Motion analysis - Calculus AB AP Study Notes

Motion analysis uses derivatives to examine how objects move through space over time. In calculus, we analyze motion through three fundamental functions:

Position function s(t) or x(t): describes the location of an object at time t, measured from a reference point (origin). Units are typically meters, feet, or any distance measure.

Velocity function v(t): the first derivative of position, v(t) = s'(t) = ds/dt. Velocity indicates both speed (magnitude) and direction (sign). Positive velocity means moving in the positive direction; negative velocity indicates motion in the opposite direction. Units: m/s, ft/s.

Speed: the absolute value of velocity, |v(t)|, representing how fast an object moves regardless of direction.

Acceleration function a(t): the first derivative of velocity or the second derivative of position, a(t) = v'(t) = s''(t) = d²s/dt². Acceleration measures the rate of velocity change. Positive acceleration can mean speeding up (if v > 0) or slowing down while moving backward (if v < 0). Units: m/s², ft/s².

Key Memory Aid - DVD: Displacement → Velocity (differentiate) → Acceleration (differentiate again)

Critical Relationships:

• Object at rest: v(t) = 0
• Object changes direction: v(t) = 0 and changes sign
• Object speeding up: v(t) and a(t) have the same sign
• Object slowing down: v(t) and a(t) have opposite signs
• Total distance = ∫|v(t)|dt (not just position change!)

Particle motion on a line is the most common exam scenario, where position is one-dimensional.

Motion analysis appears everywhere in physics and engineering. Consider a lift (elevator) in a building:

Position s(t): If ground floor is s = 0, the 3rd floor might be s = 9m. The position function tells you which floor you're on at any moment.

Velocity v(t): When v(t) = 2 m/s, you're moving upward at 2 m/s. When v(t) = -1.5 m/s, you're descending at 1.5 m/s. The feeling of "going up" or "going down" is velocity.

Acceleration a(t): That stomach-dropping feeling when the lift starts? That's positive acceleration (increasing upward velocity). When the lift slows to stop at your floor, you feel slightly heavier—that's negative acceleration (deceleration) while moving upward.

Crucial insight: A car with v(t) = -20 m/s and a(t) = 5 m/s² is moving backward but slowing down (since signs oppose). Within 4 seconds, it might stop and reverse direction.

Real-world context - Rocket Launch: At liftoff, s(0) = 0 (launchpad), v(0) = 0 (initially at rest), but a(0) > 0 (engines firing create upward acceleration). As time progresses, both position and velocity increase, with varying acceleration as fuel burns.

Analogy: Think of position as your location on a number line, velocity as your speed reading on a speedometer (with + or - for direction), and acceleration as pressing the gas pedal (+) or brake (-).

Speed vs Velocity distinction: A runner completing a 400m circular track returns to start. Displacement = 0, but distance traveled = 400m. This mirrors how ∫v(t)dt gives displacement while ∫|v(t)|dt gives total distance.