I’m not a (big) jumper, or hucker for that matter. Regardless, I’ve hit some good ones that seem to be dead on, take-off to landing that is. I’ve also gone too slow and too fast on jumps and suffered the consequences.

I think a lot of builders go with trial and error: ride it, reshape it, ride it… In other words, experimentation or science. By science I mean the consistency or repetition of outcome, and even prediction, which is what riders seek— science meets mathematics.

The math starts relatively simple for jumps and ladder drops (see below), but a lot depends on the rider, the bike, and how they behave on lift-off, but a little math will get riders/builders in the ball park faster so there is more science and success, and less trial-and-error witchcraft. Experienced builders that don’t break out calculators are good at witchcraft, or perhaps its craftsmanship/art; they can see the parabolic line of flight when they build because chances are they have enough flight-time to render themselves Pilots in Command. However, chances are their luck has ran out more than once, as well as their time riding because they were re/moving dirt, perhaps with elbows, knees, and face.

The likelihood of a rough landing or total miss and serious crash are lowered if a little math is done first. Unless of course you want to move earth multiple times rather than ride. Ride more or build more? Eat shit more? The answer is obvious, do your homework first. You can bet your ass that some stuntmen and record breaking jump events broke out calculators and rulers, and possibly consulted physicists.

Projectile Motion and Newton

The laws of motion determine jump flight. In particular, equations surrounding projectile motion. These facts of life made me sit up in physics class, and eventually make the calculator below.

A little math may lower the amount of “experimentation” and tweaking, and get you closer to the desired outcome faster. And lower risks, may lower insurance premiums! In spite of geology and rider behaviors throwing the best intentions a curve-ball, break out rulers and protractors with your shovels…

Parabolic Jump/Projectile Trajectories^Lines of jump (and huck) flight are parabolic. NOTICE how 30o and 60o (and other angles in light gray) yield the same distance or range, R, with the same initial velocity, but yield significantly different heights. 45o will give the farthest range– it’s not magic, it’s the Law. Try 30o and 60o degrees using the calculator below to see for yourself, or experiment by building two ramps side-by-side at the same height and sharing the same landing area (the 60o landing side should be a little steeper though, or go between 30o and 60o for both, perhaps 45o). See the Excel sheet image for a tabular representation of the parabolas.

In thinking about jumps and rollers I assume the safe bet is to make them smooth (not too jerky/bumpy) for bigger bikes as they have the biggest wheel base. For extra large 29ers this is somewhere near a 46 inch wheelbase…which led me to produce the following thought experiment (a better method is described below these circles under the “roller” heading):

If a jump, roller, or grade reversal is tall enough they can reduce speeds significantly, or to zero, which could allow designers to set up the next jump, roller, or reversal series to control speeds as they need for that series. (contact me about consulting)

The thoughts above made me think about two other things: the golden ratio (and angle) and perhaps more importantly isochronic or tautochrone curves (below), the implications of which are many for trails and bikes, and designing features based on wheel size:

In creating the image above I was reminded of a guy that emailed me about a year after this calculator was first posted to tell me he was using it to jump cars! He wanted to talk to me on the phone. I panicked because spending time in court five states away immediately crossed my mind. However, he was grateful, thanked me repeatedly, asked how I did the math, and if I could help him with the takeoff ramp. The drawing above assumes circular transitions, but the calculator assumes nothing other than the angle at takeoff. The car jumper was building dead straight transitions, or wedges, a little rough even with suspension depending on his speed. If rough enough it could possibly change the range of his flight. My guess is that he never skated vert or built ramps before to realize just how smooth a takeoff could be. I suggested an arced/curved entry that would then flatten based on his wheelbase. He also wanted me to design a ramp/rail device to help him get up onto two wheels to ski his cars like the Dukes of Hazzard or Knight Rider. I told him I’d pass on that because it’s not quite as straight forward as projectile motion. He was trying the witchcraft method before he found this page.

Rollers

The original circles I daydreamed about above taken a little more seriously or correctly (right click or hold to see bigger image):

The Brachistochrone is probably the best curve or cycloid for dirt jumping to keep things fast, and the companion curve of the cycloid makes for a really good/efficient roller, especially on dirt pump tracks where friction loss is greater than a cement or asphalt track. I need to think about the use of the cycloid for berms, but I assume the same holds true. Of course laying this type of berm out in the field would be a little tedious. Also, is it really necessary, even on a pump track? And wouldn’t it suck to have cookie cutter replicas at nearly every berm, or roller for that matter? Would the character and art be lost? These curves/shapes are probably the most important on pump tracks where the goal is easy pumping and conserving as much energy as possible rather than losing it. Nonetheless, knowing what this means and how it works can help builders understand the basic shapes and how to get more (or less) speed when needed, and how to not stray too far from efficient form, flow, and motion.

One thing to note is that rollers close to the height of the hub will probably be too peaky and probably result in the front sprocket hitting the top, the pedals as well. What size bikes are being used? Go with bigger rollers if the space and distance are available, or possibly lose the true companion shape and efficiency by carefully shaving off the top a little, albeit still keeping it sinuous. The other option is to go lower than the hub. They will probably also feel peaky. I hope to test some options with a 26 inch dirt jump bike in early 2020.

The cycloid could be cut as well, but used for dirt jumps instead (stopping before vertical (unless you want a skate/bike half pipe)), or at the launch angle desired for the corresponding height and distance desired). These companion curve cut outs will be used to shape rollers for testing on my home track. They may or may not work (or really feel) as well I hope, but I think it is worth a shot (continued below).

A bit too peaky

More rounded is better. The companion curve was too peaky, oh well, I tried. See images below. I think if this shape was 4-5 ft tall maybe it would work, or maybe not. It might be fun to test. Testing these companion curve rollers was not that fun, or it was more disappointing than fun. I had to try…and now you don’t have to hassle with it unless you want to try it yourself.

A solution could be to combine the top of the Brachistochrone with the bottom of the companion curve, but ultimately pedal strike avoidance and flow is key. Something sine wave like.

The peak of failure…except that the wood did kind of works as a frame/guide.

Softer was still not soft or round enough.

Bike Jump Calculator

Bike Jump Calculator Standard Metric enter speed or velocity and ramp/jump angle**, and see notes* Jump velocity @ launch lip, v= mi/hr Jump velocity at launch lip, v= m/s Jump angle, θ= degrees Jump angle, θ= degrees Horizontal range, R= feet Horizontal range, R= m Air time, t= seconds Air time, t = seconds Height, h= feet Height, h= m

*Notes:

If you find this helpful, or not, let me know with a comment or email. Please kick down a donation if you found this useful, or want the Excel sheet. Jumping is dangerous, and may cause serious injury or death (see this) Build and jump at your own risk, no guarantees are promised by the numbers output by the ramp calculator or spreadsheet, but they can get you in the ball park (if you know what average launch speeds will be– don’t forget speeds will slow as you climb up ramp, and drag friction will reduce R, sometimes significantly, leaving you short) Read #3 and 4 again, then this: THIS CALCULATOR ASSUMES same launch and landing height! (see black and red diagram). **”Jump angle” or θ = the angle of the bike the instant the back tire leaves the ramp, which should be very close to the last foot of the ramp under the lip. IMPORTANT: the bike jump calculator is for a “point” of mass. In turn, R will have to be shortened by: (wheelbase in feet or meters) x sin(jump angle) = distance to shorten R. For example, perhaps err for extra large 29er wheelbases that are somewhere around 46 inches: 46in/12in/ft= 3.83 ft, meaning 3.83 feet x sin(jump angle) = distance to shorten R for XL 29ers with 46 inch wheelbases. Launch and landing speeds will be the same (or negligibly different) if the launch and landing heights are the same (also see 5). For a slower/softer landing, landing ramps can be elevated, but this will change the range, R, to a shorter distance. If landing at a lower height than takeoff then R, or the the landing lip, can be pushed farther from the takeoff lip. Please see the “safer jump” pdf documents linked below for details about the best landing ramp design. WIND and rider behaviors like pumping, lifting, braking, spinning, flipping etc. could change any of the variables (h, R, t, and the angle of launch)– in other words the parabolic line of flight as as depicted in the red and black image will be altered, altering R Wind and drag may become a factor if strong enough or H and/or R is great enough The calculator above is for RIGID forks AND RIGID tails, i.e. hard-tails. Suspension can change the angle of launch, thus H and R. Recoil may also affect the outcome. Test your creation and measure carefully. How fast were you going, what exactly is the lip angle, and did you pull up? Was it windy? Adjust your creation accordingly, perhaps with a correction coefficient.

“How do I know my speed at takeoff?” A bike computer or gps unit could work, but may not be exact, or are dangerous to look at on takeoff; of course the jump could be bypassed or skipped to get the approximate speed the jump will be hit. Someone could help with a tape measure and stop watch as well, and this method may be better than the computer, but several trials may have to be done to determine the speed more accurately, or to get an average, in case the time keeper doesn’t hit the start and stop at exactly the right moment. A radar gun, or triggers, or lasers and an arduino could work too. Most cheaper radar guns might not be accurate enough near 10 mph, and usually have +/-1-2 mph accuracy issues.

GOOD LUCK. HAVE FUN. BE SAFE.

*props to Greg from dirtycentury for helping with the script