Motion Graphs and Kinematics
Key Takeaways
- Motion questions on the Physical Science: Physics Regents often ask for a model from a graph, table, or cluster context before any substitution is useful.
- Slope and area are not generic graph features: position-time slope gives velocity, velocity-time slope gives acceleration, and velocity-time area gives displacement.
- The 2025 Physics Reference Tables include constant-acceleration relationships, but students must decide whether constant acceleration is a reasonable model.
- Projectile situations separate horizontal and vertical motion: horizontal velocity stays constant without air resistance, while vertical velocity changes because of gravity.
- Direction, units, and sign conventions are part of the answer for velocity, acceleration, displacement, and projectile calculations.
Why Motion Models Matter
Motion is one of the clearest places where the Physical Science: Physics Regents tests reasoning instead of formula hunting. A cluster may give a stopwatch table, a video-analysis graph, or a real performance context such as a jump or rolling cart. The first job is to identify the object, time interval, direction convention, and kind of motion.
The 2025 Physics Reference Tables give relationships for average speed or velocity, acceleration, and constant-acceleration motion. They are tools, not instructions. Before using vf = vi + at or d = vit + 1/2 at^2, decide whether the motion is modeled with constant acceleration over the interval.
Distance, Displacement, Speed, and Velocity
Use distance for total path length and displacement for change in position from start to finish. Distance has no direction. Displacement does. That difference is a common Regents trap when an object moves forward, stops, and comes back.
| Quantity | What it uses | Direction? | Typical Regents cue |
|---|---|---|---|
| Average speed | total distance / time | No | path length, odometer, how fast overall |
| Average velocity | displacement / time | Yes | final position, east-west, up-down |
| Acceleration | change in velocity / time | Yes | speeding up, slowing down, turning, free fall |
| Displacement | final position - initial position | Yes | area under velocity-time graph |
If a runner travels 30 m east and then 10 m west, the distance is 40 m, but displacement is 20 m east. A speed calculation uses 40 m. A velocity calculation uses 20 m east.
Reading Position-Time Graphs
On a position-time graph, slope represents velocity. A straight slanted line means constant velocity. A horizontal line means position is not changing, so velocity is zero. A curved line means velocity is changing because the slope changes.
The sign of the slope matters. A positive slope means motion in the positive direction. A negative slope means motion in the negative direction. A steeper line has greater speed because the magnitude of the velocity is larger.
For constructed response, do not write only that the graph goes up. State the evidence: the position increases at a constant rate, so the velocity is constant and positive. That wording connects the graph feature to the physics claim.
Reading Velocity-Time Graphs
Velocity-time graphs carry two different meanings. The slope is acceleration. The area between the graph and the time axis is displacement. If the graph crosses the time axis, areas above and below the axis have opposite signs.
A horizontal line above zero means constant positive velocity and zero acceleration. A sloped line means acceleration. A line below zero means the object is moving in the negative direction, even if its speed may be increasing or decreasing.
| Velocity-time feature | Meaning |
|---|---|
| Positive slope | positive acceleration |
| Negative slope | negative acceleration |
| Area above axis | positive displacement |
| Area below axis | negative displacement |
| Larger magnitude of velocity | greater speed |
A Regents item may ask for displacement from a trapezoid or triangle under a velocity-time graph. Use geometry carefully and keep the sign if the area is below the axis.
Constant-Acceleration Equations
The reference tables list the usual constant-acceleration relationships: acceleration from velocity change over time, final velocity from initial velocity plus at, displacement from initial velocity and time, and the relationship involving vf^2, vi^2, acceleration, and displacement.
Choose the equation by the missing quantity, not by which equation looks familiar. If time is not given, the vf^2 relationship may be cleaner. If displacement is not given, the final-velocity relationship may be enough.
Write known values with units before substituting. A vertical motion problem near Earth often uses a = 9.8 m/s^2 downward. If upward is positive, the acceleration is negative. If downward is positive, it is positive. The physics is the same, but the sign convention must be consistent.
Projectile Motion in Clusters
Projectile motion separates into horizontal and vertical components when air resistance is ignored. Horizontal velocity remains constant because no horizontal net force is modeled. Vertical velocity changes because gravity provides a downward acceleration.
The sampler-style cluster approach may provide time in air, horizontal speed, or vertical height. For horizontal distance, use horizontal velocity times time. For vertical motion, use the acceleration due to gravity and the appropriate constant-acceleration model.
A common mistake is to use the total launch speed where only a component is needed. If a table gives horizontal and vertical velocity components separately, keep them separate until the prompt asks for a resultant.
Constructed-Response Routine
For a motion calculation, use this routine:
- Define positive direction.
- List known values with units.
- Select a relationship from the reference tables or graph meaning.
- Substitute with signs and units.
- Report the answer with a unit and direction when needed.
- Tie the number back to the motion claim.
This routine also helps with evidence questions. If a prompt asks whether data support a claim, calculate the relevant slope, area, or acceleration, then compare it directly to the claimed value.
Final Motion Checks
Ask three questions before moving on. Does the unit match the requested quantity? Does the sign or direction match the model? Does the answer make sense compared with the graph or table? A displacement larger than the maximum possible path length, or an upward acceleration during ordinary free fall, usually signals a setup error.
Motion questions reward careful reading. The graph, table, and text are part of the model. Use them before choosing the formula.
A position-time graph for a sensor cart rises in a straight line from 0 to 4 seconds, is horizontal from 4 to 6 seconds, and then falls in a steeper straight line from 6 to 8 seconds. Which interpretation is best supported?