Kinematics

Choose an interval before selecting an equation. Motion before and after an impact usually requires separate calculations because velocity can change abruptly during contact. A curved velocity–time graph also signals changing acceleration, so its signed area or a numerical method is more suitable than one constant-a substitution. Constant acceleration does not require constant speed or motion in the positive direction. A ball thrown upward has downward acceleration throughout its ideal flight, and the same vertical constant-a equations can describe both its ascent and descent when signs are consistent.

The useful connection between the two columns is time, not equality of displacement or speed. A horizontal launch starts with zero vertical velocity, whereas an angled launch generally does not. State a vertical sign convention and keep the launch and landing heights distinct. The familiar range formula assumes equal launch and landing levels; it should not be used unchanged for a launch from a balcony. If a time equation gives two positive roots, the projectile may pass a chosen height twice, so identify which event the question describes.

Electronic release and detection reduce human start–stop timing uncertainty, but the distance must still match the points at which the timing begins and ends. A delay in release or an initial push introduces a systematic mismatch with the assumed initial conditions. Use several distances and inspect the shape of the graph rather than relying on one calculated value of g. Longer fall times can reduce percentage timing uncertainty, although the apparatus and safety limits matter. Repeated readings estimate scatter; they do not by themselves identify a consistently incorrect distance origin.

Begin by separating the graph at every crossing of the time axis. Calculate the area of each section using a sign for displacement and no sign for distance. A horizontal segment below the axis represents constant negative velocity, not negative speed or continuing negative acceleration. If the graph has a positive gradient there, the object is slowing while moving in the negative direction. These distinctions are easier to follow when you imagine an origin and label motion to the right positive before reading any numbers.

The force of gravity depends on the local field and the object’s mass, not on whether the object is momentarily moving upward or downward. As a result, velocity passes smoothly through zero at the top while its rate of change remains downward. For a purely vertical throw the total speed is zero there; for an angled throw only the vertical component vanishes. Confusing these two cases can give a wrong kinetic energy at the highest point. Acceleration would be zero only if the resultant force were zero, which free-flight gravity does not satisfy.

Draw a tangent that follows the local slope, then choose two well-separated points on that line to calculate rise divided by run. They do not need to be measured data points, and they should not be selected from different parts of the original curve. Include axis scale factors when converting the gradient into units. A large gradient triangle reduces the relative effect of reading the two coordinates, but it cannot correct a poorly drawn tangent. If the tangent slopes downward, preserve the negative sign because it describes the direction of motion.