Solve the Jump

The physics of every stunt in #UNIFYSTL Season One. Eleven chapters, one for each stunt, for grades 6 to 12. Students predict the numbers before each stunt, and the live telemetry settles it.

Download the workbookPDF · 11 chapters · letter size, prints in black and white Answer keyTeacher edition · PDF Watch the stunts9brt.com/episodes

★ Rookie grades 6 to 8, no trig★★ Pro grades 9 to 12★★★ Engineer puzzles with more than one way in

Solve it. Don't try it.Every stunt is done by a professional rider, with engineers, permits and safety systems. Students do the math.

The chapters

Start here · every jump

The Ramp Kit

Why does every jump this season use the same two ramps?

When the lip and the landing are the same height, the wheel lands at the same angle it took off, and the top of the arc sits exactly halfway across the gap. That makes every jump predictable: practice the exact setup in a parking lot until the speed at the lip repeats, then take it to the venue. The season climbs a speed ladder, a little faster each jump, up to 40 mph for the finale.

The physicsGap = speed² × sin(2 × 20°) ÷ g, with both lips at the same height

Lip
The top edge of the launch ramp, where the wheel leaves the ground.
Landing zone
The stretch of the landing ramp that's safe to touch down on.
Speed ladder
The plan to go a little faster each jump, proving each step before the next.

Worked example. Make the kit's table: what speed at the lip clears each gap?

  1. 10 ft = 3.05 m → √(3.05 × 9.8 ÷ 0.643) = 6.82 m/s = 15.2 mph.
  2. 13 ft = 3.96 m → √(3.96 × 9.8 ÷ 0.643) = 7.77 m/s = 17.4 mph.
  3. 20 ft = 6.10 m → √(6.10 × 9.8 ÷ 0.643) = 9.64 m/s = 21.6 mph.
  4. 26 ft = 7.92 m → √(7.92 × 9.8 ÷ 0.643) = 10.99 m/s = 24.6 mph.
  5. 33 ft = 10.06 m → √(10.06 × 9.8 ÷ 0.643) = 12.38 m/s = 27.7 mph.
  6. And the finale: 40 mph flies about 69 ft.

Your turn

  1. ★ Rookie Clue: a training video runs at 60 frames per second. The wheel crosses a 10-foot chalk box in 10 frames. How fast is it going, in feet per second and in mph? (1 mph = 1.467 ft/s.)
  2. ★ Rookie The table jumps from 26 ft to 33 ft. Estimate the speed for a 30-ft gap from the table, then check it with the formula.
  3. ★★ Pro Clue: the launch ramp rises 5 ft over a 13.7 ft run along the ground. What angle is it?
  4. ★★ Pro The speed ladder goes 25, 30, 35, then 40 mph. What gap goes with each step on the kit? When the speed doubles, what happens to the gap?
  5. ★★★ Engineer Engineer: rider and wheel together weigh about 275 lb (125 kg). Holding speed up a ramp face takes mass × g × speed × sin(angle) watts, plus the power to push through the air (½ × 1.2 × 0.6 × speed³). Can the wheel hold 40 mph up the 20° face if it can give 8.4 kW and keep its reserve? What's the fastest it can hold? What about a 15° face at 40 mph?

What gets measured on the night

  • Speed at the lip, every run
  • Where the wheel touches down
  • Motor power on the ramp face

The engineer's question

"Same blueprint, same angle, same height, at every venue." This has to be true before the attempt. Why?

See every episode All chapters

Chapter 00 · Episode 00 · Now · Grades 6 to 12

Také Style

Does standing side-on really use less energy?

How far a wheel goes is the battery's energy divided by the energy it uses per mile. A big share of that energy goes into pushing air out of the way, and the push grows with the area you present to the wind.

The physicsDrag force = ½ × air density × drag coefficient × frontal area × speed²

Watt-hour (Wh)
A unit of energy. One watt for one hour, or 3,600 joules.
Frontal area
How big you look to the air coming at you.
Fair test
An experiment where only one thing changes, so you know what caused the result.

Worked example. A 2,000 Wh battery uses 30 Wh per mile. How far does it go? How far at 27 Wh per mile?

  1. Range = battery ÷ energy per mile = 2,000 ÷ 30 = 66.7 miles.
  2. At 27 Wh per mile: 2,000 ÷ 27 = 74.1 miles.
  3. Saving 3 Wh per mile adds 7.4 miles.

Your turn

  1. ★ Rookie Using the same 2,000 Wh battery, how many Wh per mile would the wheel have to use to go 10 miles farther than it does at 30 Wh per mile? What percent saving is that?
  2. ★★ Pro Find the drag force at 25 mph (11.18 m/s) facing forward and standing side-on. Air density is 1.2 kg/m³ and the drag coefficient is 1.0. The rider's frontal area is 0.55 m² facing forward and 0.40 m² side-on.
  3. ★★ Pro The energy to push through the air for one mile is the drag force times the distance (1 mile = 1,609 m). Find it for both stances in Wh (divide joules by 3,600). How much does side-on save per mile, and is that enough for problem 1?
  4. ★ Rookie List five things that must stay the same in both runs for the test to be fair.

What gets measured on the night

  • Watt-hours per mile in each stance
  • Speed held on the same loop
  • Tire pressure, temperature and wind, so the comparison is fair

The engineer's question

"The result gets published even if the stance loses." This has to be true before the attempt. Why?

See the episode All chapters

Chapter 01 · Episode 01 · November 30 · Grades 6 to 12 (the jump math is 9 to 12)

The Empty Bus

When is a bus the cheapest way to move people, and when would a Lamborghini be cheaper?

Two problems. The jump: clearing the roof is a height problem, like the Zamboni. The bus: it costs about the same to run a bus with 2 people on board or 60, so the cost of each ride is the cost of running the bus divided by the riders. How full the bus is changes everything.

The physicsCost per rider per mile = cost to run the vehicle one mile ÷ riders on board

Cost per rider-mile
What it costs to carry one person one mile: vehicle cost per mile ÷ riders on board.
Load
How many people are on board on average.
Peak demand
The busiest time, like rush hour, that the system has to be sized for.

Worked example. In 2024 it cost Metro St. Louis $13.44 to run a bus one mile, with 6.4 riders on board on average. What does each rider's mile cost?

  1. Riders on board = passenger miles ÷ bus miles = 76,719,531 ÷ 11,973,112 = 6.41.
  2. Cost per rider-mile = $13.44 ÷ 6.41 = $2.10.
  3. Fares paid for about 8 cents of every dollar it cost to run the buses.

Your turn

  1. ★★ Pro The jump. The bus is about 3.2 m (10.5 ft) tall. The season's 20° ramp kit sits on 3.0 m (10 ft) towers, and the wheel should clear the roof by 0.5 m at the top of the arc. What speed does it take at the lip?
  2. ★★★ Engineer With identical ramps at the same height, the top of the arc is exactly halfway across. At the speed from problem 1, how wide is the gap between the lips? The bus is 2.6 m wide and parked in the middle. How much room is there over the roof's edges?
  3. ★ Rookie The bus. Using $13.44 a mile, how many riders does a bus need for each rider's mile to cost under 50 cents? Under $1?
  4. ★★ Pro Now the Lamborghini. Suppose a chauffeured Urus works like this: it costs $281,000, is driven 45,000 miles a year for 5 years, then sells for $50,000. It gets 20 mpg with gas at $3.20. Insurance is $12,000 a year. Repairs and tires are 50 cents a mile. The driver costs $40 an hour, and city traffic averages 14 mph. What does it cost per mile? Per rider-mile with 4 riders?
  5. ★★ Pro How many riders does the bus need before each one rides for less than in the full SUV? Then rush hour: 60 people at one stop. What does it cost per mile to move them in SUVs, 4 to a car, compared with one bus?
  6. ★ Rookie So why do cities run big buses all day? Give two reasons. Then name one thing a city could do when a route runs nearly empty.

What gets measured on the night

  • Speed at the lip
  • Clearance over the roof
  • Landing force in g

The engineer's question

"Landed over a bus-sized frame at ground level first." This has to be true before the attempt. Why?

See the episode All chapters

Chapter 02 · Episode 02 · December · Grades 6 to 12

79 in One

What's the shortest route that touches all 79, and does the battery make it?

Visiting many places by the shortest route is one of the most famous problems in math: the traveling salesperson problem. Then comes an energy budget. The miles times the energy per mile has to fit in the battery, or you plan the charging stops.

The physicsEnergy needed = route miles × watt-hours per mile

Traveling salesperson problem
Finding the shortest loop that visits every point once.
Energy budget
Adding up the energy a trip needs before you start.
Reserve
Energy you plan never to use, in case something goes wrong.

Worked example. A 2,000 Wh battery at 30 Wh per mile. How far per charge? How far if you always keep 20% in reserve?

  1. Full battery: 2,000 ÷ 30 = 66.7 miles.
  2. Keeping 20% in reserve leaves 80% of 2,000 = 1,600 Wh to use.
  3. 1,600 ÷ 30 = 53.3 miles per charge.

Your turn

  1. ★ Rookie Five checkpoints, A to E, are this many miles apart: A–B 4, A–C 7, A–D 6, A–E 3, B–C 3, B–D 8, B–E 5, C–D 4, C–E 6, D–E 5. Start and finish at A and visit each checkpoint once. There are 12 different loops. Find the shortest.
  2. ★ Rookie Suppose the full 79-neighborhood route is 120 miles at 30 Wh per mile. How much energy does it need? Keeping a 20% reserve (1,600 Wh usable per charge) and starting full, how many charging stops does it take?
  3. ★★ Pro Sunrise to midnight is 17 hours. The wheel averages 14 mph while moving. Each of the 79 checkpoints takes 3 minutes for its photo. Each charging stop puts back 1,600 Wh from an 800-watt charger. Does the day fit, and how much time is left over?

What gets measured on the night

  • The GPS track, proving all 79
  • Total miles, ride time and stopped time
  • Energy used and every charge stop

The engineer's question

"Daylight for the worst pavement." This has to be true before the attempt. Why?

See the episode All chapters

Chapter 03 · Episode 03 · New Year's · Grades 9 to 12

The Keg Line

How fast at the lip to clear ten kegs?

Between two ramps of the same height, the wheel is a projectile. How far it flies depends only on its speed at the lip and the ramp's angle. Distance grows with the square of speed, so a little speed buys a lot of distance.

The physicsDistance = speed² × sin(2 × ramp angle) ÷ g

Projectile
Anything flying with only gravity acting on it.
Takeoff angle
The angle of the ramp's lip above horizontal.
Range
The horizontal distance from takeoff to landing.

Worked example. Ten kegs, each about 16 inches wide, between two 20° ramps. How fast at the lip?

  1. Distance = 10 × 16 in = 160 in = 4.06 m.
  2. Rearrange: speed = √(distance × g ÷ sin(2 × angle)).
  3. sin(40°) = 0.643. Speed = √(4.06 × 9.8 ÷ 0.643) = 7.87 m/s.
  4. 7.87 m/s ÷ 0.447 = 17.6 mph.

Your turn

  1. ★★ Pro Twelve kegs off the same 20° ramps. What speed at the lip?
  2. ★★ Pro Ten kegs again, but with 30° ramps. What speed now? Give one reason the rider might still choose 20°.
  3. ★★ Pro How long is the wheel in the air in the worked example? Time in the air = 2 × speed × sin(angle) ÷ g.
  4. ★★★ Engineer The rider arrives 1 mph slow on the ten-keg jump at 20°. How far does the wheel go, and how short does it land?

What gets measured on the night

  • Speed at the lip
  • Time in the air and distance cleared
  • Landing force in g

The engineer's question

"The count only goes up after a clean landing." This has to be true before the attempt. Why?

See the episode All chapters

Chapter 04 · Episode 04 · January · Grades 9 to 12

Jump the Zamboni

How high does the jump have to go, and can a tire grip ice hard enough to get there?

Two problems at once. How high the arc goes depends on the upward part of the speed at the lip. Getting to that speed depends on grip, and ice gives a rubber tire almost nothing to push against, which is why the tire gets studs.

The physicsPeak height above the lip = (speed × sin(ramp angle))² ÷ (2 × g)

Vertical component
The part of the speed pointing straight up: speed × sin(angle).
Coefficient of friction
How hard a surface lets you push sideways compared with your weight.
Run-up
The distance used to get up to speed before the lip.

Worked example. On the season's 20° ramp kit, how high above the lip does the arc peak at 20 mph?

  1. 20 mph = 8.94 m/s. Upward part = 8.94 × sin(20°) = 3.06 m/s.
  2. Peak height = 3.06² ÷ (2 × 9.8) = 0.48 m, about 1.6 feet.

Your turn

  1. ★★ Pro Suppose the machine is 2.0 m tall and the kit's lips are 1.5 m (5 ft) off the ice. The wheel should clear the machine by 0.3 m at the top of the arc. How high above the lip must the arc peak, and what speed does that take at the lip?
  2. ★★★ Engineer The quickest a tire can speed you up is about (friction coefficient × g). Say plain rubber on ice is 0.1 and a studded tire is 0.3. The run-up needed = speed² ÷ (2 × acceleration). The machine sits at center ice, about 30 m from the end boards. Which tire can reach the speed from problem 1 in time?
  3. ★★★ Engineer The rider arrives 1 mph slow. Does the wheel still clear the machine, and by how much?
  4. ★★ Pro Where does the machine go? With identical ramps the top of the arc is halfway across. At the speed from problem 1, how wide is the gap between the lips, and how far from the launch lip is the top of the arc?

What gets measured on the night

  • Speed and wheel slip on the run-up
  • Peak height over the machine
  • Landing force on ice

The engineer's question

"The studded tire is tested for grip before any ramp goes down." This has to be true before the attempt. Why?

See the episode All chapters

Chapter 05 · Episode 05 · Mardi Gras · Grades 6 to 12

The Parade

Why is it harder to balance slow than fast?

A rider on a wheel is an inverted pendulum, like a broomstick balanced on your palm. It starts to fall the moment it's upright, and the taller it is, the slower it falls. The wheel's motor catches the front-to-back fall over and over, many times a second. Side to side, the rider steers under the fall.

The physicsA feedback loop: sense the lean, correct, repeat

Inverted pendulum
A weight balanced above its pivot. It falls unless something keeps correcting it.
Feedback loop
Sense, correct, sense again. The faster the loop, the smaller each correction.
Fall time
Roughly √(height of the balance point ÷ g). Taller things fall more slowly.

Worked example. Why is a broomstick easier to balance than a pencil?

  1. Broomstick, balance point about 1.5 m up: √(1.5 ÷ 9.8) = 0.39 s.
  2. Pencil, about 0.15 m: √(0.15 ÷ 9.8) = 0.12 s.
  3. The pencil falls about 3.2 times faster, faster than a hand can react.

Your turn

  1. ★ Rookie A rider's balance point is about 1.0 m above where the tire touches the ground. What's the fall time? If the wheel's controller corrects 1,000 times a second, how many corrections fit inside it?
  2. ★★ Pro Side to side, the rider stays up by steering. The sideways push a turn gives you is speed² ÷ turn radius. Compare 3 mph and 12 mph through the same 3 m turn.
  3. ★ Rookie Experiment: balance a broomstick on your palm, then a 30 cm ruler, five tries each, and time each try. Average them. Which lasted longer, and by about how much does the fall-time formula say it should?

What gets measured on the night

  • Slowest speed held without a foot down
  • Corrections per second from the wheel's own sensors
  • Crowd noise in decibels at the stunt

The engineer's question

"No kids inside the stunt zone." This has to be true before the attempt. Why?

See the episode All chapters

Chapter 06 · Episode 06 · April · Grades 9 to 12

The Bend

How far do you have to lean to hold a turn at speed?

In a turn, the tire has to push you toward the inside, and you lean so that push and gravity line up through you. Faster or tighter means more lean, until the tire runs out of grip.

The physicstan(lean angle) = speed² ÷ (turn radius × g)

Lean angle
How far the wheel and rider tip from vertical in a turn.
Turn radius
The radius of the circle the turn is part of.
Grip limit
The most sideways push the tire can give before it slides.

Worked example. 30 mph around a 50 m radius. How far does the rider lean?

  1. 30 mph = 13.41 m/s. speed² = 179.9.
  2. tan(lean) = 179.9 ÷ (50 × 9.8) = 0.367.
  3. Lean = 20.2°.

Your turn

  1. ★★ Pro The same 50 m turn at 20 mph. What's the lean?
  2. ★★★ Engineer A tire slides when tan(lean) would have to be more than the friction coefficient. With 0.8 on dry pavement, what's the most lean, and the fastest speed around the 50 m turn? With 0.5 on a damp deck?
  3. ★★★ Engineer Suppose a map shows the bridge's bend turning 22° across 120 m of deck. Radius = arc length ÷ angle in radians. Find the radius, then the lean at 30 mph. Is the bend the hard part?

What gets measured on the night

  • Speed entering, in and leaving the bend
  • Lean angle from the wheel's sensors
  • Time over the full mile

The engineer's question

"The deck inspected seam by seam." This has to be true before the attempt. Why?

See the episode All chapters

Chapter 07 · Episode 07 · May · Grades 9 to 12

The Spirit

Why does going twice as fast take eight times the power?

Air drag grows with the square of speed, and power is force times speed, so the power to beat the air grows with the cube. Twice the speed takes eight times the power. A one-wheel has a second problem: the motor that moves you also balances you, so it can never run out.

The physicsPower to beat drag = ½ × air density × drag coefficient × frontal area × speed³

Power
How fast energy is used, in watts.
Drag area
Drag coefficient × frontal area, in m². Smaller is slipperier.
Reserve
Power held back on purpose so the wheel can still balance.

Worked example. With a drag area of 0.6 m², how much power does it take to beat the air at 25 mph? At 50 mph?

  1. 25 mph = 11.18 m/s. Power = ½ × 1.2 × 0.6 × 11.18³ = 503 W.
  2. 50 mph = 22.35 m/s. Power = ½ × 1.2 × 0.6 × 22.35³ = 4,020 W.
  3. Twice the speed, 8 times the power.

Your turn

  1. ★★ Pro How much power does it take at 40 mph?
  2. ★★★ Engineer The motor can put 3,000 W into fighting the air, but 30% has to stay in reserve for balance. What's the top speed?
  3. ★★★ Engineer Tucking cuts the drag area to 0.45 m². With the same usable power, what's the new top speed? What drag area would it take to gain a full 5 mph over problem 2?
  4. ★ Rookie With the wind behind him the timing trap reads 44 mph. Against the wind it reads 38 mph. What speed goes in the record, and why run both directions?

What gets measured on the night

  • Speed through a timed trap, both directions, averaged
  • Motor power and how much reserve was left
  • Wind speed and direction

The engineer's question

"A hard cap on speed that keeps motor power in reserve." This has to be true before the attempt. Why?

See the episode All chapters

Chapter 08 · Episode 08 · June · Grades 9 to 12

Art Hill

Why do big jumps land on a down-ramp?

Falling builds downward speed. Land on flat ground and the legs have to stop all of it at once. Land on a down-ramp that matches the flight path and the ground only has to stop the small part of the speed pointed into the ramp. That's why every big jump lands on a slope.

The physicsDownward speed at landing = √(2 × g × height fallen)

Downward speed
How fast you're falling at the moment you land.
Landing angle
The angle of your path below horizontal as you land.
g-force
Acceleration measured in multiples of gravity. 1 g is standing still.

Worked example. The wheel falls 1.5 m. How fast is it moving downward when it lands?

  1. Downward speed = √(2 × 9.8 × 1.5) = √29.4 = 5.42 m/s.
  2. 5.42 ÷ 0.447 = 12.1 mph straight down.

Your turn

  1. ★★ Pro What if the drop is 3.0 m? How much faster is that than 1.5 m?
  2. ★★ Pro The wheel is moving forward at 8.0 m/s and falls 1.5 m. At what angle below horizontal is it moving when it lands?
  3. ★★★ Engineer Land on a 30° down-ramp instead of flat ground. Only the speed into the ramp has to be stopped: (downward × cos 30°) − (forward × sin 30°). Find it. What share of the flat-ground hit is left?
  4. ★★★ Engineer Bending the knees gives about 0.30 m to stop in. Average stopping acceleration = speed² ÷ (2 × distance). Find it in g for the flat landing and the ramp landing.

What gets measured on the night

  • Takeoff speed and angle
  • Distance and height
  • Landing force, compared with each team's prediction

The engineer's question

"Crowd lines set well back from the landing." This has to be true before the attempt. Why?

See the episode All chapters

Chapter 09 · Episode 09 · July · Grades 6 to 12 (the runway math is 9 to 12)

Center Stage

If you throw a ball while you're riding, how fast is it really going?

Speeds add when one moving thing is carried by another. A ball thrown from a moving wheel leaves with the wheel's speed plus the throw; thrown backward, they subtract. The jump uses the ramp kit: the speed at the lip sets how far the wheel flies, and the runway has to be long enough to get up to that speed and to stop afterward.

The physicsBall speed over the ground = wheel speed + throw speed (thrown backward: wheel speed − throw speed)

Relative motion
How fast something moves depends on who's measuring: the rider, or the crowd standing still.
Frame of reference
The point of view you measure speed from.
Braking distance
How far it takes to stop: speed² ÷ (2 × braking acceleration).

Worked example. Riding at 20 mph, he throws the first pitch forward at 50 mph. How fast does the ball cross the plaza? How fast does it look to him?

  1. To the crowd: the wheel's speed and the throw add. 20 + 50 = 70 mph.
  2. To the rider: he's moving with the ball's starting speed, so it leaves him at the 50 mph he threw it.

Your turn

  1. ★ Rookie Riding 15 mph, he throws forward at 45 mph. What's the ball's speed over the ground?
  2. ★ Rookie The radar on the plaza reads the pitch at 68 mph. The wheel's GPS says it was doing 21 mph. How hard did he actually throw?
  3. ★ Rookie Riding 20 mph forward, he tosses the ball backward at 20 mph. What does the crowd see the ball do? What does the rider see?
  4. ★★ Pro The jump: 35 mph at the lip on the 20° kit. How wide is the gap, how long is the wheel in the air, and how high above the lip does it go?
  5. ★★★ Engineer The runway. Say the wheel speeds up at 0.3 g and brakes at 0.4 g. How much runway does it take to reach 35 mph? To stop after landing? Add about 15 ft for each ramp and the gap from problem 4. How long does the runway have to be?

What gets measured on the night

  • Speed at the lip
  • Pitch speed by radar, thrown from the wheel
  • Landing force in g

The engineer's question

"Room to stop before the stage." This has to be true before the attempt. Why?

See the episode All chapters

Chapter 10 · Episode 10 · Finale · Physics

The Wash Ave Gap

There's no second try. What is the window?

There's no second try. Too slow and the wheel lands on the deck before the landing slope; too fast and it overshoots the landing zone. At 40 mph every mile an hour moves the landing about three and a half feet, so the landing zone is the margin, and the math says how long it has to be.

The physicsGap = speed² × sin(2 × ramp angle) ÷ g, with both lips at the same height

Landing zone
The stretch of the landing ramp that's safe to touch down on.
Window
Every speed at the lip that still lands in the zone.
Target speed
The speed the rider aims for: the middle of the window.

Worked example. On the season's 20° ramp kit, how far does 40 mph at the lip fly, how high, and for how long?

  1. 40 mph = 17.88 m/s.
  2. Gap = 17.88² × sin 40° ÷ 9.8 = 20.97 m, about 69 ft.
  3. Height above the lip = (17.88 × sin 20°)² ÷ 19.6 = 1.91 m, about 6.3 ft.
  4. Air time = 2 × 17.88 × sin 20° ÷ 9.8 = 1.25 s.
  5. Every 1 mph changes the distance by about 3.4 ft.

Your turn

  1. ★ Rookie Every 1 mph moves the landing about 3.4 ft. If the rider hits the lip 3 mph slow, about how far short of 69 ft does he come down? Check it with the gap formula at 37 mph.
  2. ★★ Pro The safe landing zone runs from 64 ft to 74 ft past the lip. What speed window does that give?
  3. ★★★ Engineer How long would the landing zone have to be for a ±3 mph window around 40 mph?
  4. ★★★ Engineer Design the finale kit. Compare a 20° kit and a 15° kit at 40 mph: the gap, the height above the lip, the air time, and the power to hold 40 mph up the ramp face (275 lb of rider and wheel, drag area 0.6 m²; see The Ramp Kit). If 8.4 kW is all the wheel can give while keeping its reserve, which do you pick and why?

What gets measured on the night

  • Speed at the lip, live
  • Flight time and touchdown point
  • Wind on both roofs
  • Landing force

The engineer's question

"If any one of those is a no, it doesn't happen." This has to be true before the attempt. Why?

See the episode All chapters