KAHIBARO
Table of Contents
Constant acceleration as a special case
When acceleration stays constant, motion becomes much easier to describe. Instead of needing calculus at every step, we can use a small set of equations that connect position, velocity, acceleration, and time.
These equations are called the kinematic equations for constant acceleration. They are valid for motion in one dimension when the acceleration does not change with time.
If we use the symbols $x$ for position, $x_0$ for initial position, $v$ for velocity, $v_0$ for initial velocity, $a$ for constant acceleration, and $t$ for elapsed time, then we can build the full description of the motion.
Starting from constant acceleration
Constant acceleration means
$$
a = \text{constant}
$$
Since acceleration is the rate of change of velocity, the velocity changes by equal amounts in equal time intervals. This gives a linear relation between velocity and time:
$$
v = v_0 + at
$$
This is the first and most basic kinematic equation. It tells us how the velocity changes after time $t$.
If acceleration is positive, velocity increases in the positive direction. If acceleration is negative, velocity decreases if the object is moving in the positive direction.
For constant acceleration, the velocity after time $t$ is
$$
v = v_0 + at
$$
This equation is valid only when $a$ is constant.
Position equation
To find position, we use the fact that position changes according to velocity. For constant acceleration, the position after time $t$ is
$$
x = x_0 + v_0 t + \frac{1}{2}at^2
$$
This equation has a simple meaning. The term $x_0$ is where the object starts. The term $v_0 t$ is how far it would move if velocity stayed constant at its initial value. The term $\frac{1}{2}at^2$ is the extra change caused by acceleration.
If $a=0$, this equation becomes
$$
x = x_0 + v_0 t
$$
which is the familiar formula for motion at constant velocity.
For constant acceleration, position is given by
$$
x = x_0 + v_0 t + \frac{1}{2}at^2
$$
The term $\frac{1}{2}at^2$ appears because velocity is changing during the motion.
Average velocity under constant acceleration
For constant acceleration, the average velocity over a time interval is the simple average of the initial and final velocities:
$$
v_{\text{avg}} = \frac{v_0 + v}{2}
$$
Then the displacement can be written as
$$
x - x_0 = v_{\text{avg}} t = \frac{v_0 + v}{2} t
$$
So another useful kinematic equation is
$$
x = x_0 + \frac{v_0 + v}{2}t
$$
This form is especially useful when time is known, and both initial and final velocities are involved.
When acceleration is constant,
$$
v_{\text{avg}} = \frac{v_0 + v}{2}
$$
and therefore
$$
x = x_0 + \frac{v_0 + v}{2}t
$$
Equation without time
Sometimes time is not given, or you want to eliminate it. In that case, a very useful equation is
$$
v^2 = v_0^2 + 2a(x - x_0)
$$
This equation relates velocity directly to displacement.
It is often used when an object speeds up or slows down over a known distance.
The constant acceleration equation that does not contain time is
$$
v^2 = v_0^2 + 2a(x - x_0)
$$
Use it when time is unknown or unnecessary.
The full set of common kinematic equations
The most common constant acceleration equations are shown below.
| Quantity relation | Equation |
|---|---|
| Velocity and time | $v = v_0 + at$ |
| Position and time | $x = x_0 + v_0 t + \frac{1}{2}at^2$ |
| Position, time, and velocities | $x = x_0 + \frac{v_0 + v}{2}t$ |
| Velocity and displacement | $v^2 = v_0^2 + 2a(x - x_0)$ |
These equations are not separate laws of nature. They are different forms of the same constant acceleration motion.
Choosing the right equation
A good strategy is to list the known quantities and identify the unknown one. Then choose the equation that contains the needed variable and avoids variables you do not know.
| If you know | Often use |
|---|---|
| $v_0$, $a$, $t$ | $v = v_0 + at$ |
| $x_0$, $v_0$, $a$, $t$ | $x = x_0 + v_0 t + \frac{1}{2}at^2$ |
| $v_0$, $v$, $t$ | $x = x_0 + \frac{v_0 + v}{2}t$ |
| $v_0$, $a$, $x-x_0$ | $v^2 = v_0^2 + 2a(x-x_0)$ |
This helps prevent unnecessary algebra.
Sign convention matters
These equations work only if you keep a consistent sign convention. In one dimensional motion, you must choose a positive direction first.
If motion to the right is positive, then motion to the left is negative. If upward is positive, then downward is negative. Acceleration must also follow this same convention.
For example, if an object moves upward while gravity acts downward, then $v_0$ may be positive but $a$ is negative.
A common mistake is to insert only positive numbers because they are sizes. In kinematics, direction matters, so signs matter.
Always choose one positive direction and use it consistently for $x$, $v$, and $a$.
A wrong sign can completely change the result, even if the formula is correct.
A visual picture
For constant acceleration, the velocity changes linearly with time, while the position changes in a curved way.
The straight line on the left shows that velocity changes by the same amount each second. The curve on the right shows that position changes faster and faster when the object is speeding up.
Example of use
Suppose a car starts with velocity $v_0 = 10\ \text{m/s}$ and accelerates at $a = 2\ \text{m/s}^2$ for $t = 4\ \text{s}$.
The final velocity is
$$
v = v_0 + at = 10 + 2(4) = 18\ \text{m/s}
$$
The displacement is
$$
x - x_0 = v_0 t + \frac{1}{2}at^2
$$
$$
x - x_0 = 10(4) + \frac{1}{2}(2)(4^2) = 40 + 16 = 56\ \text{m}
$$
So after 4 seconds, the car is moving at $18\ \text{m/s}$ and has traveled $56\ \text{m}$ from its starting position.
When these equations do not apply
These equations are very powerful, but only in the special case of constant acceleration. If acceleration changes with time, then these simple formulas are no longer generally valid.
For example, motion with changing air resistance usually does not have constant acceleration. In such cases, other methods are needed.
Do not use the constant acceleration kinematic equations unless acceleration is constant, or can be treated as constant over the interval of motion.
Final idea
The kinematic equations are a compact toolkit for describing one dimensional motion with constant acceleration. Each equation connects the same core quantities, $x$, $x_0$, $v$, $v_0$, $a$, and $t$, but in a different way. Learning when to use each one is just as important as memorizing the formulas.