Kinematic Equations: Formulas, Derivations, and Examples
byKinematic equations connect displacement, initial velocity, final velocity, acceleration, and time when acceleration is constant. They describe how motion changes without asking which force caused the acceleration.
The formulas are not a menu to memorize blindly. Define a positive direction, list the known quantities with signs and units, identify the missing variable, and choose the equation that does not contain it. That method is faster and safer than trying every formula.
The Four Kinematic Equations
For one-dimensional motion with constant acceleration \(a\), let \(u\) be initial velocity, \(v\) final velocity, \(t\) elapsed time, and \(\Delta x=x-x_0\) displacement. The standard equations are
$$v=u+at$$
$$\Delta x=ut+\frac12at^2$$
$$v^2=u^2+2a\Delta x$$
$$\Delta x=\frac{u+v}{2}t$$
| Equation | Quantity omitted | Use it when |
|---|---|---|
| \(v=u+at\) | Displacement | You know velocities, acceleration, and time |
| \(\Delta x=ut+\frac12at^2\) | Final velocity | You need position after a known time |
| \(v^2=u^2+2a\Delta x\) | Time | Time is not given or needed |
| \(\Delta x=(u+v)t/2\) | Acceleration | Average velocity is easy to obtain |
These equations are equivalent under the constant-acceleration condition. They are not four separate laws. Each is another way to eliminate one variable from the same motion model.
When Kinematic Equations Are Valid
The four kinematic equations are exact only when acceleration is constant over the interval. They also apply component by component in two or three dimensions if each acceleration component is constant. Near Earth’s surface, projectile motion is often approximated with constant downward acceleration and negligible air resistance.
- Valid model: a cart with nearly constant acceleration, a freely falling object over modest height, or a vehicle braking at an approximately constant rate.
- Piecewise valid: a trip with different constant accelerations in separate time intervals. Solve each interval and pass its final state into the next.
- Not directly valid: drag proportional to speed, spring motion, orbital motion, or any case where acceleration changes continuously with position or velocity.
- Average acceleration warning: inserting an average value into a constant-acceleration formula does not generally reproduce the exact displacement.
If acceleration varies, use the definitions \(a=dv/dt\) and \(v=dx/dt\), then integrate or solve the relevant differential equation. The Newton’s laws guide explains how forces determine that acceleration.
How To Choose the Right Kinematic Equation
Write a five-column inventory before calculating. Include \(u\), \(v\), \(a\), \(t\), and \(\Delta x\). Mark the target, fill the known values with signs, and choose the equation that omits the one unused unknown.
If you want to check a value before working it by hand, use the kinematic equations calculator. It keeps the unit conversion beside the result, which is where most avoidable mistakes happen.
- Draw the motion and choose a positive axis.
- Convert every quantity to a consistent unit system.
- Write the initial and final event clearly.
- List \(u,v,a,t,\Delta x\), including negative signs.
- Select the equation that contains the target and known values but omits the extra unknown.
- Solve symbolically first, then substitute numbers and check dimensions.
OpenStax derives the same constant-acceleration relations from velocity-time graphs in its motion with constant acceleration guide. The graphical derivation is the quickest way to see why the displacement formula contains one-half.
Where the Equations Come From
Constant acceleration means velocity changes linearly with time. Integrating \(a=dv/dt\) from \(0\) to \(t\) gives the first equation.
$$\int_u^v dv=\int_0^t a\,dt\quad\Rightarrow\quad v=u+at$$
Velocity is the slope of the position-time graph. Integrating \(v=u+at\) gives displacement.
$$\Delta x=\int_0^t (u+at)\,dt=ut+\frac12at^2$$
Alternatively, the area under a velocity-time graph is displacement. For a straight velocity line, the area is a rectangle plus a triangle, or average velocity times time. This gives \(\Delta x=(u+v)t/2\). Eliminating \(t=(v-u)/a\) then gives \(v^2=u^2+2a\Delta x\).
Worked Example: Accelerating Car
A car starts at 5.0 m/s and accelerates at 2.0 m/s² for 6.0 s. Find its final velocity and displacement.
$$v=u+at=5.0+(2.0)(6.0)=17.0\,\text{m/s}$$
$$\Delta x=ut+\frac12at^2=(5.0)(6.0)+\frac12(2.0)(6.0)^2=66\,\text{m}$$
The units and scale make sense: the average velocity is \((5+17)/2=11\) m/s, and \(11\times6=66\) m.
Worked Example: Braking Distance
A car moving at 30 m/s brakes with constant acceleration \(-8.0\) m/s². Find the braking distance after the brakes begin acting. Time is absent, so use the time-free equation.
$$0^2=(30)^2+2(-8.0)\Delta x$$
$$\Delta x=\frac{900}{16}=56.25\,\text{m}$$
This is braking distance, not total stopping distance. A real driver travels farther during reaction time. At 30 m/s, even a 0.75 s reaction adds 22.5 m before braking starts.
Worked Example: Free Fall
Drop a ball from rest and ignore air resistance. Choose upward as positive, so \(u=0\) and \(a=-g\approx-9.81\,\text{m/s}^2\). After 2.0 s,
$$v=0-(9.81)(2.0)=-19.6\,\text{m/s}$$
$$\Delta y=0+\frac12(-9.81)(2.0)^2=-19.6\,\text{m}$$
Both negative signs mean downward relative to the chosen axis. If you choose downward as positive, the numbers become positive but the physical answer is unchanged. Try the free-fall calculator to check other heights and times.
Kinematics in Two Dimensions
Split vector motion into perpendicular components. The time is shared, but the horizontal and vertical equations are solved independently. For ideal projectile motion, horizontal acceleration is zero and vertical acceleration is \(-g\).
$$x=x_0+u_x t,\qquad y=y_0+u_y t-\frac12gt^2$$
The initial components are \(u_x=u\cos\theta\) and \(u_y=u\sin\theta\). Once the time is known from the vertical motion, use it in the horizontal equation. The projectile motion calculator helps verify range, flight time, and peak height under the same no-drag model.
Angular Kinematic Equations
Angular kinematic equations are the rotational versions of the linear constant-acceleration equations. They apply when angular acceleration \(\alpha\) is constant. Replace displacement with angular displacement \(\Delta\theta\), velocity with angular velocity, and acceleration with angular acceleration.
$$\omega=\omega_0+\alpha t$$
$$\Delta\theta=\omega_0t+\frac12\alpha t^2$$
$$\omega^2=\omega_0^2+2\alpha\Delta\theta$$
$$\Delta\theta=\frac{\omega_0+\omega}{2}t$$
| Linear quantity | Angular analogue | SI unit |
|---|---|---|
| Displacement \(\Delta x\) | Angular displacement \(\Delta\theta\) | radian (rad) |
| Initial velocity \(u\) | Initial angular velocity \(\omega_0\) | rad/s |
| Final velocity \(v\) | Final angular velocity \(\omega\) | rad/s |
| Acceleration \(a\) | Angular acceleration \(\alpha\) | rad/s² |
For a point at distance \(r\) from the rotation axis, the linear and angular quantities are connected by \(s=r\theta\), \(v_{\mathrm t}=r\omega\), and \(a_{\mathrm t}=r\alpha\). Use radians in these relations. Centripetal acceleration \(a_{\mathrm c}=r\omega^2\) is separate and can exist even when \(\alpha=0\).
Graphs and Physical Meaning
| Graph | Slope means | Area means under the curve |
|---|---|---|
| Position vs time | Velocity | No standard kinematic quantity |
| Velocity vs time | Acceleration | Displacement |
| Acceleration vs time | Rate of change of acceleration | Change in velocity |
Constant acceleration appears as a horizontal line on an acceleration-time graph, a straight line on a velocity-time graph, and a parabola on a position-time graph. Reading those shapes gives a useful check before any algebra.
Common Mistakes
- Mixing distance and displacement: displacement can be negative and depends on endpoints; distance counts the path length.
- Losing the sign of acceleration: acceleration is not automatically negative. Its sign comes from the chosen axis.
- Using speed as velocity: velocity includes direction.
- Applying constant-acceleration formulas to variable acceleration: a changing force usually needs calculus or numerical methods.
- Using different time intervals: every variable in one equation must refer to the same initial and final events.
- Ignoring roots: a quadratic time equation can yield two mathematical times; keep only the times that fit the physical interval.
Related Physics Guides
Use these next if you want to connect this result with the surrounding physics:
Key Takeaways
- The four kinematic equations describe constant-acceleration motion.
- Choose an axis first and carry signs consistently.
- Select the formula by identifying the variable you do not know and do not need.
- Graphs explain the equations: slope gives a rate, while area accumulates a change.
- For variable acceleration, return to derivatives and integrals instead of forcing these formulas.
Frequently Asked Questions
What are the four kinematic equations?
For constant acceleration they are v = u + at, Δx = ut + 1/2 at², v² = u² + 2aΔx, and Δx = (u + v)t/2.
When can I use kinematic equations?
Use them when acceleration is constant over the interval, or component by component when each acceleration component is constant. They are not generally exact for drag, springs, or orbital motion.
How do I choose a kinematic equation?
List u, v, a, t, and displacement. Mark the target and known values, then choose the equation that does not contain the remaining unknown variable.
Is acceleration always negative when an object slows down?
No. Acceleration is negative only relative to your chosen positive direction. An object slows when velocity and acceleration have opposite signs.
What is the difference between distance and displacement?
Distance is the total path length and is nonnegative. Displacement is final position minus initial position, so it can be positive, negative, or zero.
Do kinematic equations work for projectile motion?
Yes, in the standard no-drag model. Apply the equations separately to horizontal and vertical components, using the same elapsed time for both.
When a solution looks wrong, do not add more algebra. Recheck the axis, interval, units, and constant-acceleration assumption first.
Tell Google you want more of this.
Add Gaurav Tiwari as a preferred sourceOne tap, and this site shows up more often in your own Top Stories, AI Overviews and AI Mode. Remove it any time.
Browse more in Physics.