Motion Graphs & Reference Frames

The question behind the experiment

The same cart can be described by position, velocity, and acceleration graphs, but those graphs encode different questions. Position tells where the cart is relative to a chosen origin. Velocity describes how position changes, including direction. Acceleration describes how velocity changes. This investigation links the graphs to a cart moving on a track and asks you to predict before revealing relationships. Start by stating the positive direction and origin so the signs have a consistent meaning.

Plan a useful comparison

Compare a launched cart with a cart released from rest while keeping the track tilt explicit. Record the initial condition with each trace. A cart traveling uphill can slow down even though its velocity still points uphill; later it can move downhill under an acceleration with the same sign. Inspect the turning point as part of the complete trace rather than drawing a conclusion from a still image. Being momentarily stationary does not by itself establish zero acceleration.

Connect the observation to the model

Use a tangent on the position-time graph to reason about velocity and the slope of the velocity-time graph to reason about acceleration. The area under velocity versus time represents displacement, with signs included. Contrast that with total distance traveled when the cart reverses direction. A round trip can cancel displacement while accumulating distance. Relate every area or slope to the interval you selected, because an average over a long interval and an instantaneous quantity need not agree.

Interpret the result carefully

The incline provides a test of constant-acceleration relationships. Compare launches with different initial speeds at the same tilt and in the same lab mode. Measure the uphill distance before the cart turns, and consider plotting that distance against the square of the launch speed. Keep collisions with track stops separate from uninterrupted motion. An end stop changes the physical situation, so a run that reaches it cannot be interpreted as an ordinary free turning point on the incline.

Build an explanation from your evidence

The moving-observer comparison changes the velocity record without necessarily changing the acceleration. State whether a graph belongs to the track observer or the moving dolly. Subtracting the observer's steady speed gives a different measured cart velocity in that reference frame. The course also distinguishes sampled sensor readings from exact model quantities. Near a launch or impact, a sampling interval can contain an event, so interpret sharp graph features with the measurement interval and event timing attached.

A reference case from this course

For an Ideal launch at a 3.0° track tilt, the course reference acceleration is −0.5128923712 m/s². Its sign follows the selected uphill positive direction. Use the reference as a check on a matched setting, and distinguish this model value from a rounded or interval-based sensor reading near a launch event.

Keep the investigation reproducible

Before changing the setup, write a prediction that names the measured quantity and the expected direction of change. Retain the original trial as a reference. Record the apparatus conditions, units, and whether you used Ideal or Real lab mode. If the selected activity includes seeded variation, keep the seed and run identity with the result so another comparison does not silently mix different conditions.

Separate the observation from its explanation. A reading or a counted object belongs in the evidence; the relationship you use to interpret it belongs in the reasoning. When a result differs from the reference case, check the settings and measurement procedure before deciding that the model relationship has failed. Describe a specific source of uncertainty and where it enters your calculation. Finish with a claim supported by the recorded comparison, and identify an additional trial that could distinguish your explanation from an alternative.

Continue the comparison

Connect the changing acceleration conditions to Newton’s Second Law, or separate horizontal and vertical graph reasoning in Projectile Motion. Keep the coordinate convention visible when moving between the investigations.