Representing Motion

Learning Goals

After this lesson, you should be able to:

• translate among narrative, diagram, table, equation, and graph representations;
• use graph slope to relate position, velocity, and acceleration;
• use signed graph area to obtain displacement or velocity change;
• select and apply a constant-acceleration equation from known quantities;
• check that graphs, equations, signs, and units describe the same motion; and
• qualitatively represent nonuniform acceleration without treating it as constant.

1. Five Views of the Same Motion

No representation is automatically “more real” than the others. They are useful only when their features remain mutually consistent.

2. Read a Motion Diagram

In a strobe-style motion diagram, each dot marks the object's position after the same amount of time. Equal spacing means constant speed. Increasing spacing means increasing speed; decreasing spacing means decreasing speed. Direction comes from the order of the dots or the velocity arrows, not from spacing alone.

An acceleration arrow describes how velocity changes. It need not point in the direction of motion.

3. Slopes Connect the Time Graphs

The slope of a position–time graph is velocity. The slope of a velocity–time graph is acceleration.

For a curve, a secant line gives an interval average, while a tangent line estimates the instantaneous value. A horizontal \(x\)-graph means zero velocity; a horizontal \(v\)-graph means zero acceleration, not zero velocity.

4. Signed Areas Accumulate Change

The signed area between a velocity–time graph and the time axis is displacement. The signed area under an acceleration–time graph is change in velocity.

Area above the time axis is positive and area below it is negative. Adding absolute areas would give distance only when each interval's direction and path are treated correctly; it would not give displacement.

5. Constant-Acceleration Equations

When acceleration is constant and elapsed time is measured from \(t=0\), the following equations describe one-dimensional motion:

Select an equation containing the desired unknown and the known quantities. These are not three unrelated rules: they are compatible consequences of constant acceleration. Do not apply one equation across an interval where acceleration changes.

7. Worked Graph Example: A Reversal

Question: A velocity–time graph is a straight line from \(v_{x0}=-2.0\,\mathrm{m\,s^{-1}}\) at \(t=0\) to \(v_x=+4.0\,\mathrm{m\,s^{-1}}\) at \(t=3.0\,\mathrm s\). Find acceleration and displacement.

The line crosses \(v_x=0\) at \(t=1.0\,\mathrm s\). The first triangular area is negative and the second is positive:

Answer: \(a_x=+2.0\,\mathrm{m\,s^{-2}}\) and \(\Delta x=+3.0\,\mathrm m\). The zero crossing marks a direction reversal, not zero acceleration.

8. The Same Motion in Equation Form

Suppose the previous object begins at \(x_0=5.0\,\mathrm m\). At \(t=3.0\,\mathrm s\):

Thus \(x-x_0=+3.0\,\mathrm m\), matching the signed graph area. Agreement between two independent representations is a powerful check.

9. Vertical Motion Near Earth's Surface

Near Earth's surface, a free object's gravitational acceleration is approximately constant and downward, with magnitude \(g\approx10\,\mathrm{m\,s^{-2}}\) in the AP Physics 1 model. If upward is positive, \(a_y=-g\); if downward is positive, \(a_y=+g\).

A ball moving upward can have \(v_y>0\) while \(a_y<0\). At its highest point, \(v_y=0\) for an instant, but \(a_y\) remains downward. The graph and equation must preserve that distinction.

10. Nonuniform Acceleration: Qualitative Reasoning

If the slope of \(v_x(t)\) changes, acceleration is nonuniform. Describe where the slope is positive, negative, zero, steepening, or flattening, and sketch a compatible \(a_x(t)\). AP Physics 1 expects qualitative analysis of such motion rather than applying one constant-acceleration equation to the entire interval.

Piecewise constant acceleration may be modeled by splitting the motion at every change and carrying the final state of one segment into the next.

11. AP Translation Routine

1. Anchor: mark known values, time interval, origin, and positive direction.
2. Describe: state direction and whether speed increases, decreases, or remains constant.
3. Connect: use slope or signed area to link adjacent representations.
4. Calculate: use an equation only if its assumptions apply.
5. Cross-check: compare signs, units, endpoints, graph shape, and motion diagram.

A correct numerical value paired with an incompatible graph is not a complete physical model.

12. Investigation: Build Three Graphs from One Cart Run

Record a cart moving along a straight track beside a visible meter scale. Extract \(x\) at equal time intervals and include estimated position uncertainty.

1. Plot \(x\) versus \(t\) with labeled units and a sensible scale.
2. Estimate interval velocities from successive position differences and plot them at interval midpoints.
3. Estimate interval accelerations from successive velocity differences.
4. Compare the slope trend of each graph with the next graph.
5. Compare signed velocity-graph area with measured displacement.
6. Identify whether a constant-acceleration equation is justified by the data.

13. Common Traps and Fast Checks

• Trap: reading graph height as slope. Check: identify the vertical variable before interpreting.
• Trap: treating all area as positive. Check: retain the sign relative to the time axis.
• Trap: saying a horizontal \(v(t)\) line means the object is stopped. Check: it means constant velocity; only \(v=0\) means stopped.
• Trap: saying \(v=0\) implies \(a=0\). Check: inspect the slope of \(v(t)\) at that instant.
• Trap: using an unlabeled sketch as quantitative evidence. Check: include variables, units, scale, and interval.
• Trap: using constant-acceleration equations for a curved \(v(t)\) graph. Check: constant acceleration requires constant velocity-graph slope.

14. Key Takeaways

• Motion diagrams, tables, equations, graphs, and words must describe one consistent history.
• Slopes move from position to velocity to acceleration.
• Signed areas move from acceleration to velocity change and from velocity to displacement.
• The three kinematic equations require constant acceleration.
• Graph shape, sign, units, and physical narrative provide independent checks.