1. Motion in a Plane — The Big Picture
Motion in a plane is motion described in two dimensions. The essential difficulty is not the number of formulas; it is keeping track of position, direction, components and the observer.
Physical situation → Object / system → Reference frame → Changing quantity → Useful directions → Motion model → Relation → Check
Physics-First chain — recognize the situation before choosing the equation.
2. Chapter in One View
Foundation: Vectors → components → reference frame
Motion: Position → displacement → velocity → acceleration
Special models: Projectile motion → relative motion → circular motion / changing direction
Exam decision: Recognize the model → select the relation → check conditions.
3. Vector Language — The Foundation
Before describing two-dimensional motion, we need a language that can carry direction. A vector is a quantity described by magnitude and direction.
| Scalar | Vector |
|---|---|
| Magnitude only | Magnitude + direction |
| Distance, speed, time, mass, temperature, energy | Displacement, velocity, acceleration, force, momentum |
A vector can be resolved into components along two perpendicular directions. The components preserve directional information in a form that is easier to calculate.
Ax = A cos θ ; Ay = A sin θ
Use these when θ is measured from the chosen x-axis. Always identify the actual reference axis before choosing sine or cosine.
Vector addition and subtraction
Vector addition gives the combined effect of successive vector changes. Head-to-tail construction joins the initial point to the final point. Vector subtraction must respect direction; subtracting magnitudes alone can be wrong.
Recognition cue: If direction matters, think vector. If different directions behave differently, think components.
4. Position → Displacement → Velocity → Acceleration
Position
To describe where a particle is, choose an origin, coordinate directions and a reference frame. The position vector specifies the particle's location relative to that origin.
Displacement
Displacement is the net change in position. It depends only on initial and final positions, not on the actual path.
Δr = r₂ − r₁
Distance is the total path length. Distance and displacement are therefore not generally equal.
Velocity
v = dr/dt
Velocity is the rate of change of position and is a vector because position is a vector.
Acceleration
a = dv/dt
Acceleration describes the rate at which velocity changes. Velocity can change because speed changes, direction changes, or both.
Key exam trap: Constant speed does not imply zero acceleration. If direction changes, velocity changes and acceleration exists.
Motion chain: Position → rate of change of position → rate of change of velocity
r → v → a
5. Making 2D Motion Simple — Components
A two-dimensional motion can often be understood by separating it into useful directions. This is not merely a mathematical trick: different directions may obey different physical behaviour.
Component description: Study each component according to the physics acting in that direction. The complete 2D motion is then reconstructed from the component description.
r = rx i + ry j
v = vx i + vy j
a = ax i + ay j
Component equations follow naturally when the motion behaves differently along different directions.
Choosing axes
- Choose horizontal/vertical axes when gravity naturally separates the motion.
- Choose parallel/perpendicular directions when the geometry or constraint makes those directions simpler.
- The diagram does not decide the axes; the physical requirement of the question does.
Formula-selection cue: If the motion is complicated in vector form but simple along selected directions, resolve first and solve component by component.
6. Projectile Motion — The Main 2D Model
After projection, if air resistance is neglected, gravity is the relevant acceleration. The projectile model is built from one simple fact: horizontal and vertical motions can be treated separately.
ax = 0 ; ay = −g
Model: uniform horizontal motion + accelerated vertical motion.
Initial velocity components
ux = u cos θ ; uy = u sin θ
Position and velocity
x = (u cos θ)t
vx = u cos θ
vy = u sin θ − gt
Highest point
At the highest point, the projectile has stopped moving upward:
vy = 0
But vx generally remains non-zero, so the projectile generally still has velocity. Acceleration due to gravity also remains −g.
Main trap: A projectile is not acceleration-free. The absence of horizontal force does not remove the vertical acceleration due to gravity.
7. Projectile Formulas — Use Only Under Their Conditions
For a projectile launched and landing at the same level, with air resistance neglected:
T = 2u sin θ / g
Time of flight — same launch and landing level.
H = u² sin²θ / 2g
Maximum height — same standard projectile model.
R = u² sin 2θ / g
Horizontal range — same launch and landing level.
Conditions:
- Launch and landing are at the same level.
- Air resistance is neglected.
- g is constant.
- The object follows projectile motion after projection.
Do not force the standard same-level formulas when launch and landing heights differ. Use the component position equations and the actual initial and final heights.
Trajectory and recognition
Eliminating time between the horizontal and vertical equations gives a quadratic relation between x and y. Hence the ideal projectile trajectory is parabolic.
For the same initial speed and the same launch/landing level, two complementary projection angles give the same range.
When a projectile meets an inclined surface, choose axes according to the physical requirement. The standard same-level range expression should not be applied merely because the path looks like a projectile.
8. Relative Motion — Observer First
Motion is described relative to a reference frame. The same object can be at rest relative to one observer and moving relative to another.
vA/B = vA − vB
Observer first. Formula second. The velocity of A relative to B compares the motion of A with the motion of B.
Rain may have a vertical velocity relative to the ground while a person moves horizontally. The rain relative to the person is found by subtracting the person's ground velocity from the rain's ground velocity. The direction of this relative velocity gives the apparent direction from which the rain reaches the person.
Same physics, many situations: rain and a moving person; swimmer and river; boat and river; aircraft and wind; two vehicles moving relative to each other.
Deep recognition: Relative motion is not a new kind of motion. It is vector subtraction applied to velocities.
9. Relative Motion — A Reliable Solving Method
- Who is the object being observed? Identify A.
- Who is the observer? Identify B.
- Relative to which frame? Usually the ground or another stated frame.
- What velocities are given? vA, vB, with directions.
- Which relative velocity is required? vA/B, vB/A, etc.
Vector subtraction matters: If two velocities are not along the same line, do not subtract their magnitudes directly. Resolve into components or use vector geometry, keeping the direction of each velocity clear.
Powerful recognition rule: Observer first. Formula second.
Do not mix velocities described in different frames. First express the required velocities consistently in one reference frame, then subtract.
10. Circular Motion — When Direction Changes
Acceleration exists whenever velocity changes. In circular motion, speed may remain constant while velocity direction changes continuously.
ac = v²/r
Acceleration associated with changing velocity direction; it points toward the centre of the circular path.
| SEE THIS | THINK THIS |
|---|---|
| Circular path | Velocity direction changes |
| Constant speed | Velocity can still change |
| Acceleration toward centre | Centripetal / normal acceleration |
| Speed v and radius r | ac = v²/r |
The detailed treatment of circular and rotational motion belongs to the separate Rotational Motion lesson. For Motion in a Plane, the essential recognition is that changing direction means changing velocity and therefore acceleration.
11. Choosing the Right Physical Description
Do not judge the Physics from the diagram alone. A curved path does not automatically mean circular motion. An inclined surface does not automatically determine the axes. A projectile-like path does not automatically guarantee the standard projectile formulas.
| ASK | DECISION |
|---|---|
| What is happening? | Identify the physical situation. |
| What is the system? | Identify the object or objects. |
| Relative to whom? | Set the reference frame. |
| What is changing? | Position, velocity, speed or direction? |
| Why is it changing? | Gravity, relative motion, changing direction, or another cause. |
| Which directions are useful? | Horizontal/vertical, parallel/perpendicular, or another pair. |
| Which motion pattern fits? | General 2D, projectile, relative motion, changing direction. |
| Which equation describes it? | Only now choose the equation. |
SYSTEM → REFERENCE FRAME → PROCESS → ASSUMPTIONS → DIRECTIONS → VALIDITY
Core rule: Having the variables required by a formula does not make the formula valid. The physical conditions must also be satisfied.
12. Formula Selection Guide
| SEE THIS SITUATION | THINK THIS | USE / CHECK |
|---|---|---|
| Net change in position | Displacement | Δr = r₂ − r₁ |
| Rate of change of position | Velocity | v = dr/dt |
| Rate of change of velocity | Acceleration | a = dv/dt |
| Different behaviour along directions | Components | Resolve into useful axes |
| Gravity-only motion after projection | Projectile | ax = 0, ay = −g |
| Same-level projectile | Standard projectile results | Check all conditions first |
| “A relative to B” | Relative velocity | vA/B = vA − vB |
| Circular path / changing direction | Centripetal acceleration | ac = v²/r |
| Situation changes | New physical stage | Stop and rebuild the model |
13. Common Misconceptions & Exam Traps
| WRONG IDEA | CORRECT PHYSICS |
|---|---|
| Distance and displacement are the same | Distance is path length; displacement is net change in position. |
| Velocity is just speed | Velocity includes direction. |
| Constant speed means zero acceleration | Direction change also changes velocity. |
| At the highest point of a projectile, velocity is zero | Only the vertical component is zero; horizontal velocity generally remains. |
| No horizontal force means no projectile acceleration | Gravity still provides vertical acceleration. |
| Relative motion depends only on the moving object | Relative velocity depends on both object and observer. |
Exam traps: using a formula before identifying the physical model; subtracting velocity magnitudes instead of vectors; mixing reference frames; using same-level projectile formulas for different launch/landing conditions; choosing axes because of the picture rather than because they simplify the physics; assuming a familiar diagram guarantees a familiar formula.
14. See This → Think This
| SEE THIS | THINK THIS |
|---|---|
| “Relative to” | Observer first |
| “Highest point” | vy = 0, not generally v = 0 |
| “Constant speed” + curved path | Velocity direction changes → acceleration |
| Gravity after projection | Vertical acceleration remains −g |
| Two different directional behaviours | Resolve into components |
| Changed physical condition / new stage | Stop and rebuild the model |
15. Quick Checks
Q: A particle returns to its starting point. What is its displacement?
Answer: Zero.
Q: Can a particle have constant speed and non-zero acceleration?
Answer: Yes, if its velocity direction changes.
Q: At the highest point of an ideal projectile, which velocity component is zero?
Answer: The vertical component.
Q: What remains constant in ideal projectile motion?
Answer: The horizontal velocity component.
Q: What should you identify first in a “velocity of A relative to B” question?
Answer: The observer B.
Q: What is the basic relative-velocity relation?
Answer: vA/B = vA − vB.
Q: What is the direction of centripetal acceleration?
Answer: Toward the centre.
Q: When should standard same-level projectile formulas be used?
Answer: Only when their physical conditions are satisfied.
16. Mini Exam Reasoning
Example 1
Clue: Launch + gravity only.
Physics situation: Projectile motion.
Model: Independent horizontal and vertical components.
Example 2
Clue: One object is described relative to another.
Physics situation: Relative motion.
Example 3
Clue: Circular path at constant speed.
Physics situation: Changing velocity direction.
17. One-Minute Recall
- Vector → magnitude + direction.
- Position → where the object is relative to the chosen origin/frame.
- Displacement → net change in position.
- Velocity → rate of change of position.
- Acceleration → rate of change of velocity.
- Components → separate motion into useful directions.
- Projectile → ax = 0, ay = −g.
- Highest point → vy = 0; velocity generally not zero.
- Relative motion → observer first; vector subtraction.
- Circular motion → direction changes → acceleration toward centre.
- Formula use → recognize model → check conditions → apply.
Situation → Model → Condition → Relation → Sign → Units → Physical sense
Final Exam Scan
Final Physics-First message: Do not start with the formula. Start with the physical situation. Once the model is recognized, the mathematics becomes much simpler.
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