Chapter Introduction
Motion in a Straight Line is the foundation of one-dimensional kinematics. It connects position, displacement, velocity, acceleration, motion models and graphs.
Chapter in One View
| Step | Think |
|---|---|
| Reference system | Choose origin and positive direction. |
| Position | Where is the object? |
| Displacement | How did position change? |
| Velocity | How is position changing? |
| Acceleration | How is velocity changing? |
| Model | What physical condition governs the motion? |
| Representation | What does the equation or graph show? |
| Observer | Relative to whom? |
| Verification | Do sign, units and Physics make sense? |
PHYSICS SITUATION → MODEL → FORMULA → CONDITION → APPLY → VERIFY
Master Exam Chain
Must-Know Concepts
| Concept | Essential Idea |
|---|---|
| Reference / coordinate | Origin and positive direction are choices. Negative coordinate does not automatically mean negative motion. |
| Position x | Tells where the object is relative to the chosen reference. |
| Displacement Δx | Net change in position: Δx = xf − xi. |
| Distance | Total path length. Distance ≥ |displacement|. |
| Speed | Magnitude of velocity. |
| Velocity | Rate of change of position; its sign gives direction. |
| Acceleration | Rate of change of velocity. |
Speeding Up / Slowing Down
v and a same sign → speeding up. v and a opposite signs → slowing down.
Mathematical & Graph Language
| Idea | Motion Meaning |
|---|---|
| Derivative / slope | Rate of change. |
| Integral / area | Accumulated change. |
| x–t slope | Velocity. |
| v–t slope | Acceleration. |
| v–t area | Displacement. |
| a–t area | Change in velocity. |
v = dx/dt
Recognize: Instantaneous velocity.
a = dv/dt
Recognize: Instantaneous acceleration.
Δx = ∫v dt
Recognize: Displacement accumulated from velocity.
Δv = ∫a dt
Recognize: Velocity change accumulated from acceleration.
Average & Instantaneous Quantities
| Quantity | Relation | Recognize |
|---|---|---|
| Average speed | Total distance / total time | Whole journey; use distance. |
| Average velocity | Displacement / total time | Whole interval; use signed displacement. |
| Instantaneous speed | |v| | Speed at one instant. |
| Instantaneous velocity | dx/dt | Velocity at one instant. |
| Average acceleration | Δv/Δt | Velocity change over an interval. |
| Instantaneous acceleration | dv/dt | Acceleration at one instant. |
Fast Check
Returns to starting point → displacement = 0, but distance may be non-zero.
Motion Model: Constant Acceleration
Model Condition
Use the standard UAM equations only when acceleration is constant throughout the interval.
| See This | Use This |
|---|---|
| u, a, t → v | v = u + at |
| u, a, t → displacement | Δx = ut + ½at² |
| u, v, Δx → no time | v² = u² + 2aΔx |
| u, v, t → displacement | Δx = ½(u + v)t |
Motion Graphs — See This → Think This
| See This | Think This |
|---|---|
| x–t graph with positive slope | Positive velocity. |
| x–t graph with negative slope | Negative velocity. |
| Horizontal x–t graph | Zero velocity. |
| Changing slope on x–t | Changing velocity. |
| Positive slope on v–t | Positive acceleration. |
| Negative slope on v–t | Negative acceleration. |
| Horizontal v–t | Zero acceleration. |
| Area under v–t | Displacement. |
Graph Trap
Signed v–t area gives displacement. If the graph crosses the time axis, total distance requires adding the magnitudes of separate areas.
Special Cases
Free Fall
Free fall is UAM with gravitational acceleration when air resistance is neglected.
| Choice | Acceleration |
|---|---|
| Upward positive | a = −g |
| Downward positive | a = +g |
Highest Point
v = 0, but acceleration remains downward. Zero velocity does not mean zero acceleration.
Relative Motion
| Relation | Meaning |
|---|---|
| vA/B = vA − vB | Velocity of A measured by observer B. |
| aA/B = aA − aB | Relative acceleration of A with respect to B. |
- “How does A appear to B?” → relative velocity.
- Identify the observer and sign convention before subtracting.
- For meeting/catching questions, think in terms of relative position and relative velocity.
Multi-Stage Motion
- Split the motion where physical conditions change.
- Choose the valid model for each stage.
- Find the transition state.
- Use that state as the initial state of the next stage.
- Verify the final result.
Formula Selection Guide
| See This | Think This | Use This |
|---|---|---|
| Need instantaneous velocity from x(t) | Rate of change of position | v = dx/dt |
| Need instantaneous acceleration from v(t) | Rate of change of velocity | a = dv/dt |
| Velocity known over time | Accumulated velocity | Δx = ∫v dt |
| Acceleration known over time | Accumulated acceleration | Δv = ∫a dt |
| Constant a; need v after t | UAM | v = u + at |
| Constant a; need displacement after t | UAM | Δx = ut + ½at² |
| Constant a; time not needed | UAM | v² = u² + 2aΔx |
| Observer/reference changes | Relative motion | vA/B = vA − vB |
| Conditions change | Multi-stage motion | Solve stage by stage |
Common Misconceptions & Exam Traps
Wrong Idea → Correct Physics
| Wrong Idea | Correct Physics |
|---|---|
| Negative position means moving backward. | Position sign depends on the reference; velocity gives direction. |
| Negative velocity means slowing down. | Speeding/slowing depends on the signs of v and a. |
| Acceleration means speed is increasing. | Acceleration means velocity is changing. |
| v = 0 means a = 0. | Velocity can be zero while acceleration is non-zero. |
| Distance = displacement. | Distance is path length; displacement is net position change. |
| Speed = velocity. | Speed is the magnitude of velocity. |
| UAM equations work for every motion. | They require constant acceleration. |
| Relative velocity is just subtraction. | First identify the observer and sign convention. |
See This → Think This
| Clue | Recognize |
|---|---|
| Negative coordinate | Check the chosen positive direction. |
| x–t graph | Slope gives velocity. |
| v–t graph | Slope gives acceleration; area gives displacement. |
| Constant acceleration | Use UAM relations. |
| v = 0 at highest point | Acceleration is still −g if upward is positive. |
| “With respect to B” | Relative motion; B is the observer. |
| Different conditions in different intervals | Split into stages. |
| v and a same sign | Speeding up. |
| v and a opposite signs | Slowing down. |
Visual memory: reference → position → slope → area. Use the graph to identify the Physics situation before choosing a formula.
Quick Checks
Q: A coordinate is negative. Does that prove negative motion?
Answer: No. Position sign and velocity sign are different.
Q: What does x–t slope give?
Answer: Velocity.
Q: What does v–t area give?
Answer: Displacement.
Q: When is v = u + at valid?
Answer: When acceleration is constant over the interval.
Q: At the highest point of an upward throw?
Answer: v = 0, but acceleration is still downward.
Q: Two objects and “relative to B”?
Answer: Use relative velocity with B as observer.
Q: A body returns to its starting point?
Answer: Displacement is zero; distance need not be zero.
Q: Conditions change during motion?
Answer: Split into stages and connect transition states.
Mini Exam Reasoning
Example 1
Clue: Constant acceleration + u and t are given.
Physics situation: Uniformly accelerated motion.
Model: UAM.
Example 2
Clue: One object is described relative to another.
Physics situation: Relative motion.
Model: Relative velocity.
One-Minute Recall
- Reference system → origin + positive direction.
- Position → where the object is.
- Displacement → final position − initial position.
- Distance → total path length.
- Velocity → rate of change of position.
- Acceleration → rate of change of velocity.
- x–t slope → velocity; v–t slope → acceleration.
- v–t area → displacement.
- Constant acceleration → UAM equations.
- Free fall → UAM with a = ±g.
- Relative velocity → identify observer first.
- Changing conditions → solve stage by stage.
Situation → Model → Condition → Relation → Sign → Units → Physical sense
Final Exam Scan
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Practice MCQs
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