2.1 Motion in One Dimension

Kinematics in One Dimension

Kinematics in one dimension describes motion along a straight line. It gives you the tools to talk precisely about where something is, how far it's moved, and in which direction. These concepts are the foundation for everything else in mechanics.

Position-time graphs and straightforward calculations let you visualize and quantify that motion. Once you can read these graphs and work through the math, you can predict where an object will be in the future or reconstruct where it's been.

Distance vs. Displacement

These two terms sound similar but mean very different things. Mixing them up is one of the most common early mistakes in physics.

Distance is a scalar quantity. It measures the total path length traveled, no matter which direction you went. It's always positive or zero, and it depends on the path you took. Think of it like an odometer reading on a car.

Displacement is a vector quantity. It measures only the straight-line difference between your final position and your initial position. It can be positive, negative, or zero, and it doesn't care about the path you took. Think of it like the difference between two GPS coordinates.

The relationship between them: distance is always greater than or equal to the magnitude of displacement. They're only equal when you move in a straight line without ever changing direction. For example, if you walk 3 blocks north and then 1 block south, your distance is 4 blocks, but your displacement is only 2 blocks north.

Frame of Reference

A frame of reference is the coordinate system you choose to describe position and motion. It can be fixed (like the ground) or moving (like the inside of a car). Every measurement of position, velocity, or acceleration depends on which frame you're using.

This matters because the same motion looks different from different frames. A passenger walking forward at 1 m/s inside a train moving at 20 m/s has a velocity of 1 m/s relative to the train, but 21 m/s relative to the ground. Neither answer is "wrong"; they just use different reference frames. Picking a clear frame of reference before you start a problem keeps your signs and values consistent.

Calculations for One-Dimensional Motion

Position is represented by a coordinate $x$, measured relative to an origin at $x = 0$. You can think of it like a number line: positions to the right of the origin are positive, positions to the left are negative.

Sign conventions: By default, rightward (or upward) is the positive direction, and leftward (or downward) is negative. You can choose differently, but you must stay consistent throughout a problem.

Displacement is the difference between final and initial positions:

$\Delta x = x_f - x_i$

If you start at $x_i = 2 \text{ m}$ and end at $x_f = -3 \text{ m}$, your displacement is $-5 \text{ m}$. The negative sign tells you the direction.

Distance is the sum of the absolute values of each individual segment of motion:

$\text{Distance} = |x_1 - x_0| + |x_2 - x_1| + \ldots$

Using the same example, if you first move from $x = 2 \text{ m}$ to $x = 5 \text{ m}$, then back to $x = -3 \text{ m}$, your distance is $|5 - 2| + |-3 - 5| = 3 + 8 = 11 \text{ m}$, while your displacement is still $-5 \text{ m}$.

Position-Time Graph Interpretation

Position-time graphs plot time on the horizontal axis and position on the vertical axis. They're one of the most useful tools for understanding one-dimensional motion at a glance.

How to read the slope:

• Positive slope (line going up to the right): the object is moving in the positive direction.
• Negative slope (line going down to the right): the object is moving in the negative direction.
• Zero slope (horizontal line): the object is at rest.

The slope of a position-time graph gives you the velocity:

$v = \frac{\Delta x}{\Delta t}$

A steeper slope means a faster speed. A straight line means constant velocity. A curved line means the velocity is changing, which indicates acceleration.

To create a position-time graph, plot the object's position at each recorded time, then connect the points. The shape of the resulting line tells you the full story of the motion.

One related idea: on a velocity-time graph (which you'll encounter soon), the area under the curve represents displacement. Don't confuse this with the position-time graph, where it's the slope that gives you velocity.