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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 $\mathrm{\Delta }x$ , time interval $t$, initial velocity ${v}_{0}$, final velocity $v$, and acceleration $a$.
Kinematic formulaFormula that describes the relationships between kinematic variables when acceleration is constant.

## Equations

1. $v={v}_{0}+at$
2. $x={x}_{0}+{v}_{0}t+\frac{1}{2}a{t}^{2}$
3. ${v}^{2}={v}_{0}^{2}+2a\left(x-{x}_{0}\right)$
4. $x-{x}_{0}=\frac{1}{2}\left({v}_{0}+v\right)t$
Symbols
• ${x}_{0}$ is
• $x$ is the
• $t$ is the
• ${v}_{0}$ is initial velocity
• $v$ is final velocity
• $a$ is acceleration
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 $v={v}_{0}+at$ to solve for the variables $v$, ${v}_{0}$, $a$, 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 ${v}_{0}=0$, "dropped" often means ${v}_{0}=0$, and "comes to a stop" means $v=0$.
Also, the magnitude of the acceleration due to gravity on all objects in free fall on Earth is usually assumed to be $g=9.8\frac{\text{m}}{{\text{s}}^{2}}$, 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: ${a}_{g}=-9.81\frac{\text{m}}{{\text{s}}^{2}}$. 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 ${v}_{0}$ 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, $x={x}_{0}+{v}_{0}t+\frac{1}{2}a{t}^{2}$, might require using the
.

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?

• Is there a trick to memorize these formulas?
• 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.
• I dont know how these equations even work. I am so confused and not sure what to do
• First, memorise the formulas
Usually, I’ll memorise an acronym SUVAT where S stands for displacement, U stands for initial velocity, V stands for final velocity, A stands for acceleration and T stands for time taken.

Thus, I’ll memorise the formula like this:
1. v=u + at
2. s=ut + 1/2at^2
3. v^2=u^2 + 2as

Secondly, practice.
Tip: try drawing out the scenario to help you better understand the question.
Tip: before attempting the question, list out the values for the different variables in SUVAT. This will help you identify which variables are presented/not presented in the question so that you can identify which formula to use given the variables, instead of trying to picture the whole scenario inside your mind. Moreover, you should also strive to indicate the direction (e.g. if a car is moving right to left, indicate it so that you would know if the velocity/acceleration/displacement is positive or negative) or (e.g. a stone dropping from the sky, indicate stone, top to bottom (vertical), then you would know if you are dealing with positive or negative vectors. ).

Third, keep trying even if you don’t manage to get the answer immediately. Practice makes perfect.

All the best. Hope this helps.
• Do all the kinematic equations apply for constant acceleration only?
• Yes because kinematics only requires algebra. To solve problems with changing acceleration you need Calculus( Langrarian mechanics or EOM's)
• This lesson was by far the worst... like how the hell was the last two video suppose to help in solving to get the answer if it didn't even go over how to apply the formulas.
• The strategy they showed in the last 2 videos were as follows.

We have 5 variables, acceleration; final velocity; initial velocity; time; displacement.

Now what we will do is fill in all the information into these 5 variables. For example the question provides us only the information for: final velocity; initial velocity; time and asks us to find out the displacement.

(Generally in some question where lets say a ball has been thrown from the top of a building, you know that the acceleration would be negative 9.81. In this question pretend that we don't know that acceleration)

We will find a formula which best fits our case. We need to pick a formula which contains all the variables we know and a single variable (displacement) that we are trying to find out. The formula best fit is the equation 4 shown above => displacement = ((final velocity + initial velocity) / 2) ✖ time
• How do I know when to use the quadratic formula for the second kinematic equation?
• When trying to calculate time you need to use the quadratic formula (or completing the square) because there is a "t^2" term and a "t" term.
• 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 :)
• 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.
• 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?
• 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
(1 vote)
• Initial velocity = 0m/s
Final velocity = 28m/s
Time taken = 4s
Acceleration = (Final Velocity - Initial Velocity) / 2 = (28-0)/4 = 7m/s^2

Hope this helps.
• What's a easier way to memorize these formulas?
• Usually, I’ll memorise an acronym SUVAT where S stands for displacement, U stands for initial velocity, V stands for final velocity, A stands for acceleration and T stands for time taken.

Thus, I’ll memorise the formula like this:
1. v=u + at
2. s=ut + 1/2at^2
3. v^2=u^2 + 2as

Hope it helps.