# Kinematic Equations for Constant Acceleration Calculator

This kinematics calculator will help you to solve constant acceleration problems using kinematic equations

As you may know, there are two main kinematic equations of motion for uniform, or constant, acceleration.

$V=V_0+at\\\\S=V_{avg}t=\frac{V_0+V}{2}t=V_0t+\frac{at^2}{2}$

Thus, we have five parameters of motion: initial velocity V₀, final velocity V, acceleration a, time t, and displacement, or distance, S, and two equations. Therefore, to use these equations, we need three known parameters and two unknown parameters. Also, as Combinatorics – combinations, arrangements and permutations tells us, the number of combinations of 3 from 5 is 10, so there are ten types of kinematic equations problems at all; each has a different set of known parameters.

This calculator allows you to enter any three known parameters and clear the parameters that should be found, and it kindly finds them. Kinematic equations for each set of parameters are listed below the calculator. BTW, by default, the acceleration has the value of gravity constant g, making a free-fall problem.

#### Kinematic Equations Calculator

Digits after the decimal point: 2
Initial velocity, Vo

Final velocity, V

Acceleration, a

Time, t

Displacement, S

### Kinematic equations

Below are ten types of problems along with the solution formulas.

#### Case 1. Find the unknowns given initial velocity, acceleration, and time

Example problem: An airplane accelerates down a runway at n m/s² for m seconds until it finally lifts off the ground. Determine the final velocity and the distance traveled before takeoff.
Solution: Although the problem specifies only the acceleration and the time, the third parameter is known implicitly. The problem assumes that an airplane starts from the rest, hence, its initial velocity is zero. Thus, we can use our kinematic equations as is

$V=V_0+at\\\\S=V_0t+\frac{at^2}{2}$

#### Case 2. Find the unknowns given initial velocity, final velocity, and time

Example problem: An airplane accelerates down a runway for n seconds until it finally lifts off the ground having a speed of m m/s. Determine the acceleration and the distance traveled before takeoff.
Solution: Again we know that the initial velocity is zero. To solve the problem, we need to rearrange our kinematic equations like so:

$a=\frac{V-V_0}{t}\\\\S=\frac{V_0+V}{2}t$

#### Case 3. Find the unknowns given initial velocity, final velocity, and acceleration

Example problem: An airplane accelerates down a runway at n m/s² until it finally lifts off the ground having a speed of m m/s. Determine the time and the distance traveled before takeoff.
Solution: The same case with the initial velocity being zero. So, our kinematic equations will be:

$t=\frac{V-V_0}{a}\\\\S=\frac{V^2-V^2_0}{2a}$

#### Case 4. Find the unknowns given initial velocity, final velocity, and distance

Example problem: An airplane accelerates down a runway until it finally lifts off the ground having a speed of n m/s after traveling the distance of m meters. Determine the time and the acceleration.
Solution: As usual we know that the initial velocity is zero. And we use the kinematic equations like so:

$t=\frac{2S}{V_0+V}\\\\a=\frac{V-V_0}{t}$

Here it is easier to find the time first, then plug the time into the second equation.

#### Case 5. Find the unknowns given initial velocity, time, and distance

Example problem: An airplane accelerates down a runway for n seconds until it finally lifts off the ground after traveling the distance of m meters. Determine the liftoff speed and the acceleration.
Solution: In this problem, the initial velocity is zero as well. The kinematic equations for this case are:

$V=\frac{2S}{t}-V_0\\\\a=\frac{V-V_0}{t}$

Note that here it is easier to find the final velocity first, then plug it into the second equation.

#### Case 6. Find the unknowns given initial velocity, acceleration, and distance

Example problem: An airplane accelerates down a runway at n m/s² until it finally lifts off the ground after traveling the distance of m meters. Determine the liftoff speed and the time.
Solution: Yet again the initial velocity is zero. However, this case is quite complicated, because the only way to find the time is to solve the quadratic equation:

$\frac{a}{2}t^2+V_0t-S=0$

Of course, to find the time you need to pick up the positive root. After that, you can plug the time into the next equation to find the final velocity:

$V=V_0+at$

#### Case 7. Find the unknowns given final velocity, time, and distance

Example problem: An car decelerates for n seconds until it finally stops after traveling the distance of m meters. Determine the initial speed and the deceleration.
Solution: Now the initial velocity is unknown and the final velocity is zero. Also, we will get a negative value for acceleration, meaning that the car decelerates. To solve the problem, we need to use the following form of the kinematic equations:

$V_0=\frac{2S}{t}-V\\\\a=\frac{V-V_0}{t}$

Here it is easier to find the initial velocity first, then plug it into the second equation.

#### Case 8. Find the unknowns given final velocity, time, and acceleration

Example problem: An car accelerates at n m/s² for t seconds until it reaches m m/s. Determine the initial speed and the distance traveled.
Solution: Again we do not know the initial velocity. We need to use the kinematic equations like so:

$V_0=V-at\\\\S=\frac{V+V_0}{2}t$

#### Case 9. Find the unknowns given final velocity, acceleration, and distance

Example problem: An car accelerates at n m/s² for m meters until it reaches m m/s. Determine the initial speed and the time.
Solution: It is also a complicated case, there we again need to solve the quadratic equation to find the time. Our equation will be:

$-\frac{a}{2}t^2+Vt-S=0$

Then time is found, find the initial velocity:

$V_0=V-at$

#### Case 10. Find the unknowns given time, acceleration, and distance

Example problem: An car accelerates at n m/s² for m seconds and it traveled s meters. Determine the initial speed and the final speed.
Solution: Use the following kinematic equations:

$V_0=\frac{S}{t}-\frac{at^2}{2}\\\\V=V_0+at$

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PLANETCALC, Kinematic Equations for Constant Acceleration Calculator