Kinematics Calculator

About the Kinematics Calculator

Kinematics describes how objects move without asking why. In introductory physics, most problems involve motion in a straight line with constant acceleration: a car braking, a ball dropped from a bridge, a train leaving a station or a sprinter at the start of a race. These situations are governed by five quantities and four short equations, often called the SUVAT equations after their symbols.

The hard part is usually choosing the right equation, because each one leaves out a different quantity. This calculator does that choice for you. Leave exactly two of the five boxes blank, and it solves for both, using an equation that avoids the other unknown, so no intermediate rounding is carried forward. When the physics allows two answers, such as a ball passing the same height on the way up and on the way down, both are listed.

What are the equations of motion?

For motion with constant acceleration $a$ , starting with velocity $u$ and ending with velocity $v$ after time $t$ , the displacement $s$ satisfies four linked equations: $v = u + at$ , $s = ut + \frac12 at^2$ , $s = \frac12(u + v)t$ and $v^2 = u^2 + 2as$ . A fifth form, $s = vt - \frac12 at^2$ , follows from the others.

All quantities except time are vectors in one dimension, so their signs matter. Choose a positive direction, for example upward or forward, and give every displacement, velocity and acceleration a sign relative to it. A ball thrown upward at 12 m/s has $u = 12$ and, near Earth's surface, $a = -9.8$ m/s².

How to use the Kinematics Calculator

Pick a positive direction and write each known quantity with its sign.
Enter the three values you know in SI units: metres, metres per second, metres per second squared and seconds.
Leave the two unknown boxes empty.
Press Solve motion to see which equation is used for each unknown and the substitution.
Read the answers and the check; if two solutions appear, decide which one fits the situation.

Formulas

Velocity–time

v = u + at

Has no displacement s.

Displacement–time

s = ut + \tfrac12 at^2

Has no final velocity v.

Average velocity

s = \tfrac12 (u + v)t

Has no acceleration a.

Timeless equation

v^2 = u^2 + 2as

Has no time t.

Worked example: Dropped from rest for 3 s

The worked example drops an object from rest, $u = 0$ , with $a = 9.8$ m/s² downward taken as positive, and asks about the motion after $t = 3$ s. The final velocity is $v = 0 + 9.8 \times 3 = 29.4$ m/s.

The displacement is $s = 0 \times 3 + \frac12 \times 9.8 \times 3^2 = 44.1$ m. The check $v^2 = u^2 + 2as$ gives $864.36 = 0 + 2 \times 9.8 \times 44.1$ , which agrees.

The solution below is generated by the same engine that powers the calculator above.

Final answer

s = 44.1\ \text{m},\ v = 29.4\ \text{m/s}

How we read your problem

Find displacement s and final velocity v for motion in a straight line with constant acceleration.

u = 0\ \text{m/s},\ a = 9.8\ \text{m/s}^2,\ t = 3\ \text{s}

Step-by-step solution

List the known quantities
Unknown: displacement s and final velocity v.
$u = 0\ \text{m/s},\quad a = 9.8\ \text{m/s}^2,\quad t = 3\ \text{s}$
Final velocity
$v = u + at = 0 + (9.8)(3) = 29.4$
Displacement
$s = ut + \tfrac12at^2 = 44.1$

Formulas used

Velocity
$v = u + at$
Displacement
$s = ut + \tfrac12 at^2$
Average velocity
$s = \tfrac12 (u + v)t$
Timeless equation
$v^2 = u^2 + 2as$

With constant acceleration, any three of displacement, initial velocity, final velocity, acceleration and time fix the other two. Each pair of unknowns is solved with the equation that does not contain the other unknown.

Verification

Check v² = u² + 2as: 864.36 = 864.36.

Detailed explanation

Choosing the equation

Every equation of motion contains four of the five quantities. To find an unknown, use the equation that contains the three values you know and that unknown, and leaves out the other unknown. If a question gives distance, initial speed and final speed and asks for time, the average velocity equation $s = \frac12(u + v)t$ is ideal because it does not involve acceleration.

For example, a car speeds up from 10 m/s to 30 m/s over 100 m. With $s = \frac12(u + v)t$ , the time is $t = 2s/(u + v) = 200/40 = 5$ s, and then $a = (v - u)/t = 4$ m/s². The timeless equation confirms it: $30^2 = 10^2 + 2 \times 4 \times 100$ .

When there are two answers

Equations containing $t^2$ or $v^2$ can have two solutions. A ball thrown upward at 20 m/s with $a = -9.8$ m/s² is 10 m above the launch point twice: at about 0.583 s on the way up, moving at +14.28 m/s, and at about 3.498 s on the way down, moving at −14.28 m/s. Both are physically correct; the question decides which one you need.

Similarly, finding the initial velocity from $v^2 = u^2 + 2as$ gives a plus and a minus root. The calculator keeps only roots that lead to a positive time, and lists each valid pair, so you can see why a negative starting velocity can also end at the same place.

Braking, free fall and units

A braking car has acceleration opposite to its velocity. A car moving at 25 m/s that stops ( $v = 0$ ) with $a = -5$ m/s² needs $t = (0 - 25)/(-5) = 5$ s and travels $s = (0 - 625)/(2 \times -5) = 62.5$ m. Stopping distance grows with the square of speed, which is why doubling speed roughly quadruples it.

Convert units before entering values. Speeds in km/h are divided by 3.6 to give m/s, so 72 km/h is 20 m/s. Mixing km/h with seconds or centimetres with metres is one of the most common sources of answers that are wrong by a factor of 3.6 or 100.

Common mistakes

Ignoring signs, for example using $a = +9.8$ for a ball thrown upward when up is positive.
Using these equations when the acceleration is not constant.
Entering speeds in km/h while the time is in seconds.
Rounding the first unknown and then using it to find the second.
Treating distance travelled and displacement as the same when the object reverses direction.

Tips

Write the list s, u, v, a, t, fill in what you know and circle what you need before choosing an equation.
Starting from rest means $u = 0$ ; coming to rest means $v = 0$ .
Free fall near Earth uses $|a| = 9.8$ m/s² (9.81 for more precision), directed downward.
For two-dimensional motion such as a thrown ball, treat horizontal and vertical directions separately; the projectile motion calculator does this automatically.
Sketch a velocity–time graph: its slope is the acceleration and the area under it is the displacement.

Frequently asked questions

What do s, u, v, a and t stand for?

s is displacement, u is initial velocity, v is final velocity, a is acceleration and t is time. Together they give the name SUVAT equations.

How many values do I need to solve a kinematics problem?

Three. With constant acceleration, any three of the five quantities determine the remaining two, provided the time is positive and the values are consistent.

Which equation should I use if time is not given?

Use v² = u² + 2as, the only equation without time. It connects velocities, acceleration and displacement directly.

Why does the calculator show two solutions?

Some equations are quadratic in time or velocity. A thrown object can pass the same position twice, once rising and once falling, so both times are valid answers to the equation.

Can acceleration be negative?

Yes. A negative acceleration points opposite to your chosen positive direction. It slows an object moving in the positive direction and speeds up one moving in the negative direction.

Do the SUVAT equations work for circular motion or air resistance?

No. They assume straight-line motion with constant acceleration. Circular motion, springs and motion with drag have changing acceleration and need other methods, often calculus.

How do I convert km/h to m/s?

Divide by 3.6, because 1 km/h is 1000 m per 3600 s. So 90 km/h is 25 m/s, and to go back from m/s to km/h multiply by 3.6.

Solve your motion problem

Enter three of the five quantities above and leave two blank to get both unknowns with every equation shown.

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