Linearizing Worksheet 1 – Stopping Distance vs Speed

Graph Overview

• Situation Illustrated: Relationship between a car’s speed and the minimum stopping distance once brakes are applied.
• Visual Summary: Up-sloping curve beginning at the origin; steeper at higher speeds.
  • Horizontal axis (x): Speed.
  • Vertical axis (y): Stopping Distance.

Axis Units

• x-axis (Speed): $\text{m\,/\,s}$
• y-axis (Stopping Distance): $\text{m}$

Qualitative Observations

• Non-linear trend: Curve bows upward—suggesting stopping distance grows faster than speed.
• Example Point: At $v = 10\, \text{m\,/\,s}$, graph shows a certain stopping distance (read from the graph in class activity).
• Doubling Speed (Prompt c):
  • When speed doubles from $10\, \text{m\,/\,s}$ to $20\, \text{m\,/\,s}$, the stopping distance is more than double according to the curve.
  • Indicates disproportionate increase—hints at a quadratic dependence.

Linearity Check (Prompt d)

• Linear (proportional) relationship would satisfy $\frac{d<em>2}{d</em>1}=\frac{v<em>2}{v</em>1}$ for any two points.
• Graph violates this: ratio of distances exceeds ratio of speeds ⇒ not linear / not directly proportional.

Candidate Functional Forms (Prompt e)

• i. $d = k v$ (linear) ⟶ rejected by curvature.
• ii. $d = k v^2$ (quadratic) ⟶ favoured: upward-curving shape fits visually.
• iii. $d = k \sqrt{v}$ ⟶ would flatten out at high v → not observed.
• iv. $d = \frac{k}{v}$ ⟶ decreasing curve → opposite behavior.
• Best choice: $d = k v^2$.
  • The constant $k$ encodes friction coefficient, brake efficiency, road condition, etc.

Linearization Strategy (Prompt f)

• Goal: Convert curved $d$ vs $v$ plot into a straight line.
• If $d = k v^2$, take $d$ as dependent variable and $v^2$ as independent:
  • Plot $d$ vs $v^2$.
  • Expected outcome: Straight line through origin (intercept $≈0$ if data ideal).
• Slope of linearized graph:
  • $\text{slope} = k$ (units: $\frac{\text{m}}{(\text{m\,/\,s})^2}=\frac{\text{s}^2}{\text{m}}$).
• Intercept:
  • Should be ≈$0$; a non-zero intercept indicates measurement errors, braking reaction distance, or systematic offset.
• Resulting Linear Equation:
  • $d = k v^2$
  • After linearization: $d = (\text{slope})\,v^2 + (\text{intercept})$

Physical & Practical Significance

• Safety Implication: Stopping distance quadruples when speed doubles (since $d\propto v^2$).
• Driver Education: Highlights why high speeds dramatically increase risk.
• Engineering Context: Informs design of road signs, speed limits, runway lengths, autonomous braking systems.

Quick Reference

• To test data in lab/homework:
  1. Compute $v^2$ for each recorded speed.
  2. Plot $d$ (y) vs $v^2$ (x).
  3. Fit a straight line; slope ≈ $k$.
• If graph is linear with $R^2 \approx 1$, quadratic model is validated.
• Typical classroom data give $k\approx 0.3{-}0.4\,\text{s}^2/\text{m}$ under dry asphalt conditions.