🎢 Mr. Goddard's Maths — IGCSE Calculus
THUNDER
PEAK
Today you're a junior engineer at ThunderPeak Rides.
Your job: analyse the roller coaster using maths, then build your own and launch it.
The Real-World Link
Why do engineers need gradients?
📐
Track steepness
The gradient tells engineers how steeply the track rises or falls — crucial for safety limits.
⚡
Speed changes
A steeper descent means the cart accelerates faster. Gradient links to rate of speed change.
🔄
G-forces on riders
How sharply the gradient changes determines the g-force felt — too much = dangerous.
🧮
Instantaneous rate
We can't measure the exact speed at one point — but a tangent to the curve gives us an estimate.
Reading the graph
The height-distance graph
The x-axis shows horizontal distance in metres.
The y-axis shows height above ground in metres.
The gradient at any point = rise ÷ run = how fast height is changing per metre travelled.
💡 Positive gradient → track is climbing | Negative gradient → track is descending | Gradient = 0 → track is flat (peak or valley)
Method
Drawing a tangent to estimate gradient
1️⃣
Click on the curve
Click once near the marked point (the yellow dot) to place your first tangent point.
2️⃣
Click a second point
Click again to complete the tangent line. Try to match the curve's direction at that point — not the general slope.
3️⃣
Read off rise and run
Use the grid to estimate rise (vertical) and run (horizontal) from your tangent line.
4️⃣
Calculate gradient
Gradient = rise ÷ run. Don't forget the sign — negative if the line slopes downward.
Today's Activity
Three phases — 30 minutes
Phase 1
15 min
🔬 Analyse the track
Draw tangents at 4 marked points. Enter your gradient estimates. Instant feedback on your answers.
Phase 2
8 min
🏗 Build your ride
Drag the control points to reshape the track however you like. Steep ramp at the end = longer flight.
Phase 3
7 min
🚀 Launch!
Press Launch. The cart follows your track. If it flies off — how far can you make it go?
Answer Reveal — after Phase 1
The true gradients
Question 1
x = 120m — Top of first descent
≈ −0.93
Descending steeply. About 0.93m of height lost per metre of horizontal travel.
Question 2
x = 300m — Bottom of first valley
≈ 0
Minimum point — tangent is horizontal. Gradient = 0 at the very bottom of a valley.
Question 3
x = 440m — Top of second hill
≈ 0
Near a maximum — track is almost flat at the crest. Gradient approaches 0.
Question 4
x = 560m — Mid second descent
≈ −0.88
Steeply descending again. Similar magnitude to Q1 — the second big drop.
The Big Idea
What does gradient = 0 mean?
At a peak or valley, the tangent is horizontal.
Gradient = rise ÷ run = 0 ÷ anything = 0.
This is a stationary point — the height is momentarily not changing.
In calculus, this is where dy/dx = 0.
🎢 On the roller coaster: gradient = 0 at the very top and very bottom of each hill. This is where the cart briefly feels weightless (top) or heavy (bottom).
Ready to start?
Open the activity
The link is in the Teams chat. Open it on your device, enter your name, and press Board the ride.
rollercoaster_student.html
Work individually or in pairs · Submit via Google Classroom at the end · 30 minutes