Uniform Motion

What this formula says

A body is at position $x_0$ at time $t = 0$. It moves with a constant velocity $v$. After time $t$, its position is:

$x = x_0 + vt$

Its new position is its starting position plus how far it has travelled. That is all.

What uniform motion means

Uniform motion means the velocity is constant — same magnitude, same direction, at every instant. This requires zero net force and no change in direction.

A car cruising on a straight highway at a steady speed is approximately uniform motion. A planet in a circular orbit moves at constant speed but is not uniform motion — its direction changes continuously.

Derivation

Velocity is defined as the rate of change of position:

$v = \frac{dx}{dt}$

Since $v$ is constant, separate variables:

$dx = v \, dt$

Integrate both sides — position from $x_0$ to $x$, time from $0$ to $t$:

$\int_{x_0}^{x} dx = \int_0^t v \, dt$

Left side: $x - x_0$

Right side ($v$ constant, comes outside): $vt$

$x - x_0 = vt$

$\boxed{x = x_0 + vt}$

Without calculus

If you travel at constant velocity $v$ for time $t$, distance covered is $vt$. Add this to starting position:

$\text{final position} = \text{initial position} + \text{distance covered} = x_0 + vt$

Sign convention

Example: Body starts at $x_0 = -5$ m, moves at $v = +3$ m/s for $t = 4$ s:

$x = -5 + 3 \times 4 = +7 \text{ m}$

Started 5 m left of origin, ended 7 m right of it.