In Activity 4-2-2 you measured the fraction f of its height a marble returned in a valley. That worked because every valley had the same shape. A coaster is different. It might be long and winding or short and direct. The longer the marble travels, the more friction acts on it, so a fraction measured on one track can’t be borrowed for another.
Remember from Activity 4-2-2 that the thermal energy friction produces is F × d: a force acting over a distance. For a coaster, d is the length of track the marble rolls along. So the right question isn’t “what fraction comes back?” It’s “how much energy does one meter of this tubing cost?”
🔑The three lengths in the diagram do different jobs. h₀ is a height: it sets the energy you start with. s is a length along the tube: it sets how much friction acts. k describes how steep the hill is.
BMath
Calibrating λ: Height Lost per Meter
Because the marble is at rest at both turning points of your calibration valley, the PE difference between them is exactly the thermal energy friction made. Divide the height lost by the track length traveled, and you get a number that belongs to the tubing, not to any one track:
λ = (h₁ − h₂) ÷ s
λ (the Greek letter lambda) is measured in meters of height per meter of track. Multiply it by mg and you get the energy lost per meter: E per meter = mgλ.
Calibrating a valley
Given
Marble mass m = 0.005 kg
Released from h₁ = 0.30 m; peak at h₂ = 0.255 m
String measured along the path: s = 2.22 m
Solve
λ = (0.30 − 0.255) ÷ 2.22 = 0.045 ÷ 2.22
λ ≈ 0.0203 m per m
Energy per meter = m g λ = 0.005 × 9.8 × 0.0203
≈ 0.00099 J per m = 0.99 mJ per m
✓ Each meter of this tubing costs the marble about 1 mJ, or about 2 cm of height.
💡Measure s with a string laid along the inside of the tube, then straighten the string and measure it against a meter stick. A ruler can’t follow a curve, but a string can.
CMath
Predicting the Climb
Now follow the marble from the release point to the highest point it reaches on the hill. It starts with PE of mgh₀ and ends, at rest, with mgh. The difference is friction’s bill. Friction acts over two stretches: the track before the hill (length s), and the climb itself, whose length is k × h, where k = track length ÷ rise for your hill.
Using λ to turn track length into lost height:
h₀ − h = λ × (s + k h)
Solving for h gives the formula you’ll use in the project:
h = (h₀ − λ s) ÷ (1 + λ k)
Predicting a climb
Given
h₀ = 0.50 m, marble 0.005 kg
λ = 0.020 m/m (from your calibration)
Track before the hill: s = 2.4 m
Hill: 0.80 m of track rises 0.20 m, so k = 0.80 ÷ 0.20 = 4.0
Solve
h = (h₀ − λs) ÷ (1 + λk)
h = (0.50 − 0.020 × 2.4) ÷ (1 + 0.020 × 4.0)
h = (0.50 − 0.048) ÷ (1 + 0.08) = 0.452 ÷ 1.08
h ≈ 0.419 m
Budget: PE start = 24.5 mJ PE at peak = 20.5 mJ
Friction before the hill = 2.35 mJ on the climb = 1.64 mJ total ≈ 4.0 mJ
✓ The marble should climb to about 0.42 m. The budget balances: 24.5 mJ = 20.5 mJ + 4.0 mJ.
⚠️The λk term in the denominator is easy to forget. It accounts for the friction the marble meets while climbing. Leave it out and you’ll predict a climb that is too high, because you’ve ignored part of the bill.
DExample
Coaster Budget Calculator
Try it before you build. Which matters more for your climb: a taller start, a smoother track, or a shorter track? Notice the range at the bottom, too.
Design a coaster on paper. The marble’s mass is fixed at 0.005 kg. Change the release height, the friction, the track length, and the hill’s slope, and see where the marble ends up.
h = (0.5 − 0.02 × 2.4) ÷ (1 + 0.02 × 4)
Predicted climb0.419 m
Start 24.5 mJFriction before hill 2.35 mJFriction on climb 1.64 mJPE at peak 20.5 mJ
If your λ is really 20% bigger, the climb is 0.404 m. If it’s 20% smaller, it is 0.434 m. A prediction is a range, not a single number.
EMath
Measuring and Judging Your Prediction
A prediction is only useful if you can say how good it was. The standard way is percent error, which compares the gap between predicted and actual to the actual value:
✓ Case 1 is a strong prediction. Case 2 is off by a fifth, so something in the model (probably λ or a length) needs a second look.
💡Everything you measure has some uncertainty: heights to about ±1 cm, lengths to about ±2 cm, and λ most of all, since it comes from a small height difference. A prediction within about 10% is strong. A larger gap isn’t a failure. It’s information about where your model is weakest.
To track down a gap, ask which input was least certain. Re-measure s and k. Check the joints for kinks. If your coaster has tight curves, remember that curves usually cost extra, so calibrate λ on a curved test track and make a second prediction.
FConcept
Why Predict Heights, Not Speeds?
A rolling marble holds some of its kinetic energy in its spin, not just in its forward motion. That makes speed along the track harder to predict from a simple budget. But at the top of the climb the marble stops for an instant. It isn’t moving or spinning, so all of its remaining energy is PE, and the budget stays clean. That’s why the project asks for a climb height.
🔑Whenever you can, compare energies at moments when the object is at rest. They’re the easiest to measure and the easiest to budget.
GWatch Out
Three Common Mix-Ups
⚠️“A taller start gives a taller finish by the same amount.” Not quite. Friction takes its share along the way, so each extra centimeter at the start gives you slightly less than a centimeter at the finish.
⚠️“A good prediction has to match exactly.” No real measurement does. What matters is that you made the prediction first, gave a range, measured honestly, and explained the gap.
⚠️“Measure the track length with a ruler across the top.” The marble travels along the curve, so you have to measure along the tube. A straight-line distance is too short and makes your friction look smaller than it is.
HConcept
Quick Check
Four questions. They are not graded. Answer them to find out what still needs work.
Q1A marble is released from 0.30 m and rises to 0.25 m after traveling 2.0 m of track in total. What is λ?
Q2Why do we predict how high the marble climbs, rather than its speed at the bottom?
Q3You predicted a climb of 0.42 m and the actual climb was 0.40 m. What is the percent error (relative to the actual)?
Q4Your coaster has long, tight curves, but you calibrated λ on a straight valley. What should you do before predicting?
Score: 0 of 4
IConcept
Looking Ahead
The energy budget you build here is the tool you’ll use to close out the unit.