Can the long way be the fast way? | High Road, Low Road Race

You'll build two side-by-side marble roads that start at the same height and finish at the same height. The high road runs straight from start to finish. The low road dips down to the table and climbs back up, so it is a little longer. Most people bet on the short, straight road, and most people lose. Bruce builds his tracks from wood and metal shelf track (there is a detailed build plan for that); this version uses foam pipe insulation and a stack of books.

Safety. A grown-up should do the long cuts in the foam tube. Marbles on the floor are easy to slip on, and small children can choke on them, so keep them in a cup between races and use bigger balls for little ones.

“Watching in slow motion you can see both balls start together, but look at the big lead the one ball gets as it drops to a lower position.”
Bruce Yeany, in the video (5:48)

Step by step

Build it

Observe

Try it & notice

Predict first: both marbles start at the same height and finish at the same height. Which one will reach the end first: the high road, the low road, or a tie? Tell someone your guess.

Hold the ruler across both channels just past the edge of the start stack. Set a marble behind it in each road, then lift the ruler straight up. Which marble reaches the end first? Race at least three times.

Swap the two marbles and race again. Does the winner change? (If not, it's the road, not the marble.)

Lay the string along each road from start to finish and mark it. Which road is longer? Does the longer road win or lose?

Watch the middle of the race closely, or film it in slow motion. Where does the low-road marble pull ahead? Where is it moving fastest?

Explain

What's going on?

For young scientists

When you lift a marble up high, it stores energy, like a stretched rubber band waiting to go. The higher it is, the more energy it can turn into speed as it rolls downhill. Both marbles start with the same height, so they start with the same stored energy.

The low-road marble drops much lower, so it turns much more of its height into speed, right near the start. It zooms along the bottom of the dip much faster than the high-road marble, which only creeps down its gentle slope. Climbing out at the end slows it down again, but by then it has a big lead.

The low road really is longer. But speed matters more than distance here: the low-road marble is faster for almost the whole trip, so it wins even though it goes farther.

For grown-ups & teachers

This is conservation of energy. Each ball starts with the same gravitational potential energy (mgh). As it rolls, the drop in height becomes kinetic energy, so at any point its speed depends only on how far it is below the start: for a solid ball rolling without slipping, v = sqrt(10/7 g h). The low-road ball reaches a lower point earlier, so it is faster over most of the course, and the time for a trip is the sum of distance divided by speed along the way. A modest increase in distance (Bruce measures about 5 percent on his classroom track and about 8 percent on the giant outdoor one) is easily outweighed by much higher average speed.

The common misconception is that the shortest path must be the fastest. The straight line is the shortest path, but not the quickest one under gravity. The fastest route between two points is a curve that drops steeply early, called the brachistochrone (a cycloid); Bruce's identical-curved-tracks build plan explores the related idea of a curve where the time to the bottom hardly depends on where you start.

Rolling matters. A rolling ball splits its kinetic energy between moving forward (translational) and spinning (rotational), so it is slower than a sliding block would be, but both balls share the same split, so the race stays fair. Bruce's part 2 video uses this to explain why slopes must stay gentle: if a climb is so steep that the ball slips instead of rolling, it loses energy to friction and may not make it out. Foam channels also have more rolling resistance than metal track, which is why the home version ends lower than it starts.

Ignoring friction, both balls arrive at the finish with the same speed, because they end at the same height. Only the time differs. To extend it, add hills to the low road that rise to the height of the straight road (Bruce's hill tracks still win), make the dip deeper, or time the races with slow-motion video.

NGSS performance expectations

4-PS3-1 Use evidence to construct an explanation relating the speed of an object to the energy of that object.
MS-PS3-2 Develop a model to describe that when the arrangement of objects interacting at a distance changes, different amounts of potential energy are stored in the system.
MS-PS3-5 Construct, use, and present arguments to support the claim that when the kinetic energy of an object changes, energy is transferred to or from the object.

Science words

potential energystored energy; a marble held up high has potential energy because it can fall

kinetic energythe energy something has because it is moving

speedhow far something goes in a certain amount of time

distancehow long a path is from start to finish

frictiona rubbing force that slows things down

predictionyour best guess about what will happen, made before you test it

From Bruce's notebook

Tips from 41 years of teaching it

Bruce wrote up his high road, low road tracks in 1997 and has built many versions since. Here's what his notes add.

Bruce's first, sturdier version raced two pool balls; his quick version used golf balls. Steel balls and large glass marbles work too.
His quick version is small: a 2-foot straight track, a low road about 4 inches longer, sloping down to the table over about 5 1/2 inches and climbing back up over about 2 inches.
The tracks can be made taller, but only if the low road's slope isn't so steep that the ball slips. Bruce notes that a slipping ball loses some of its rotational (spinning) energy and won't make it back to the top.
On his hill tracks, every little hill rises to the same height as the straight track. Wherever the ball is lower than the straight track, it moves faster, so the hilly track still wins.
Bruce runs fine sandpaper along the edges of the track so the balls roll smoothly; any roughness shows up as a slower ball.
The space between the two lanes depends on the balls: they must be far enough apart that the balls can't touch as they race side by side.

Bruce poses the question before he ever lets the balls go: which track will get the ball to the other side first, and why? Once he had the basic pair working, he kept raising the stakes, building low roads with one, two, three and four humps and one with a much bigger drop, and asking the same question each time.
From Bruce's classroom

More to try

From Bruce's track notes: race different types of balls against each other, such as brass against aluminum, and see if there is a difference.

Go further

Make it a challenge

Add hills

Bruce builds low roads with as many as four humps. Tape a few small books under the low road to make hills. Does it still win?

Deeper dip

Raise the start and finish stacks so the dip is deeper, keeping the curves gentle. Does the low road win by more?

Measure the difference

In part 2, Bruce lays a strip of paper along each road to compare their lengths. Measure yours with string and work out how much longer the low road is.

Different balls

Race a steel marble against a glass marble on the same road, or swap in bigger balls. Does the type of ball matter as much as the shape of the road?

Go big

Bruce built a giant version outdoors with two bowling balls that splash into a wading pool. With a grown-up, try pool noodles cut in half and golf balls on a longer, sturdier slope.