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Motion with constant acceleration review

Review the key concepts, equations, and skills for motion with constant acceleration, including how to choose the best kinematic formula for a problem.

Key terms

TermMeaning
Kinematic variableVariable that describes the motion of an object over time. Includes displacement delta, x , time interval t, initial velocity v, start subscript, 0, end subscript, final velocity v, and acceleration a.
Kinematic formulaFormula that describes the relationships between kinematic variables when acceleration is constant.

Equations

  1. v, equals, v, start subscript, 0, end subscript, plus, a, t
  2. x, equals, x, start subscript, 0, end subscript, plus, v, start subscript, 0, end subscript, t, plus, start fraction, 1, divided by, 2, end fraction, a, t, squared
  3. v, squared, equals, v, start subscript, 0, end subscript, squared, plus, 2, a, left parenthesis, x, minus, x, start subscript, 0, end subscript, right parenthesis
  4. x, minus, x, start subscript, 0, end subscript, equals, start fraction, 1, divided by, 2, end fraction, left parenthesis, v, start subscript, 0, end subscript, plus, v, right parenthesis, t
Symbols
Assumptions
  • Acceleration is constant over the time interval

Using the kinematic formulas

Choosing the best kinematic formula

To choose the kinematic formula that's right for your problem, figure out which variable you are not given and not asked to find.
For example, we could use v, equals, v, start subscript, 0, end subscript, plus, a, t to solve for the variables v, v, start subscript, 0, end subscript, a, or t if we knew the values of the other three variables. Note that each kinematic formula is missing one of the five kinematic variables.

Finding the known variables

Sometimes a known variable will not be explicitly given in a problem, but rather implied with codewords. For instance, "starts from rest" means v, start subscript, 0, end subscript, equals, 0, "dropped" often means v, start subscript, 0, end subscript, equals, 0, and "comes to a stop" means v, equals, 0.
Also, the magnitude of the acceleration due to gravity on all objects in free fall on Earth is usually assumed to be g, equals, 9, point, 8, start fraction, start text, m, end text, divided by, start text, s, end text, squared, end fraction, so this acceleration will usually not be given explicitly.

Common mistakes and misconceptions

  1. People forget that some of the kinematic variables are vectors and can have negative signs. For example, if upward is assumed to be positive, then the acceleration due to gravity must be negative: a, start subscript, g, end subscript, equals, minus, 9, point, 81, start fraction, start text, m, end text, divided by, start text, s, end text, squared, end fraction. A missing negative sign is a very common mistake, so don't forget to check which direction is defined as positive!
  2. People forget that the kinematic variables we plug into a kinematic formula must be consistent with that time interval. In other words, the initial velocity v, start subscript, 0, end subscript has to be the velocity of the object at the initial position and start of the time interval t. Similarly, the final velocity v must be the velocity at the final position and end of the time interval t.
  3. The second kinematic equation, x, equals, x, start subscript, 0, end subscript, plus, v, start subscript, 0, end subscript, t, plus, start fraction, 1, divided by, 2, end fraction, a, t, squared, might require using the
    quadratic formula
    .

Learn more

For deeper explanations, see our videos choosing kinematic equations and a worked example with kinematic equations.
To check your understanding and work toward mastering these concepts, check out our exercises choosing the best kinematic equation and solving problems with kinematic equations.

Want to join the conversation?

  • scuttlebug purple style avatar for user DNG
    Is there a trick to memorize these formulas?
    (1 vote)
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    • purple pi teal style avatar for user Nikhita
      You only need to memorize the 1st and 2nd formulas.
      The 3rd and 4th can be made from those two by using some algebra. Specifically isolating "a" in the first equation, plugging it into the second, and simplifying gives you the forth equation while doing the same with "t" gives you the third equation.
      (13 votes)
  • duskpin sapling style avatar for user Heet Shah
    Do all the kinematic equations apply for constant acceleration only?
    (5 votes)
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  • blobby green style avatar for user AJ Barak
    How do I know when to use the quadratic formula for the second kinematic equation?
    (4 votes)
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  • blobby green style avatar for user Roberto Reyes
    Could someone explain to me what the x0 stands for? Is there such a thing as initial and final displacement? Thanks in advance for any help :)
    (2 votes)
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    • aqualine ultimate style avatar for user Hazooloo
      To calculate displacement, you need to subtract the initial position from the final position:

      Displacement = Xf - Xi

      Xi = Xo = initial position.

      So in your question, you mixed Xo for being something related to displacement, whereas it is related to position and is something used to calculate displacement.
      (1 vote)
  • boggle yellow style avatar for user Zoe LeVell
    why do the equations listed here include an x0? This is not shown in the videos.
    (1 vote)
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    • starky tree style avatar for user Austin
      The equations here include the term "x0" because that is part of delta x, which is used in some cases to help solve for some kinematic problems, and sometimes we are solving for x final. The reason it might of not been shown in videos is because they didn't need to solve for a "x" value, therefore leaving it out. An example:

      v=v0+at

      In conclusion, they include "x0" because it is the initial position for x, and is needed to find the final position, and is part of the 5 variables used in kinematic equations.
      (1 vote)
  • blobby green style avatar for user lbarbour24
    What's a easier way to memorize these formulas?
    (1 vote)
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  • duskpin seedling style avatar for user Aditi
    Could someone explain to me what the x0 stands for? Is there such a thing as initial and final displacement? Thanks in advance for any help :)
    (1 vote)
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  • blobby blue style avatar for user purplebike10
    Do all of the kinematic formulas have initial velocity? All of the other variables are missing from one equation, but it looks like initial velocity is in all of them. If that's the case, how would you solve an equation where initial velocity is the variable that you are not given and not asked to find?
    (0 votes)
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  • blobby green style avatar for user ALIAR
    halima takes her car to the racetrack. it accelerates from 0 to 28 m/s in 4 seconds. what is the accerlsation of her car
    (0 votes)
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  • blobby green style avatar for user Maia
    Under the heading "Choosing the best kinematic formula" it states "To choose the kinematic formula that's right for your problem, figure out which variable you are not given and not asked to find."
    Do they mean: "..., figure out which variable you are not given and asked to find." I removed the second "not."
    Was the second "not" left in error?
    Thanks in advance for the clarification.
    (0 votes)
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    • blobby green style avatar for user Dolly Thicke
      You want an equation to contain the one variable you are trying to find, but not any variables you don't have and are not trying to find. It's a difficult point to communicate, but the EASY way to do this is to look for the one thing you DO NOT need and find the one equation that doesn't have that variable. When you figure that out, it's easy breezy!
      (2 votes)