Physics Cycle

POSITION, DISTANCE AND DISPLACEMENT — SUMMARY
1. CORE DEFINITIONS
Position is the location of an object relative to a chosen reference point (origin). It answers "WHERE is the object?" Position can be positive, negative, or zero depending on the chosen origin and direction. It is represented by:

x in one-dimensional motion

(x, y) in two-dimensional motion

(x, y, z) in three-dimensional motion

Distance is the total length of the actual path travelled by an object. It answers "HOW MUCH ground has been covered?" Distance is a scalar quantity — it has magnitude only, no direction. Distance is always positive or zero and can never be negative.

Displacement is the change in position of an object. It answers "HOW FAR and IN WHAT DIRECTION has the position changed?" Displacement is a vector quantity — it has both magnitude and direction. In one dimension: Δx = xf - xi.

2. KEY DIFFERENCES
Feature Distance Displacement
Type Scalar Vector
Depends on Actual path taken Only initial and final positions
Sign Always positive or zero Can be positive, negative, or zero
Direction Not required Essential
Path reversal Increases distance Can reduce or cancel displacement
3. FUNDAMENTAL RELATIONSHIP
distance ≥ |displacement|

Equality occurs only when motion is along a straight line without reversing direction.

4. UNITS AND DIMENSIONS
SI unit of both distance and displacement: metre (m)

Dimension of both: L (length)

Common conversions: 1 km = 1000 m; 1 cm = 0.01 m; 1 mm = 0.001 m

5. POSITION AND DISPLACEMENT FORMULAS
Displacement: Δx = xf - xi

Position vector (2D): r = xi + yj

Position vector (3D): r = xi + yj + zk

Resultant displacement (perpendicular components): R = √(Rx² + Ry²)

Direction: θ = tan⁻¹(Ry/Rx)

6. DISTANCE AND DISPLACEMENT IN SPECIAL CASES
Straight line, no reversal: distance = |displacement|

Reversal of direction: distance > |displacement|

Example: 10 m east then 6 m west → distance = 16 m, displacement = 4 m east

Round trip (return to start): distance > 0, displacement = 0

Example: 400 m lap → distance = 400 m, displacement = 0 m

Circular motion:

One full revolution: distance = 2πr, displacement = 0

Half revolution: distance = πr, displacement = 2r

Quarter revolution: distance = πr/2, displacement = r√2

7. AVERAGE SPEED VS AVERAGE VELOCITY
Average speed = total distance / total time (scalar)

Average velocity = total displacement / total time (vector)

Example: 100 m east then 40 m west in 20 s

Distance = 140 m, Displacement = 60 m east

Average speed = 7 m/s, Average velocity = 3 m/s east

8. GRAPHICAL RELATIONSHIPS
Displacement-time graph:

Gradient = velocity

Horizontal line → stationary (v = 0)

Positive gradient → positive velocity

Negative gradient → negative velocity

Curve → changing velocity (instantaneous velocity = tangent gradient)

Distance-time graph:

Gradient = speed

Steeper graph → greater speed

Distance normally does not decrease

Velocity-time graph:

Area under graph = displacement

Gradient = acceleration

9. POSITION → VELOCITY → ACCELERATION CHAIN
Velocity: v = dx/dt (rate of change of position)

Acceleration: a = dv/dt = d²x/dt² (rate of change of velocity)

Average velocity: v_avg = Δx/Δt

10. EQUATIONS OF MOTION (CONSTANT ACCELERATION)
v = u + at

s = ut + ½at²

v² = u² + 2as

x = x₀ + ut + ½at²

Where s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time.

11. CONSTANT VELOCITY CASE
Δx = vΔt

x = x₀ + vt

12. MULTIPLE JOURNEYS
Total displacement: Δx_total = Δx₁ + Δx₂ + Δx₃ + ... (using signed values)

Total distance: d_total = |d₁| + |d₂| + |d₃| + ... (adding magnitudes)

13. SIGN CONVENTION
A sign convention must be established before solving one-dimensional problems:

Right/east/up = positive

Left/west/down = negative

Once chosen, must remain consistent throughout the calculation

14. EFFECT OF CHANGING THE ORIGIN
Changing the origin changes numerical position values but does not change the actual physical distance travelled between two events. Position is coordinate-dependent; distance and displacement magnitude are not.

15. COMMON ERRORS TO AVOID
Calling displacement a scalar (it is a vector)

Using distance for average velocity (use displacement)

Using displacement for average speed (use distance)

Ignoring direction when reporting displacement

Giving a negative distance (distance is non-negative)

Assuming zero displacement means zero distance

Adding perpendicular displacements directly (use Pythagoras)

Confusing position with displacement

16. WORKED EXAMPLES SUMMARY
Example 1: 25 m east → distance = 25 m, displacement = 25 m east

Example 2: 30 m east then 10 m west → distance = 40 m, displacement = 20 m east

Example 3: 15 km north then 8 km east → distance = 23 km, displacement = 17 km

Example 4: 400 m lap → distance = 400 m, displacement = 0 m

Example 5: x = -4 m to x = +11 m → displacement = +15 m

Example 6: x = 40 m to x = 15 m → displacement = -25 m

Example 7: 120 m east then 50 m west in 34 s → distance = 170 m, displacement = 70 m east, average speed = 5 m/s, average velocity ≈ 2.06 m/s east

17. PRACTICAL APPLICATIONS
GPS navigation, vehicle tracking, robotics, aviation, satellite tracking, surveying, athletics, engineering, traffic management, autonomous vehicles, projectile analysis, astronomy, industrial automation.

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