A.1.1 Displacement, Velocity & Acceleration

Eğitimetri Digital | IB
Physics SL Curriculum

Lesson 1: Kinematics — Describing Motion & Graphical Analysis

Theme A.1 — Position, Displacement, Speed, Velocity, Acceleration & Graph Conversions

🎯 Learning Objectives

Distinguish between distance (scalar) and displacement (vector).
Define average and instantaneous speed vs. velocity.
Analyse acceleration as the time rate of change of velocity.
Interpret and convert between displacement-time (x-t), velocity-time (v-t), and acceleration-time (a-t) graphs.

A1.1. Displacement, Velocity & Acceleration

1. Position, Distance, and Displacement

Position Vector (x): Defined as the location of an object relative to a reference point (origin, O).
Distance (d): The total length of the path travelled along a route from one point to another. It is a scalar quantity and can never be negative (d >= 0).
Displacement (Δs): The vector difference between the final position and the initial position:
Δs = s_final – s_initial

Figure 1: Position vector of point A(4,4) relative to origin O(0,0) in the x-y coordinate plane.

[Buraya X-Y Koordinat Sistemindeki Position Vector Görselini Ekleyin]

Figure 2: Comparison of distance and displacement (Big Ben to Tower of London).

[Buraya Londra Haritası Rota Görselini Ekleyin]

Key Concept Note: Even if an object undergoes cyclical motion and returns exactly to its starting point, its net displacement will be zero, but its total distance covered can never be zero.

Worked Example: Cyclical Motion on a Circular Track

A runner completes one full lap on a circular track with a radius of r = 50 m, starting and ending at point A.

[Buraya Çembersel Pist (r=50m) Görselini Ekleyin]

Distance (s): The total path length covered by the runner is equal to the circumference of the circular track. Answers in formal IB exams are frequently left in terms of pi (π):
d = 2 × π × r = 2 × π × 50 m = 100π m
Displacement (Δs): Because the runner returns precisely to the initial starting position (point A), the net change in position is zero:
Δs = 0 m

2. Speed and Velocity

Speed (v): The distance travelled per unit time. Scalar quantity, cannot take negative values. (Speed = Distance / Time)
Velocity (v): The displacement per unit time. Vector quantity that includes direction. (Velocity = Displacement / Time)

3. Instantaneous vs. Average Values

In kinematics, motion parameters are analysed either at a specific clock reading (instantaneous) or over a finite time interval (average).

A. Instantaneous Speed vs. Average Speed

Instantaneous Speed: Speed at any given precise moment (e.g., car speedometer).
Average Speed: Total distance divided by total time interval for the journey.

B. Instantaneous Velocity vs. Average Velocity

Instantaneous Velocity: Velocity at a specific moment. Determined by the slope of the tangent line on an x-t graph (First derivative: v = ds/dt).
Average Velocity: Net change in position (displacement) divided by total elapsed time interval (v_avg = Δs / Δt).

C. Connecting Context: The London Journey Application

When driving along Route 1, Route 2, or Route 3 from Big Ben to the Tower of London:

Dashboard values show instantaneous speed.
Evaluating winding road length vs. total time yields average speed.
Evaluating straight-line distance (displacement of 3.5 km) vs. total time yields magnitude of average velocity.

Conceptual Case Study: Three Cars in London

Route 1: d₁ = 4.0 km in t₁ = 23 min
Route 2: d₂ = 4.5 km in t₂ = 23 min
Route 3: d₃ = 5.5 km in t₃ = 24 min

A. Analysis of Displacement: All three cars share identical displacement because initial and final positions are the same:
Δs₁ = Δs₂ = Δs₃ = 3.5 km (East-Northeast)

B. Calculation of Average Speed:
Car 1: v₁ = 4.0 / 23 ≈ 0.174 km min⁻¹
Car 2: v₂ = 4.5 / 23 ≈ 0.196 km min⁻¹
Car 3: v₃ = 5.5 / 24 ≈ 0.229 km min⁻¹

C. Calculation of Average Velocity:
Car 1 & 2: v = 3.5 / 23 ≈ 0.152 km min⁻¹
Car 3: v = 3.5 / 24 ≈ 0.146 km min⁻¹

IB Exam Tip: Car 1 and Car 2 have identical average velocities despite different paths because displacement and time intervals are identical. SI unit conversions (m/s) are frequently required in exams!

4. Describing Motion with Graphs

Case Study A: Uniform Rectilinear Motion (Constant Velocity)

An object covers equal distances in equal time intervals.

[Buraya Car Motion Fig 3 ve x-t / v-t Grafiklerini Ekleyin]

Position–Time (x–t) Graph: Straight line where the slope (tan θ) yields the magnitude of the velocity.
Velocity–Time (v–t) Graph: Horizontal straight line where the area underneath yields the magnitude of displacement (Area = base × height).

Conceptual Check: For an object executing uniform rectilinear motion in a single direction, average velocity, instantaneous velocity, average speed, and instantaneous speed all share the exact same magnitude (e.g., 10 m s⁻¹).

Case Study B: Non-Uniform Motion & Graphical Analysis

[Buraya Non-Uniform Motion Tangent Analiz Grafiklerini Ekleyin]

Average Speed/Velocity: Slope of the secant line connecting two distinct points on the curve.
Instantaneous Speed/Velocity: Slope of the tangent line drawn at a specific point on the curve.

Acceleration (a)

Definition: The rate of change of velocity over time. It is a vector quantity (a = Δv / Δt). SI Unit: m s⁻².

Uniformly Accelerated Motion & The (x, 3x, 5x…) Rule

For an object accelerating uniformly from rest, displacement in consecutive equal time intervals increases as consecutive odd numbers (x, 3x, 5x, 7x, 9x…). Total displacements scale with squares of time (1x, 4x, 9x, 16x, 25x).

[Buraya v-t ve x-t İvme Grafiklerini (Graph 5, 6, 7, 8) Ekleyin]

Velocity–Time Graph Gradient: Gives acceleration (a).
Velocity–Time Graph Area: Gives displacement (s = ½ a t²).
Acceleration–Time Graph Area: Gives change in velocity (Δv = a × t).

Case Study C: Directional Analysis & Masterclass v-t Graph

Motion in opposite directions, initial velocity scenarios, and braking/deceleration graphs (Graphs 9 through 23) demonstrate how positive/negative regions and gradients dictate velocity and displacement profiles.

[Buraya Masterclass v-t Analiz Tablosu ve Grafikleri (Graph 18 – 23) Ekleyin]

Activity Exercise: Ticker Tape Analysis on an Inclined Plane

Experimental Parameters: 50 Hz Frequency, Sampling interval every 10 ticks (Δt = 0.20 s).

[Buraya Excel Veri Tablosu ve Ticker Tape Grafiklerini (Graph 24) Ekleyin]

Displacement-Time (x-t): Quadratic parabolic profile confirming constant acceleration. Average velocity = 8.0 m / 2.0 s = 4.0 m s⁻¹.
Velocity-Time (v-t): Linear increase; triangular area confirms total displacement of 8.0 m.
Acceleration-Time (a-t): Horizontal line confirming constant acceleration of 4.0 m s⁻².

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