How to Find Initial Velocity

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1 Finding Initial Velocity with Final Velocity, Acceleration, and Time

2 Finding Initial Velocity with Distance, Time, and Acceleration

3 Finding Initial Velocity with Final Velocity, Acceleration, and Distance

4 Finding Initial Velocity with Final Velocity, Time, and Distance

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Article Summary

Co-authored by Sean Alexander, MS

Last Updated: November 12, 2024 Fact Checked

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Velocity is a function of time and defined by both a magnitude and a direction. Often in physics problems, you will need to calculate the initial velocity (speed and direction) at which an object in question began to travel. There are multiple equations that can be used to determine initial velocity. Using the information given in a problem, you can determine the proper equation to use and easily answer your question.

What formulas are used to find initial velocity?

• If you have values for the final velocity, acceleration, and time involved, use V_i = V_f - (a * t).
• If you know values for the distance, time, and acceleration, use V_i = (d / t) - [(a * t) / 2].
• If you're given the final velocity, acceleration, and distance, use V_i = √ [V_f² - (2 * a * d)].
• If you're given the final velocity, time, and distance, use V_i = 2(d/t) - V_f.

Steps

Method 1

Method 1 of 4:

Finding Initial Velocity with Final Velocity, Acceleration, and Time

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1. 1
   Know the right equation to use. In order to solve any physics problem you must know which equation to use. Writing down all of the known information is the first step to finding the right equation. If you have values for the final velocity, acceleration, and time involved, you can use the following equation:

   • Initial velocity: V_i = V_f - (a * t)
   • Understand what each symbol stands for.
     • V_i stands for “initial velocity”
     • V_f stands for “final velocity”
     • a stands for “acceleration”
     • t stands for “time”
   • Note that this equation is the standard equation used when finding initial velocity.
2. 2
   Fill in the known information. Once you have written the known information and determined the proper equation, you can fill in values for the appropriate variables. Carefully setting up each problem and writing out every step of the process is important.

   • If you make a mistake, you can easily find it by looking back at all of your previous steps.
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3. 3
   Solve the equation. With all of the numbers in place, use the proper order of operations to finish the problem. If you’re allowed, use a calculator to limit the number of simple math mistakes.

   • For example: An object accelerating east at 10 meters (32.8 ft) per second squared traveled for 12 seconds reaching a final velocity of 200 meters (656.2 ft) per second. Find the initial velocity of that object.
     • Write the known information:
     • V_i = ?, V_f = 200 m/s, a = 10 m/s², t = 12 s
   • Multiply the acceleration and time. a * t = 10 * 12 =120
   • Subtract the product from the final velocity. V_i = V_f – (a * t) = 200 – 120 = 80 V_i = 80 m/s east
   • Write your answer correctly. Include a unit of measurement, usually meters per second or m/s, as well as a direction the object was traveling in. Without providing information about the direction, you only have a measurement of speed rather than velocity.
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Method 2

Method 2 of 4:

Finding Initial Velocity with Distance, Time, and Acceleration

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1. 1
   Know the right equation to use. In order to solve any physics problem you must know which equation to use. Writing down all of the known information is the first step to finding the right equation. If you know values for the distance, time, and acceleration, you can use the following equation:

   • Initial velocity: V_i = (d / t) - [(a * t) / 2]
   • Understand what each symbol stands for.
     • V_i stands for “initial velocity”
     • d stands for “distance”
     • a stands for “acceleration”
     • t stands for “time”
2. 2
   Fill in the known information. Once you have written the known information and determined the proper equation, you can fill in values for the appropriate variables. Carefully setting up each problem and writing out every step of the process is important.

   • If you make a mistake, you can easily find it by looking back at all of your previous steps.
3. 3
   Solve the equation. With all of the numbers in place, use the proper order of operations to finish the problem. If you’re allowed, use a calculator to limit the number of simple math mistakes.

   • For example: An object accelerating west at 7 meters (23.0 ft) per second squared traveled a distance of 150 meters (492.1 ft) within 30 seconds. Calculate the initial velocity of that object.
     • Write the known information:
     • V_i = ?, d = 150 m, a = 7 m/s², t = 30 s
   • Multiply the acceleration and time. a * t = 7 * 30 = 210
   • Divide the product by two. (a * t) / 2 = 210 / 2 = 105
   • Divide the distance by the time. d / t = 150 / 30 = 5
   • Subtract your first quotient from the second quotient. V_i = (d / t) - [(a * t) / 2] = 5 – 105 = -100 V_i = -100 m/s west
   • Write your answer correctly. Include a unit of measurement, usually meters per second or m/s, as well as a direction the object was traveling in. Without providing information about the direction, you only have a measurement of speed rather than velocity.
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Method 3

Method 3 of 4:

Finding Initial Velocity with Final Velocity, Acceleration, and Distance

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1. 1
   Know the right equation to use. In order to solve any physics problem you must know which equation to use. Writing down all of the known information is the first step to finding the right equation. If you are given the final velocity, acceleration, and distance, you can use the following equation:

   • Initial velocity: V_i = √ [V_f² - (2 * a * d)]
   • Understand what each symbol stands for.
     • V_i stands for “initial velocity”
     • V_f stands for “final velocity”
     • a stands for “acceleration”
     • d stands for “distance”
2. 2
   Fill in the known information. Once you have written the known information and determined the proper equation, you can fill in values for the appropriate variables. Carefully setting up each problem and writing out every step of the process is important.

   • If you make a mistake, you can easily find it by looking back at all of your previous steps.
3. 3
   Solve the equation. With all of the numbers in place, use the proper order of operations to finish the problem. If you’re allowed, use a calculator to limit the number of simple math mistakes.

   • For example: An object accelerating north at 5 meters (16.4 ft) per second squared traveled 10 meters (32.8 ft), ending up at a final velocity of 12 meters (39.4 ft) per second. Calculate the object's initial velocity.
     • Write the known information:
     • V_i = ?, V_f = 12 m/s, a = 5 m/s², d = 10 m
   • Square the final velocity. V_f²= 12² = 144
   • Multiply the acceleration by the distance and the number two. 2 * a * d = 2 * 5 * 10 = 100
   • Subtract this product from your previous one. V_f² - (2 * a * d) = 144 – 100 = 44
   • Take the square root of your answer. = √ [V_f² - (2 * a * d)] = √44 = 6.633 V_i = 6.633 m/s north
   • Write your answer correctly. Include a unit of measurement, usually meters per second or m/s, as well as a direction the object was traveling in. Without providing information about the direction, you only have a measurement of speed rather than velocity.
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Method 4

Method 4 of 4:

Finding Initial Velocity with Final Velocity, Time, and Distance

Download Article

1. 1
   Know the right equation to use. In order to solve any physics problem you must know which equation to use. Writing down all of the known information is the first step to finding the right equation. If you are given the final velocity, time, and distance, you can use the following equation:

   • Initial velocity: V_i = 2(d/t) - V_f
   • Understand what each symbol stands for.
     • V_i stands for “initial velocity”
     • V_f stands for “final velocity”
     • t stands for “time”
     • d stands for “distance”
2. 2
   Fill in the known information. Once you have written the known information and determined the proper equation, you can fill in values for the appropriate variables. Carefully setting up each problem and writing out every step of the process is important.

   • If you make a mistake, you can easily find it by looking back at all of your previous steps.
3. 3
   Solve the equation. With all of the numbers in place, use the proper order of operations to finish the problem. If you’re allowed, use a calculator to limit the number of simple math mistakes.

   • For example: An object with a final velocity of 3 meters (9.8 ft) traveled south for 15 seconds and covered a distance of 45 meters (147.6 ft). Calculate the object's initial velocity.
     • Write the known information:
     • V_i = ?, V_f = 3 m/s, t = 15 s, d = 45 m
   • Divide distance by time. (d/t) = (45/15) = 3
   • Multiply that value by 2. 2 (d/t) = 2 (45/15) = 6
   • Subtract final velocity from the product. 2(d/t) - V_f = 6 - 3 = 3 V_i = 3 m/s south
   • Write your answer correctly. Include a unit of measurement, usually meters per second or m/s, as well as a direction the object was traveling in. Without providing information about the direction, you only have a measurement of speed rather than velocity.
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Expert Q&A

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• Question

  What are the difficulties some people have in understanding acceleration?

  

  Sean Alexander, MS
  Academic Tutor

  Sean Alexander is an Academic Tutor specializing in teaching mathematics and physics. Sean is the Owner of Alexander Tutoring, an academic tutoring business that provides personalized studying sessions focused on mathematics and physics. With over 15 years of experience, Sean has worked as a physics and math instructor and tutor for Stanford University, San Francisco State University, and Stanbridge Academy. He holds a BS in Physics from the University of California, Santa Barbara and an MS in Theoretical Physics from San Francisco State University.

  

  Sean Alexander, MS

  Academic Tutor

  Expert Answer

  One of the difficulties people face is the unit for acceleration, which is meters per second squared. To understand this with a real-world example, think about a rock being dropped off a cliff. Acceleration due to Earth's gravity is 9.8 meters per second squared, so we can round up to 10 for this example. Neglecting friction, when you first let go of the rock, its velocity is zero. After one second of falling, the speed of the stone will be 10 meters a second. After two seconds of falling, the speed of the stone will be 20 meters a second. After three seconds of falling, the speed of this will be 30 meters per second. It gains 10 meters a second of speed for each second that it is falling.

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• Question

  A bullet of 60 gm is fired using a rifle of mass 12kg. Rifle recoils with a velocity of 2.5m/s. What is the initial velocity?

  

  Community Answer

  Kinetic energy -> Kinetic energy 0.5mv^2 -> 0.5mv^2 0.5 x 12 x 2,5^2 = 0.5 x 0.06 x v^2 6 x 6.25 = 0.03 x v^2 37.5 = 0.03 x v^2 sqrt(37.5/0.03) = v sqrt(1250) = v v = 35.3 m/s

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• Question

  If displacement and time are related as s = 3.5t + 5t2, what is the initial velocity?

  

  Community Answer

  Initial velocity is 3.5. The equation is s = ut + 1/2at^2, where s - distance, u - inititial velocity, and a - acceleration.

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  Thank you for your feedback.
  If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow

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