Math practice · Unit #07 · 7th grade · Newton's laws

F = ma, No Appeals

The second law is one equation with three disguises. Learn to rearrange it and you can predict crashes, rockets, and shopping carts. No exceptions, no excuses — show your work.

skills: solving equations inverse relationships nonlinear patterns ≈ 35 min
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Name Period Date
PART A

The triangle, drilled

F = m × a  ·  a = F ÷ m  ·  m = F ÷ a
One law, three arrangements. Two minutes each — setup, then answer, with units.

  1. A 1,200 kg car accelerates at 3 m/s². What force is the engine delivering?

  2. A 0.2 kg phone is pushed with 5 N. Acceleration?

  3. A rocket's engines produce 180,000 N and it accelerates at 6 m/s² at liftoff. Mass of the rocket?

  4. A 60 kg skater pushes off the boards with 240 N. Her acceleration?

  5. Crash test: a 75 kg dummy decelerates at 350 m/s² (about 36 g). What force did the seatbelt deliver? Put a box around the answer — that number is why Part D exists.

PART B

Same push, different mass

One spring pusher, 24 N every time. Heavier and heavier carts. Complete the table, then answer below it.

Force held constant: F = 24 N
Mass (kg)Acceleration (m/s²)
212
4
8
12
24

  1. Every time the mass doubles, the acceleration does what? This relationship has a name: inverse. Write the rule as a sentence: "When force is constant, acceleration is inversely proportional to ___."

  2. Without computing: a 48 kg cart on the same 24 N spring — will its acceleration be more or less than 1 m/s²? How do you know from the pattern?

PART C

The ×4 rule (stopping distance)

From the stopping-distance investigation: braking distance grows with the square of speed. Model numbers below — double the speed, and the distance doesn't double. It does something worse.

Braking distance model (same car, same road)
Speed (mph)Braking distance (m)Speed multiplierDistance multiplier
2018——
40×2
60×3

  1. Fill the table: if distance scales with speed squared, ×2 speed means ×___ distance, and ×3 speed means ×___ distance. Compute the meters.

  2. Your older sibling says "60 mph is only three times 20 mph, so I need three times the following room." Correct them with the table.

PART D

Why crumple zones are math

Same car, same crash — the only difference is how long the stop takes. Force = mass × (speed change ÷ stopping time). A 60 kg passenger comes to rest from 15 m/s.

  1. Dashboard stop: speed change of 15 m/s in 0.1 s → deceleration = 15 ÷ 0.1 = ? Then force on the passenger = 60 kg × that = ?

  2. Crumple zone + airbag stop: same 15 m/s but spread over 0.5 s. Redo it.

  3. The stopping time got 5× longer. What happened to the force? Finish the slogan: "Crumple zones don't reduce the speed you have to lose — they reduce the ___ you lose it with."

Answer key — teachers

Part A

  1. F = 1,200 × 3 = 3,600 N
  2. a = 5 ÷ 0.2 = 25 m/s²
  3. m = 180,000 ÷ 6 = 30,000 kg
  4. a = 240 ÷ 60 = 4 m/s²
  5. F = 75 × 350 = 26,250 N — huge; this is why we engineer crashes.

Part B

6, 3, 2, 1 m/s².

  1. Halves. "…inversely proportional to mass."
  2. Less than 1 — 48 kg is past 24 kg (which gave exactly 1), and acceleration only shrinks as mass grows. (48 kg → 0.5 m/s².)

Part C

×2 speed → ×4 distance → 72 m · ×3 speed → ×9 distance → 162 m.

  1. ×4 and ×9; 18 × 4 = 72 m, 18 × 9 = 162 m.
  2. Three times the speed needs NINE times the room (162 m vs 18 m) — the square fools everyone who thinks in multiples.

Part D

  1. Deceleration = 150 m/s²; F = 60 × 150 = 9,000 N
  2. Deceleration = 30 m/s²; F = 60 × 30 = 1,800 N
  3. Force dropped 5× too (9,000 → 1,800). Slogan: "…the force you lose it with."