AP Calculus: Motion Along a Line (Position, Velocity, Acceleration, Speed)

Motion Along a Line (Position, Velocity, Acceleration, Speed)

Concept Summary

• Position (s(t)): A function s(t) giving an object’s location on a number line at time t; the foundation for all motion analysis.
• Velocity (v(t)): The derivative of position, v(t) = s'(t), indicating both speed and direction of motion.
• Acceleration (a(t)): The derivative of velocity (second derivative of position), a(t) = v'(t) = s''(t), describing how velocity changes over time.
• Speed: The absolute value of velocity, |v(t)|, representing how fast an object moves regardless of direction.
• Displacement: The net change in position, s(b) – s(a), over a time interval [a, b]; distinct from total distance traveled.

Core Questions

WHAT (definitional)

Q: What is instantaneous velocity? A: The derivative of position at a specific time, v(t) = s'(t), representing the object’s speed and direction at that exact moment. Trap/Clarification: Instantaneous velocity-average velocity; the latter is ?s/?t over an interval.

Q: What does negative velocity indicate? A: The object is moving in the negative direction of the chosen coordinate system (e.g., left or downward). Trap/Clarification: Negative velocity-slowing down; acceleration determines speed changes.

WHY (causal/explanatory)

Q: Why is acceleration the derivative of velocity? A: Acceleration measures how fast velocity changes, so it’s the rate of change (derivative) of velocity with respect to time. Trap/Clarification: Constant velocity-zero acceleration; a(t) = 0 only if velocity isn’t changing.

Q: Why is speed the absolute value of velocity? A: Speed is a scalar (magnitude only), while velocity is a vector (magnitude + direction); absolute value removes direction. Trap/Clarification: Speed can increase while velocity decreases (e.g., moving left faster).

HOW (process/application)

Q: How do you find when an object changes direction? A: Solve v(t) = 0 and check sign changes of v(t) around those times; direction changes where velocity crosses zero. Trap/Clarification: v(t) = 0 alone doesn’t guarantee a direction change (e.g., v(t) = t² at t = 0).

Q: How is total distance traveled calculated? A: Integrate |v(t)| over the interval: ? |v(t)| dt, or sum absolute displacements between direction changes. Trap/Clarification: Displacement (?v(t) dt)-total distance unless v(t) never changes sign.

Q: How do you determine if an object is speeding up or slowing down? A: Compare signs of v(t) and a(t): - Same sign-speeding up. - Opposite signs-slowing down. Trap/Clarification: Zero acceleration-zero speed; it means constant velocity.

CAN (conditions/possibilities)

Q: Can an object have zero velocity but non-zero acceleration? A: Yes; e.g., a ball at its peak height (v = 0) under gravity (a = –9.8 m/s²). Trap/Clarification: Zero velocity-zero acceleration; acceleration is the rate of change of velocity.

Q: Under what conditions is displacement equal to total distance traveled? A: Only if the object never changes direction (i.e., v(t) does not change sign) during the interval. Trap/Clarification: Displacement can be zero while total distance is positive (e.g., round trip).

Quick Facts & Traps

• Fact: a(t) > 0 means velocity is increasing, but the object could still be moving left (v(t) < 0).
• Trap: "Acceleration is negative"-Reality: Negative acceleration means decreasing velocity (if v(t) > 0) or increasing velocity (if v(t) < 0).
• Fact: The Fundamental Theorem of Calculus links motion: s(b) – s(a) = ? v(t) dt.
• Trap: "Speed is the derivative of position"-Reality: Speed is |v(t)|, not s'(t) (which is velocity).
• Fact: If v(t) and a(t) have the same sign, the object is speeding up; opposite signs mean slowing down.
• Trap: "Direction changes when a(t) = 0"-Reality: Direction changes when v(t) crosses zero (not necessarily when a(t) = 0).

Rapid-Fire True/False

• Statement: If v(t) = 0 at t = c, the object must change direction at t = c. Answer: FALSE Why the common mistake happens: Students assume v(t) = 0 always implies a turnaround, but v(t) could touch zero without changing sign (e.g., v(t) = t²).

• Statement: Total distance traveled is always greater than or equal to displacement. Answer: TRUE Why the common mistake happens: Students confuse displacement (vector) with distance (scalar) and forget distance accounts for all motion.

• Statement: If a(t) = 0 for all t, the object is at rest. Answer: FALSE Why the common mistake happens: Zero acceleration means constant velocity, not necessarily zero velocity.