Uniform Linear Motion

Motion Without Changing Velocity

Uniform linear motion is motion along a straight path with constant velocity. Constant means the speed stays the same and the direction also does not change.

Picture a small rail cart moving on a straight track. If, during each $1\text{ s}$ interval, it moves $2\text{ m}$ and then another $2\text{ m}$ during the next second, the motion is uniform linear motion.

In uniform linear motion, the object may keep moving. What stays unchanged is its velocity.

\begin{aligned} \Delta x &= v\Delta t \\ x(t) &= x(t_0)+v(t-t_0) \end{aligned}

The symbols $x(t_0)$ and $x(t)$ are the positions at the initial and final times, $v$ is the constant velocity, $\Delta t=t-t_0$ is elapsed time, and $\Delta x$ is displacement.

Position Marks With Equal Spacing

If an object's position is recorded at equal time intervals, uniform linear motion produces equally spaced marks. This is like a ticker tape: marks are made regularly, so the gaps between marks show how far the object moves during each time interval.

Suppose the initial position is $x_0=1\text{ m}$ and the velocity is $+2\text{ m/s}$ . That means the position increases by $2\text{ m}$ every second.

Time	Position
$0\text{ s}$	$1\text{ m}$
$1\text{ s}$	$3\text{ m}$
$2\text{ s}$	$5\text{ m}$
$3\text{ s}$	$7\text{ m}$
$4\text{ s}$	$9\text{ m}$
$5\text{ s}$	$11\text{ m}$

The position value changes, while the difference between consecutive positions stays equal. From $1$ to $3$ , from $3$ to $5$ , and so on, the difference is always $2\text{ m}$ .

Uniform Motion Car Marks

Change the constant velocity to see the distance between marks stay equal every second.

Constant velocity: $4\text{ m/s}$
Mark interval: $1\text{ s}$
Distance between marks: $4\text{ m}$
Motion time: $5\text{ s}$

Calculating Final Position

Suppose an object starts from $x_0=1\text{ m}$ and moves right with constant velocity $v=2\text{ m/s}$ for $5\text{ s}$ . Its displacement is:

\begin{aligned} \Delta x &= v\Delta t \\ &= 2(5) \\ &= 10\text{ m} \end{aligned}

Its final position is:

\begin{aligned} x_t &= x_0+\Delta x \\ &= 1+10 \\ &= 11\text{ m} \end{aligned}

This calculation works because the velocity does not change. If the velocity changes, we cannot use one fixed value of $v$ for the whole motion.

Recognizing Uniform Motion on Graphs

Two graphs identify uniform linear motion. The velocity-time graph must be horizontal, while the position-time graph must be a straight line with constant slope.

Graph	Shape in uniform linear motion	Meaning
Velocity against time	Horizontal line	Velocity is constant
Position against time	Slanted straight line	Position changes regularly

The signed area between a velocity-time graph and the time axis equals displacement. For constant velocity, that area is a rectangle, so $\Delta x=v\Delta t$ . An area below the time axis contributes negative displacement.

On a position-time graph, the slope of the line represents velocity. The sign of the slope gives direction, while a larger absolute steepness means a larger speed. For uniform linear motion, the line stays straight because the slope does not change from start to finish.

Direction Distinguishes Constant Speed from Constant Velocity

Uniform linear motion needs constant velocity. Both the size and direction of motion must stay fixed. Constant speed only fixes the size.

A car moving with constant speed on a circular road is not in uniform linear motion, because the direction of its velocity keeps changing. A rail cart moving straight with constant speed and direction is in uniform linear motion.