instantaneous velocity measures the velocity of an object at a specific moment along its path. ^{[29]
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Make sure your units match by measuring distance in meters (m), time in seconds (s), and acceleration in meters per second per second (m/s ² ).

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How to Find Velocity: Formulas & Units for Average, Initial, & More

Velocity Formulas

Steps

Calculating Average Velocity Using Position and Time

Final Velocity

Circular Velocity

Instantaneous Velocity

Velocity from Kinetic Energy

Velocity vs. Speed vs. Acceleration

How to Add Velocities

Velocity Units

Calculator, Practice Problems, and Answers

Community Q&A

Video

Tips

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