Chapter Boss: Loop Designer

Why it matters

Roller-coaster engineers get one push — the first lift hill. Every loop after that is paid for in height. Overspend and riders black out; underspend and they hang upside down. Welcome to the job.

The idea in plain words

The chapter in one design rule. Tap the terms.

mgh = mg(2r) + ½mv²top, v²top ≥ gr

Loop radius 10 m. Minimum start height?

h = 25 m — exactly what your slider found

• joule: mgh — the budget: all the energy the drop will ever have
• joule: mg(2r) — cost of the climb to the loop top
• joule: ½mv²top — speed you MUST still have up there to hold the track
• h = 2r + r/2 = 20 + 5

Predict first

For a frictionless loop of radius r, the MINIMUM start height is about…

At the loop's top the car still needs enough KE to press outward (v² ≥ gr), so h = 2r + ½r = 2.5r. Reaching with zero speed means falling off.

• 2.5 r — the car needs speed at the top, not just height — correct
• 2 r — just reach the top
• r

What you do

Set the lift hill: too low and the car falls off the loop, too high and you wasted the client's money. Find the exact minimum.

Check yourself

A 60 kg athlete runs up 5 m of stairs in 6 s (g = 10). Her power is…

W = mgh = 3000 J; P = 3000 ÷ 6 = 500 W.

Halve a moving car's speed. Its kinetic energy…

K ∝ v². Half the speed = (½)² = ¼ the energy — and ¼ the crash damage.

A crane lifts 200 kg by 15 m, then holds it still for a minute. Work done WHILE HOLDING is…

Holding is hard on the motor but S = 0 → W = 0. Work was only done during the lift (30 kJ).

kWh and joule measure…

Both are energy units — the kWh is just joules bought in bulk.

A stone thrown UP loses KE because…

Force down, motion up: W = −mgh. Every lost joule of KE sits in the PE account, refunded on the way down.

Why does your coaster need h a bit ABOVE 25 m in real life?

Real tracks leak energy to friction and air drag — engineers pad the budget a few per cent.

• 500 W — correct
• 3000 W
• 50 W
• drops to one quarter — correct
• halves
• doubles
• zero — no displacement, no work — correct
• 30 000 J per minute
• negative
• the same thing — energy (1 kWh = 3.6 MJ) — correct
• different things: power and energy
• different things: energy and force
• gravity does negative work on it, banking it as PE — correct
• the energy is destroyed by height
• air pressure eats it
• friction converts some mechanical energy to heat along the track — correct
• gravity weakens at speed
• the formula is wrong