Math practice · Unit #06 · 8th grade · Matter is made of particles

The Smell Race

Popcorn odor reaches the back row before anyone sees the bag. Something traveled — and it had a rate. Motion math, but the moving thing is invisible. That's the whole trick of this unit.

skills: rates estimation probability intuition ≈ 30 min
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Name Period Date
PART A

Timing the invisible

Sample class data (yours will differ — we'll replace it with our own stopwatch results): a scent source opens at the front; each row records when they first smell it. Rate = distance ÷ time, same as the motion unit.

The smell race — sample data
RowDistance (m)Time (s)Rate (m/s)
128
2422
3641

  1. Compute each row's rate (round to two decimals). Does the smell travel at a constant speed? What happens to the rate as it gets farther from the source?

  2. Propose one particle-level reason the smell slows down with distance. Hint: are the scent molecules getting more crowded or more spread out? Are they still colliding?

PART B

The 500 m/s paradox

Air molecules at room temperature move at about 500 m/s — faster than a jet. So why did the smell take 41 seconds to cross 6 meters?

  1. At 500 m/s with a clear straight path, how long should 6 m take? (time = distance ÷ speed)

  2. The smell actually took 41 s. How many times slower is that than the straight-path prediction? (41 ÷ your answer)

  3. Resolve the paradox in two sentences, using the words collision and zigzag. (An air molecule collides billions of times per second — it almost never gets to go straight.)

PART C

The random walk

One scent molecule's life: flip a coin — heads = one step toward the back row, tails = one step toward the front. Here is one real trial of 10 flips: H T H H T H T T H H

  1. Count heads and tails. Net progress = heads − tails = ___ steps toward the back.

  2. To be 10 steps back after 10 flips, you'd need every flip to be heads. The chance of one specific run of 10 flips is 1 in 2¹⁰ = 1,024. In one sentence: why does a smell still eventually reach the back row even though each molecule walks randomly?

  3. Extension (fast finishers): there are countless molecules. If each has even a small chance of drifting back-row-ward, what happens to the smell as a whole over a minute?

PART D

Hot vs. cold

Published data: a drop of food coloring spreads 10 mm through still water. Cold water: 50 s. Warm water: 25 s.

  1. Compute both rates (mm/s). Warm is how many times faster?

  2. Particles move faster when they're warmer — that's what temperature is. Use the rates to finish the claim: "The warm water's particles carried the color about ___× faster, evidence that temperature measures particle ___."

Answer key — teachers

Part A

Row 1: 0.25 m/s · Row 2: ≈0.18 m/s · Row 3: ≈0.15 m/s.

  1. Not constant — the rate drops with distance.
  2. Scent molecules spread thinner (lower concentration) and keep colliding with air; fewer of them are pushing into new territory per second.

Part B

  1. 6 ÷ 500 = 0.012 s
  2. 41 ÷ 0.012 ≈ 3,400× slower
  3. Each molecule collides constantly and zigzags; almost no path is straight, so the smell's net progress is thousands of times slower than any single molecule's speed.

Part C

  1. 6 heads, 4 tails → net +2 steps toward the back.
  2. Most random runs still drift somewhere, and molecules that wander back-row-ward stay there — with countless molecules, some fraction always arrives, and they accumulate (diffusion is one-way overall because there's no smell to send back).
  3. The smell fills the room — countless random walkers guarantee it, no aiming required.

Part D

  1. Cold: 0.2 mm/s · Warm: 0.4 mm/s → 2× faster.
  2. "…about 2× faster … particle speed (energy)."