1D motion with constant acceleration

Motion along one axis

One-dimensional kinematics describes motion along a single line. The line might be horizontal, vertical, or along a track, but once the coordinate axis is chosen, position can be represented by a single function $x(t)$. Velocity and acceleration are defined by derivatives:

v(t)=rac{dx}{dt}

a(t)=rac{dv}{dt}=rac{d^2x}{dt^2}

These equations are more fundamental than the memorized constant-acceleration formulas. They say that velocity is the rate of change of position, and acceleration is the rate of change of velocity.

Constant acceleration

A special and important case occurs when acceleration is constant. Suppose

$a(t)=a$

where $a$ is a constant. Since acceleration is the derivative of velocity, integrate with respect to time:

$v(t)=int a,dt = at+C$

The constant $C$ is determined by the initial velocity. If $v(0)=v_0$, then $C=v_0$, so

$v(t)=v_0+at$

Now use velocity as the derivative of position:

rac{dx}{dt}=v_0+at

Integrating again gives

x(t)=x_0+v_0t+rac{1}{2}at^2

where $x_0$ is the initial position.

Displacement form

The displacement is

$Delta x = x-x_0$

so the position equation can be written

Delta x=v_0t+rac{1}{2}at^2

This form is useful when the initial position is not important by itself.

Another equation relates velocity and displacement without time. Start with

a=rac{dv}{dt}

Using the chain rule,

rac{dv}{dt}=rac{dv}{dx}rac{dx}{dt}=vrac{dv}{dx}

So

a=vrac{dv}{dx}

Separate and integrate:

$int_{v_0}^{v} v,dv = int_{x_0}^{x} a,dx$

This gives

rac{1}{2}(v^2-v_0^2)=a(x-x_0)

or

$v^2=v_0^2+2aDelta x$

Signs and coordinate choices

In one-dimensional motion, sign matters. If upward is chosen positive, then gravitational acceleration near Earth's surface is approximately

$a=-gapprox -9.8 m/s^2$

If downward is chosen positive, then $a=+g$. Neither choice is wrong, but the choice must be used consistently.

Velocity can be positive or negative depending on direction. Acceleration can be negative even while an object is moving upward or downward. For example, a thrown ball moving upward has positive velocity if up is positive, but its acceleration is negative because gravity points downward.

Free fall

Free fall near Earth is a common constant-acceleration case when air resistance is ignored. The acceleration is approximately constant and downward. The motion of a tossed ball, a dropped object, or a falling stone can be modeled with the same equations.

At the highest point of a vertical throw, the velocity is instantaneously zero, but acceleration is not zero. Gravity is still acting, and the acceleration remains $-g$ if up is positive.

When the formulas apply

The standard equations apply only when acceleration is constant. If acceleration depends on time, position, or velocity, the motion must be handled with more general calculus methods or numerical methods.

For example, air resistance often depends on speed, so falling with drag is not a constant-acceleration problem. A car accelerating with changing engine force may also require more advanced modeling.

The big idea

Constant-acceleration formulas are not isolated facts. They come from integrating acceleration to get velocity and integrating velocity to get position. Understanding their derivation helps you use them correctly, interpret signs, and recognize when they do or do not apply.