Math practice · Unit #08 · 7th grade · Energy

The Energy Ledger

Energy is never created or destroyed — it's bookkept. Every joule has to be somewhere. Keep the ledger balanced and the roller coaster has no secrets from you.

skills: squares & patterns formulas conservation bookkeeping ≈ 35 min
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Name Period Date
PART A

The squared-term discovery

KE = ½ × m × v² kinetic energy (J) = ½ · mass (kg) · speed² (m/s)²
A 2 kg cart at faster and faster speeds. Fill in the kinetic energy column — the first row is done — then find the pattern before anyone tells it to you.

Same cart (m = 2 kg), increasing speed
Speed v (m/s)v²KE = ½ · 2 · v² (J)
111
2
3
4
5

  1. Double the speed (1 → 2 m/s) and the kinetic energy multiplies by ___. Triple it (1 → 3) and KE multiplies by ___. Write the rule: "KE grows with the ___ of speed."

  2. You met this pattern before — the stopping-distance ×4 rule in the Newton's laws unit. In one sentence: why does a car need 4× the room to stop from twice the speed?

PART B

The beetle shelf: PE = mgh

PE = m × g × h potential energy (J) = mass (kg) · 9.8 · height (m)
Our mascot (mass 0.02 kg — a chunky 20 g) climbs the shelf. Gravity never sleeps.

  1. Beetle on the low shelf, 0.5 m up. Stored PE?

  2. Beetle on the top shelf, 1.5 m up. Stored PE? How many times bigger than the low shelf — and why is it exactly that factor?

  3. Now a 1 kg textbook on a 2 m shelf. PE? Who stores more energy per meter climbed — and what does that tell you about where the mgh formula "hides" the danger of high shelves?

PART C

Bookkeeping: every joule somewhere

A coaster car starts at the top with 100 J of potential energy. On the way down, energy moves between accounts — but the total must always be 100 J. Fill in the missing entries of the ledger.

The ledger (all values in joules)
PositionPotentialKineticHeat (friction)Total
Top of hill10000100
Halfway down502100
Bottom018100
After the flat run-out030100

  1. The 18 J of "heat" at the bottom — the energy didn't vanish. Where did it go, physically? One sentence.

  2. At the end of the ride the car stops. Write the final row of the ledger yourself (all 100 J accounted for).

PART D

The First Hill Rule, as a word problem

A coaster's lift hill is 40 m tall. The cars carry no engine — the lift hill's PE is all the energy they'll ever get. Friction costs the ride 20% of its energy before the second hill.

  1. Only 80% of the energy survives. In "energy-height" terms, that's like starting from an effective height of 40 × 0.80 = ___ m.

  2. The second hill is 35 m tall. Does the car make it over? Show the comparison that decides it.

  3. State the First Hill Rule as an inequality a roller-coaster designer could tape to the wall: "height of any later hill must be ___ than (1 − friction loss) × height of the first hill."

Answer key — teachers

Part A

v=2: 4, 4 J · v=3: 9, 9 J · v=4: 16, 16 J · v=5: 25, 25 J.

  1. ×4, ×9 — "KE grows with the square of speed."
  2. Stopping means bleeding off kinetic energy; twice the speed = 4× the KE = 4× the energy the brakes must turn into heat = 4× the distance.

Part B

  1. PE = 0.02 × 9.8 × 0.5 = 0.098 J
  2. 0.02 × 9.8 × 1.5 = 0.294 J — exactly 3× because height tripled and PE is proportional to h.
  3. 1 × 9.8 × 2 = 19.6 J — mass is the quiet multiplier; heavy things up high are the dangerous ones (≈200× the beetle's per-meter energy).

Part C

Halfway: KE = 48 J · Bottom: KE = 82 J · Run-out: heat = 70 J.

  1. Friction turned it into heat in the wheels, track, and air — moved, not lost.
  2. PE 0, KE 0, heat 100, total 100.

Part D

  1. 32 m
  2. No — 32 m of effective energy < 35 m of hill; the car stalls and rolls back.
  3. "…must be less than (1 − friction loss) × first-hill height."