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Algebra 1 9-12 / Week 10 / Wednesday
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Week 10 · Quadratic Equations

Wednesday

Loft Lab: time the flight
// When is the height zero?
⏱ about 20 min

Wednesday: Loft Lab, Time the Flight

A grown-up checks the launcher tubing while Comet fills the bottle a third full. "Lab day. Three pump counts, three flights each."

Wren holds the stopwatch. "Four pumps first. I time from launch to landing. What do you notice about the timings?"

The first flight lands in about three seconds. The six-pump flight takes four. The eight-pump flight takes five.

"Longer flight, taller arc," Comet says. "Can we write a rule for each?"

Nova projects the whiteboard rule. "Would you like a hint? Your rule lands at t = 4. What if it landed at T?"

"h = 5t(T - t)," Wren says. "Zero at t = 0 and at t = T. Our model, from our timings."

"Then when is the eight-pump rocket at 20 meters?" Comet asks. "That is a quadratic equation."

What you need

  • A plastic bottle rocket with cardboard fins, its launcher tubing and a bicycle pump.
  • Water, a stopwatch or phone timer, a tape measure and your Loft Log.
  • A grown-up, an open outdoor space and a dry place to stand well back.
Safety first
A grown-up is present for every launch and checks the launcher before each one.
Everyone stands back behind the launch line. The rocket always points up and away from people, windows and roads.
Never look down into the bottle while it is under pressure. Wear glasses or goggles if you have them.
Launch only in open space, never near power lines. Wipe up water so no one slips.

Run the lab

  1. Set the pump count, for example 4. Fill the bottle a third full and set it on the launcher.
  2. Everyone steps back. The grown-up gives the go. Pump, launch, and start the stopwatch at liftoff.
  3. Stop the watch when the rocket lands. Record the flight time T in your table.
  4. Repeat twice more at the same pump count and take the middle value.
  5. Repeat for 6 pumps and 8 pumps.
  6. For each T, write the crew's model h = 5t(T - t) and expand it to -5t² + 5Tt.

The crew's Tuesday timings

The crew's timings are made up for the Loft. Yours will differ, and that is the point of a lab.

PumpsFlight time T (s)Model h(t)Lands at
43-5t² + 15tt = 3
64-5t² + 20tt = 4
85-5t² + 25tt = 5

The 5 in the model is the crew's own fit to their timings. It is a model, not a law, and a different crew might fit a different number.

When is the eight-pump rocket at 20 meters? Set -5t² + 25t = 20. Divide everything by -5: t² - 5t + 4 = 0.

Factor: (t - 4)(t - 1) = 0, so t = 1 or t = 4. Once going up, once coming down.

SOLVE THE LAB EQUATIONS
  • Read the question.
  • Tap your answer.
The eight-pump model is -5t² + 25t. When is the height 20? Solve t² - 5t + 4 = 0.
The four-pump model is -5t² + 15t. Solve -5t² + 15t = 0 for the launch and landing times.
The six-pump model is -5t² + 20t. When is the height 15? Solve t² - 4t + 3 = 0.
The eight-pump model is -5t² + 25t. How many seconds after launch does it land? Type the number.
WHY THIS EXERCISEEach flight time gives its own quadratic. Solving it tells when the rocket is at any height.
StatementTrue or false?
A longer flight time T gives a taller arc in the crew's model.?
5 × 5 × (5 - 5) = 0?
5 × 2 × (5 - 2) = 30?
The 5 in the model is a law of nature that every crew must use.?
Dividing both sides of an equation by -5 changes its solutions.?
WHY THIS EXERCISEThe model is the crew's own fit. The algebra moves are the same ones that kept the hanger balanced in week 2.
Draw your launch table: pumps across, flight time up. Mark the middle value for each pump count.
Try it
Use your own middle flight time T for 6 pumps. Write h = 5t(T - t) and find when your rocket was half as high as its top.
Hint: the top height is h at t = T ÷ 2. Set h equal to half of that and solve.

Great lab work. Tomorrow the crew asks which heights the rocket ever reaches, and the discriminant answers.

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