Physics 1A: Motion in One Dimension (Kinematics)

Position, displacement, and the difference that matters

Physics describes motion using signed quantities. Where an object is at any moment is called its position — a location on a chosen axis. The change in position from a starting point to an ending point is displacement. Displacement is a vector: it has both a magnitude (how far) and a direction (which way).

Contrast this with distance, which is a scalar: it only records how much ground was covered, ignoring direction. If a runner jogs 100 m north and then 100 m south back to the starting point, their distance traveled is 200 m — but their displacement is zero, because they ended where they began.

The CBE will phrase questions carefully to test whether you spot the difference. "How far did the car travel?" typically asks about distance. "What is the car's displacement?" asks about the net change with a sign. When in doubt, ask: does direction matter? If yes, use displacement.

Speed vs velocity, average vs instantaneous

Two more scalar/vector pairs to keep straight:

• Speed = distance ÷ time — a scalar. It tells you how fast, not which way.
• Velocity = displacement ÷ time — a vector. It tells you how fast AND which way.

Then within each, two variants:

• Average velocity = total displacement ÷ total time. This flattens everything that happened in between into a single number.
• Instantaneous velocity = the velocity at one specific moment. On a position-vs-time graph, it is the slope of the tangent line at that point.

Consider a car that accelerates from rest, cruises at highway speed, and slows to a stop. Its instantaneous velocity varies wildly during the trip. Its average velocity is just total distance divided by total time — usually a much smaller number than the peak instantaneous velocity. Both are useful; they answer different questions.

Acceleration — the rate of change of velocity

Acceleration is how quickly velocity changes:

a = (v_f − v_i) / t

Units are m/s² (meters per second per second). If a car goes from rest to 30 m/s in 6 seconds, its average acceleration is 5 m/s² — meaning velocity increased by 5 m/s during every second of the trip.

Three subtleties the CBE tests:

• Negative acceleration is not the same as slowing down. An object moving in the negative direction with a negative acceleration is speeding up. Signs describe direction, not intent. Slowing down happens when acceleration points opposite to velocity — whatever the signs.
• Zero acceleration ≠ zero velocity. A car cruising at 25 m/s has velocity but no acceleration. This is Newton's first law playing out in real time.
• Constant velocity means straight line on a position-time graph. Constant acceleration means straight line on a velocity-time graph. If you can match a description in words to the correct graph shape, you have half the CBE motion questions handled already.

The four kinematic equations

For an object undergoing constant acceleration in one dimension, four equations relate the five variables (initial velocity v_i, final velocity v_f, displacement Δx, acceleration a, time t). You will use all four. They are:

1. v_f = v_i + at — final velocity from initial velocity, acceleration, and time. No displacement involved.
2. Δx = v_i·t + ½at² — displacement from initial velocity, acceleration, and time. No final velocity involved.
3. v_f² = v_i² + 2a·Δx — final velocity from initial velocity, acceleration, and displacement. No time involved. This is the one to reach for when a problem does not mention time.
4. Δx = ½(v_i + v_f)·t — displacement from average of initial and final velocity and time. No acceleration involved. Useful when acceleration is unknown but both velocities are given.

Strategy: read the problem, list which variables you have, and pick the equation that excludes the one you cannot get. Do not memorize them independently; recognize that each equation omits exactly one variable, and match to the problem.

A specific case that trips up students: free fall. When an object falls near Earth's surface with air resistance negligible, its downward acceleration is g = 9.8 m/s² (or often approximated as 10 m/s² for quick estimates). The kinematic equations apply directly — you just substitute a = g and be careful about signs (usually define downward as positive or negative consistently throughout a problem, and stick with it).

Motion graphs, read like a language

Motion is often given to you as a graph rather than in words. Two graphs matter for one-dimensional motion:

• Position vs time (x-t graph)
  • Slope of the line at any point = instantaneous velocity.
  • Straight line = constant velocity.
  • Curved line = acceleration (velocity is changing).
  • Flat horizontal line = object is stationary.
  • Line going up = object moving in positive direction; line going down = moving in negative direction.
• Velocity vs time (v-t graph)
  • Slope of the line at any point = instantaneous acceleration.
  • Straight sloped line = constant acceleration.
  • Horizontal line = constant velocity, zero acceleration.
  • Area between the line and the time axis = displacement during that interval.
  • Where the line crosses zero = the moment the object is momentarily at rest or reversing direction.

The area-under-a-v-t-curve rule is the single most tested graph fact on the CBE. Any time you see a v-t graph and the question asks about how far the object went, you compute the area of the region under the curve. Rectangular regions use base × height. Triangular regions use ½ × base × height. Combined shapes: break into rectangles and triangles and add.

Frames of reference — motion depends on who is watching

All the motion quantities above are measured relative to a chosen reference frame. A passenger on a moving train who walks forward at 1 m/s inside the train is:

• Moving at 1 m/s relative to the train.
• Moving at 26 m/s relative to the ground (if the train is going 25 m/s).
• Moving at 0 m/s relative to their own coffee, which is walking with them.

The physics is consistent in every frame — no experiment inside the train can tell whether the train is stationary or moving smoothly. This is a foundational idea that leads to Einstein's relativity, but for the CBE it shows up as questions like: "The velocity of A relative to B is +5 m/s. B is moving at −3 m/s relative to the ground. What is A's velocity relative to the ground?" Answer: +5 + (−3) = +2 m/s. Add the relative velocities as vectors along the line of motion.

A worked example — piece by piece

A car accelerates uniformly from 12 m/s to 30 m/s over a distance of 84 m. How long does the acceleration take?

List what you have and what you need:

• v_i = 12 m/s, v_f = 30 m/s, Δx = 84 m — these are given.
• a = ? and t = ? — these are unknown. But you only need t.

Which equation excludes acceleration? Equation 4: Δx = ½(v_i + v_f)·t.

Rearrange: t = 2·Δx / (v_i + v_f) = 2(84) / (12 + 30) = 168 / 42 = 4.0 s.

Notice you never needed acceleration. Picking the right equation saves a step.

Check yourself

Verify each item quickly before moving on:

1. Draw a position-time graph for an object at rest. What shape is it? Draw one for constant positive velocity. Now negative velocity.
2. On a velocity-time graph, sketch an object that starts fast, slows to a stop, and then reverses direction. What sign changes happen and where?
3. State the four kinematic equations from memory. For each, identify which variable is missing.
4. An object is thrown straight up. What is its acceleration at the highest point? (Trap question — think carefully.)
5. A boat crosses a river relative to the water at 3 m/s. The current is 2 m/s downstream. Describe the boat's velocity in the ground frame.

Answer to #4: the acceleration is still g = 9.8 m/s² downward, even at the moment velocity is zero at the peak. Acceleration is the rate of velocity change, not the current velocity. Missing this is one of the most common conceptual errors on the CBE.

Practice with CBE-style questions

Kinematics is the largest single topic in Physics Semester A. Work through the practice bank filtered by Motion (Kinematics) — the questions cover 1D and 2D kinematics, graph reading, and the kinematic equations. Every question has a step-by-step solution.

Independent practice content aligned to Texas Essential Knowledge and Skills (TEKS) §112.39(c)(4). Not affiliated with TTU K-12, UT High School, UT-Austin, the Texas Education Agency, or any Credit by Examination administrator. Texas CBE™ does not administer any exam or grant academic credit.