AP Physics – Linear Motion (Displacement, Velocity, Acceleration, Free Fall)

AP Physics – Linear Motion (Displacement, Velocity, Acceleration, Free Fall)

AP Physics: Linear Motion (Displacement, Velocity, Acceleration, Free Fall) – Exam-Ready Study Guide

What This Is

Linear motion is the study of how objects move in a straight line—how far they go (displacement), how fast they move (velocity), and how quickly their speed changes (acceleration). This is foundational for the AP Physics exam because it’s the basis for nearly all mechanics problems. A real-world example: When a skydiver jumps from a plane, their motion starts with free fall (acceleration due to gravity) until air resistance becomes significant. Understanding these concepts lets you predict where the skydiver will be at any moment or how fast they’re moving when they pull the parachute.

Key Terms & Concepts

• Displacement (?x or ?y): Change in position of an object (vector quantity). Not the same as distance traveled (scalar).

• Example: If you walk 3 m east, then 4 m west, your displacement is 1 m west (not 7 m).

• Average velocity (v_avg): Displacement divided by time interval: v_avg = ?x / ?t.

• Units: m/s. Vector (has direction).

• Instantaneous velocity: Velocity at a single moment in time (slope of the position-time graph at a point).

• Average acceleration (a_avg): Change in velocity divided by time interval: a_avg = ?v / ?t.

• Units: m/s². Vector (direction matters).

• Kinematic equations (for constant acceleration):

• v = v? + at (final velocity = initial velocity + acceleration × time)
• ?x = v?t + ½at² (displacement = initial velocity × time + ½ × acceleration × time²)
• v² = v?² + 2a?x (final velocity² = initial velocity² + 2 × acceleration × displacement)

• Variables: v = final velocity, v? = initial velocity, a = acceleration, t = time, ?x = displacement.

• Free fall: Motion under the influence of only gravity (ignore air resistance). On Earth, a = g = 9.8 m/s² downward (AP often uses 10 m/s² for simplicity).

• Example: A ball dropped from a cliff accelerates at 9.8 m/s² until it hits the ground.

• Position-time (x-t) graph: Slope = velocity. Curved line = acceleration.

• Velocity-time (v-t) graph: Slope = acceleration. Area under the curve = displacement.

• Acceleration due to gravity (g): 9.8 m/s² (downward). Always acts downward, even if an object is moving upward (e.g., a thrown ball slows down on the way up).

• Vector vs. scalar:

• Vector: Has magnitude and direction (e.g., velocity, displacement).
• Scalar: Only magnitude (e.g., speed, distance).

Step-by-Step: Solving Linear Motion Problems

1. Read the problem carefully. Identify given quantities (v?, v, a, ?x, t) and what’s asked (e.g., "Find the time to reach max height").
2. Draw a diagram. Sketch the motion (e.g., a ball thrown upward, a car braking).
3. Choose a coordinate system. Define positive/negative directions (e.g., upward = +, downward = –).
4. Pick the right kinematic equation. Use the one that matches your knowns/unknowns:
5. Missing ?x?-Use v = v? + at.
6. Missing v?-Use ?x = v?t + ½at².
7. Missing t?-Use v² = v?² + 2a?x.
8. Plug in values (with signs!). If upward is +, then a = –9.8 m/s² for free fall.
9. Solve and check units. Ensure answers make sense (e.g., time can’t be negative).

Common Mistakes

• Mistake: Confusing displacement with distance.

• Correction: Displacement is the straight-line change in position (vector); distance is the total path length (scalar). Example: Walking in a circle returns you to start (displacement = 0, distance = circumference).

• Mistake: Forgetting that g is always downward (even for upward motion).

• Correction: If upward is +, then a = –9.8 m/s² for free fall. A ball thrown upward slows down at 9.8 m/s².

• Mistake: Using kinematic equations when acceleration isn’t constant.

• Correction: Kinematic equations only work for constant acceleration. For non-constant acceleration (e.g., air resistance), use calculus or energy methods.

• Mistake: Mixing up velocity and acceleration signs.

• Correction: Velocity and acceleration can have opposite signs (e.g., a ball moving upward (+v) but accelerating downward (–a)).

• Mistake: Ignoring initial velocity (v? = 0) for dropped objects.

• Correction: If an object is dropped (not thrown), v? = 0. If it’s thrown, v?-0.

AP Exam Insights

• Free fall is a favorite. Expect multiple-choice and FRQs about objects thrown upward/downward or dropped. Trick: At max height, v = 0 (but a = –9.8 m/s² still!).
• Graph interpretation. AP loves x-t and v-t graphs. Remember:
• Slope of x-t = velocity.
• Slope of v-t = acceleration.
• Area under v-t = displacement.
• Sign conventions. AP graders deduct points for incorrect signs. Always define +/– directions.
• FRQ traps:
• Asking for displacement when students calculate distance.
• Using 10 m/s² instead of 9.8 m/s² (check the problem!).
• Forgetting to include units in answers.

Quick Check Questions

1. A ball is thrown straight upward with an initial velocity of 20 m/s. What is its velocity after 3 seconds? (Use g = 10 m/s².)
2. A) –10 m/s
3. B) 0 m/s
4. C) 10 m/s
5. D) –20 m/s

6. Answer: A) –10 m/s. Explanation: Use v = v? + at-v = 20 + (–10)(3) = –10 m/s (negative = downward).

7. A car accelerates from rest at 4 m/s² for 5 s. How far does it travel?

8. Answer: 50 m. Explanation: Use ?x = v?t + ½at²-?x = 0 + ½(4)(5)² = 50 m.

9. On a velocity-time graph, a straight line with a negative slope represents:

10. A) Constant positive acceleration
11. B) Constant negative acceleration
12. C) Increasing acceleration
13. D) Zero acceleration
14. Answer: B) Constant negative acceleration. Explanation: Slope of v-t graph = acceleration; negative slope = negative acceleration.

Last-Minute Cram Sheet

1. Displacement (?x) = final position – initial position (vector).
2. Average velocity = ?x / ?t (vector).
3. Average acceleration = ?v / ?t (vector).
4. Kinematic equations only work for constant acceleration.
5. Free fall: a = g = 9.8 m/s² downward (AP may use 10 m/s²).
6. At max height in free fall, v = 0 but a = –9.8 m/s².
7. Slope of x-t graph = velocity; slope of v-t graph = acceleration.
8. Area under v-t graph = displacement.
9. Signs matter! Define +/– directions and stick to them.
10. Dropped objects: v? = 0; thrown objects: v?-0.