5.1 Describing Motion: Kinematics and Motion Graphs

Why Kinematics Gets Its Own Section

Motion is the entry point to Domain II, and the framework asks specifically that teachers can "measure, graph and describe" changes in motion. Exam items in this space are heavily graph-based, because interpreting a motion graph is the skill that distinguishes a teacher who can diagnose student thinking from one who can only recite a formula. The single most common student error — reading a distance-time graph as if it were a picture of the path — is directly testable.

Scalars Versus Vectors

Distance is always greater than or equal to the magnitude of displacement. A student who walks 3 m east, then 4 m north has traveled a distance of 7 m but has a displacement of 5 m northeast (the 3-4-5 right triangle). Exam items exploit this pairing constantly.

Speed, Velocity, and Acceleration

Average speed = total distance ÷ total time. A bus that covers 240 km in 3 h averages 80 km/h even if it stopped twice; average speed says nothing about what happened in between. Instantaneous speed is the reading on a speedometer at one moment.

Acceleration is defined as a = Δv / Δt = (v_f − v_i) / t, with units of m/s². Three situations all count as acceleration:

• Speeding up — a bicycle going from 2 m/s to 8 m/s in 3 s accelerates at (8 − 2)/3 = +2 m/s².
• Slowing down — a car going from 20 m/s to 0 in 4 s accelerates at (0 − 20)/4 = −5 m/s². Negative acceleration in this case means deceleration; it does not automatically mean "moving backward."
• Changing direction — a car circling a track at a constant 15 m/s has changing velocity because the direction changes, so it is accelerating even though the speedometer never moves.

The Constant-Acceleration Equations

For motion with uniform acceleration:

1. v = v₀ + at
2. Δx = v₀t + ½at²
3. v² = v₀² + 2aΔx

Worked example. A skateboarder starts from rest and accelerates at 1.5 m/s² for 4.0 s. How fast is she going, and how far has she traveled?

• v = 0 + (1.5)(4.0) = 6.0 m/s
• Δx = 0 + ½(1.5)(4.0²) = ½(1.5)(16) = 12 m

Check with equation 3: v² = 0 + 2(1.5)(12) = 36, so v = 6.0 m/s. Consistency across two routes is a good habit to model for students.

Free Fall

Near Earth's surface, gravity accelerates all objects downward at g ≈ 9.8 m/s² regardless of mass, once air resistance is negligible. A rock dropped from rest is moving at 9.8 m/s after 1 s, 19.6 m/s after 2 s, and 29.4 m/s after 3 s. It falls ½(9.8)(1²) = 4.9 m in the first second but ½(9.8)(2²) − 4.9 = 14.7 m during the second second, because distance grows with t².

Air resistance is what breaks the pattern in a real classroom. A flat sheet of paper and a book dropped together do not land together; crumple the paper and they nearly do. As a falling object speeds up, air resistance grows until it equals the weight; at that point the net force is zero, acceleration stops, and the object descends at constant terminal velocity. This is a favorite item stem because students expect "heavier falls faster" and the correct explanation names air resistance rather than mass.

Reading Motion Graphs

Distance-time (position-time) graphs — slope is speed.

Velocity-time graphs — slope is acceleration, area is displacement.

The area under a velocity-time graph gives displacement. For an object moving at a constant 6 m/s for 5 s, the area is a rectangle: 6 × 5 = 30 m. For an object accelerating uniformly from 0 to 6 m/s over 5 s, the area is a triangle: ½(5)(6) = 15 m.

The Misconception to Plan For

Middle-school students routinely read a distance-time graph as a map of the path. Shown a distance-time graph with a rising straight line, they say "the object went up a hill." The instructional fix is to have students physically walk a graph using a motion detector: the graph is drawn in real time as they move toward and away from the sensor, which makes the axes' meaning concrete before any equations appear.