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Automobile on a Trampoline: More Kicks With Kinetic Energy

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Automobile on a Trampoline: More Kicks With Kinetic Energy

Oh, sure, you’ve seen a watermelon dropped from a balcony onto a trampoline. But what happens when you drop a car from a high tower onto a trampoline? That’s a whole new level of physics fun, and it’s exactly what happens in this video from Mark Rober and the How Ridiculous guys.First they built their…

Automobile on a Trampoline: More Kicks With Kinetic Energy

Oh, definite, you’ve viewed a watermelon dropped from a balcony onto a trampoline. Nonetheless what occurs while you plunge a automobile from a excessive tower onto a trampoline? That’s a entire fresh level of physics enjoyable, and it’s exactly what occurs in this video from Mark Rober and the How Ridiculous guys.

First they constructed their very contain monster trampoline with overlapping sheets of bulletproof kevlar for the pad, supported by a thick steel frame and 144 immense old garage door springs. Then they tested it with a bunch of various things, dropping a entire sack of watermelons, 20 bowling balls and a 66-pound Atlas stone onto a bed of water balloons. The auto plunge occurs plan the pause of the video, beginning at 9: 20.

Even while you manufacture no longer yelp here’s superior (c’mon, it’s empirically proven to be superior), it be restful a worthy offer for some physics complications you can work out at home, while we’re all doing this social distancing thing. I’m going to resolve some of these for you—and I’ll faux I’m doing them as examples. The truth? I will’t serve myself; I appropriate savor physics.

1. How excessive is the plunge?

Can you sing from the video how far the automobile falls earlier than hitting the trampoline? Right here’s the steady question, and I will shatter it by providing you with the reply. So stop here while you fancy to must raise a seek at it for your contain first.

Ready? If your physics, you realized that to search out the gap, all you wish to kind is measure the free-fall time.

Let’s open with the basics. Once an object leaves a person’s hand, the ideal force acting on it is the downward gravitational force. The magnitude of this force is the fabricated from its mass (m) and the gravitational field (g = 9.8 N/kg). Since the acceleration of an object additionally is dependent upon the mass, all free falling objects contain the identical downward acceleration of 9.8 m/s2. Nonetheless what’s the connection between fall time and height? I will fetch this—and no, I would no longer appropriate whisper “Exhaust a kinematic equation.”

The definition of acceleration in a single dimension is the swap in velocity (Δv) divided by a swap in time (Δt). If I know the elapsed time (I will glean that from the video), and I know the acceleration (due to here’s on Earth), then I will resolve for the swap in velocity. Show, I’m the utilization of detrimental g for the acceleration, since it be racy downward.

Illustration: Rhett Allain

In this expression, v1 is the beginning velocity of the article, which in this case is zero, and v2 is the final velocity. Now for one other definition—moderate velocity (in a single dimension) appears fancy this, the place (Δy) is the swap in vertical place:

Illustration: Rhett Allain

For an object with a continuing acceleration (fancy we contain came), the frequent velocity is appropriate the sum of the preliminary and final velocity divided by two—it be actually the frequent of the velocities. And since the preliminary velocity is zero, the frequent velocity is appropriate half of the final velocity. I will use this to search out the swap in place, i.e., the gap it falls:

Illustration: Rhett Allain

Yes, the swap in the y place is detrimental, since the article is racy down. All that is left is the time. I regarded on the half of the video with the dropped watermelons. One of the significant photography are in leisurely motion, but some seem to be in odd time. I will glean the fall time from these photography.

You should maybe strive to use the time tag on YouTube to kind this, nonetheless it be no longer detailed ample. I fancy to use the Tracker video evaluation system—it be my toddle-to for this more or much less thing (and it be free). From that, I glean a fall time of two.749 seconds. Plugging that into the equation above, I glean a fall height of 37.0 meters (121.5 toes). Enhance, that is one question solved.

2. What’s the impact velocity?

Within the event you plunge an object from leisure (i.e., zero preliminary velocity), how rapid will it be touring factual earlier than it hits the trampoline? Oh, you belief I was going to reply this question too? Nope. Truly, this one’s no longer too powerful. You should maybe possibly possibly also use the time and the definition of acceleration to search out this reply. You should maybe possibly possibly also kind it. I have faith in you.

3. What’s the efficient spring constant?

Let’s toddle thru this entire motion. The auto drops. Whereas falling, the gravitational force pulls on it, inflicting it to urge up, more and more, until it contacts the trampoline. At this level, the springs on the trampoline stretch and kind an upward pushing force on the automobile. The farther the springs stretch, the increased the upward pushing force.

Have in mind that in negate for an object to leisurely down, there wants to be a get force pushing in the wrong manner as the motion. When the automobile first hits the trampoline, the backwards pushing force is LESS than gravity, so the get force is restful downward, and the automobile keeps speeding up. Right here’s one thing that college students tend to no longer contain a true intuition for. Have in mind, it be the get force that determines acceleration.

It’s no longer until the spring force turns into increased than the downward pushing gravity force that the automobile begins slowing down. Obviously, it be restful racy down, so the springs stretch noteworthy more, and this will increase the spring force. One way or the opposite the automobile stops falling and begins racy aid up.

Now, how kind we quantify that? One manner to mannequin the force from a spring is with Hooke’s laws. This says that the spring force (Fs) is proportional to the gap (s) that the spring stretches or compresses. This proportionality constant is is known as the spring constant, k. You should maybe possibly possibly also yelp of k as the stiffness of the spring.

Illustration: Rhett Allain

Truly, we can’t apply this mannequin on to our trampoline, due to it assumes that the springs are per the motion of the automobile. In actual fact, if the automobile strikes down 10 cm, the springs stretch even better than that, as a result of the geometry of the effort. Nonetheless manufacture no longer dismay, we can appropriate faux all the pieces is in a single dimension, and that will give us an total efficient spring constant. That makes the difficulty seek fancy this:

Illustration: Rhett Allain

Now we can fetch an expression for the spring constant k by the utilization of the work-energy precept. This says that the work carried out on a tool is the identical as the swap in energy in that diagram. So if we outline our diagram to include Earth, the automobile, and the spring, there are no external interactions in the diagram and thus no work carried out. Which implies the entire energy ought to be constant—energy is conserved.

For this plan, there are essentially appropriate three kinds of energy eager. Listed below are the equations for these energies at the side of explanations under:

Illustration: Rhett Allain
  • Kinetic energy (Ample): Right here’s the energy an object has when it’s racy. The kinetic energy is dependent upon both the mass of the article and its velocity.

  • Gravitational likely energy (Ug): When two objects are gravitationally interacting (fancy the automobile and the Earth), there’s a capacity energy related to their place. On Earth’s flooring we can approximate this as being proportional to the automobile’s mass and a few arbitrary vertical place. (Don’t dismay about this place; it be handiest the swap in place that essentially matters.)

  • Elastic likely energy (Us): Most regularly acknowledged as spring likely energy. This depends on both the quantity the spring is compressed or stretched and the spring constant. Enhance—here’s how we can glean an expression for the spring stiffness.

what’s so worthy about the utilization of the work-energy precept? I will appropriate seek at changes from one suppose to one other and ignore your entire stuff in between. This implies I will open with the automobile at leisure (on the stay of the plunge) and pause with the automobile on the bottom of the spring (once more at leisure). I manufacture no longer must know the device rapid the automobile is racy at facets in the center—that appropriate does no longer topic. Placing all of this collectively, I glean the next.

Illustration: Rhett Allain

Factual some notes. I’m the utilization of the 1 subscript for the place and velocity on the stay of the plunge, and the three subscript for the bottom. (Stage 2 is when it hits the spring). At both of these positions, the kinetic energy is zero. Which implies the swap in kinetic energy is additionally zero. The swap in height (y3y1) is appropriate –h (from the map above). For the stretch first and foremost of the plunge (s1), here’s appropriate zero, since the spring hasn’t yet been compressed. Now I will use this (at the side of my notation from the map) to resolve for the spring constant, k.

Illustration: Rhett Allain

That’s making some development. All we need now is the stretch distance s (how far the trampoline strikes down) and the mass of the automobile. The stretch distance mustn’t be too powerful to estimate—it appears to be roughly 1.5 meters.

Nonetheless what about the mass? Mark mentioned he adjusted the mass of the automobile, but he did no longer whisper what the resulting mass used to be. Oh, I would appropriate seek info from him? No. The place’s the enjoyable in that? Strive to return up with a true wager for the mass to operate the question.

4. Calculate the steady trampoline spring force.

OK, we assumed above that the springs are per the automobile’s motion, but that’s clearly no longer the case. The chilly thing a pair of trampoline is that the springs stretch a distance that is various than the gap the trampoline strikes down. Let’s manufacture an awfully simplified trampoline so we can behold what’s occurring.

This version has a horizontal bar supported by two horizontal springs. When a mass is on prime of the bar, it strikes down and stretches the springs. Right here’s a map:

Illustration: Rhett Allain

A pair of things we contain got to set up in mind: First, if the trampoline strikes down a distance of y, how noteworthy does a spring (with an unstretched length of L0) stretch? That is no longer too powerful to resolve out from the map.

2nd, what component of this spring force is in the upward direction? The spring on the left exerts a force pulling up and to the left, while the one on the factual pulls up and to the factual. If the springs are equal, the horizontal substances of these spring forces cancel, and we’re left with appropriate the upward component. Nonetheless how noteworthy that is is dependent upon the perspective of the spring with appreciate to the horizontal (θ in my map).

Right here’s what you can kind subsequent: Factual scheme stop some values for the spring constant and the unstretched length. Now location the get vertical spring force as a feature of vertical place. Is this location linear? That’s what it is probably going you’ll maybe possibly possibly quiz for a single Hooke’s laws spring. In actual fact, I’m no longer definite what you can glean—for that reason here’s a worthy homework question.

Though I derived an expression for the efficient spring constant of the trampoline, I didn’t glean a numerical cost. Within the event you fancy to must glean a rough estimate of this cost, you can open with 144 garage door springs. You should maybe possibly possibly also estimate the unstretched length (possibly about 75 centimeters). I’m no longer definite about the garage door spring constant. They are saying these are “450-pound” springs, nonetheless it be no longer definite what which capacity that. Factual wager.

If it is probably going you’ll maybe just contain the efficient spring constant (or a force as a feature of distance), you can then return to the old narrate and resolve for the mass of the automobile. This will maybe possibly be worthy. Don’t cheat and seek info from Mark.

5. The place is the automobile’s heart of mass?

I essentially don’t contain any belief what more or much less automobile they dropped. Per chance it’s some Australian mannequin? Nonetheless I kind know they changed the mass, and my suspicion is that they did so by eliminating the engine. Doing that would possibly maybe just manufacture this stunt more uncomplicated to drag off—without an engine it would possibly maybe possibly possibly be more at possibility of fall in a “wheels up” place without rotating.

Why kind I feel that? Thanks to the heart of mass. An object’s heart of mass is the level at that you simply can faux there’s a single gravitational force acting on it. Obviously the automobile is made of a bunch of small pieces, and every interacts gravitationally with the Earth. Nonetheless it without a doubt’s more shimmering to address all of these forces as appropriate one force. And if it is probably going you’ll maybe just contain a single force, you wish a single space for that force—that is the heart of mass.

Most automobiles contain a heart of mass that is no longer in the heart. That is as a result of this very big auto half known as the engine, which shifts the heart of mass towards the front. Nonetheless what while you grasp a automobile from a cable? To preserve it from rotating, both the stress force from the cable and the gravitational force must toddle thru the identical level, so that they manufacture no longer exert a torque. Which implies you can scheme a line from the cable extending thru the automobile and this will maybe toddle thru the heart of mass.

Right here’s a shot of that inserting automobile:

Courtesy of Mark Rober

Within the event you make use of three attachment facets (as viewed in the photo), the automobile can restful rotate slightly bit to contain the heart of mass per the significant cable, nonetheless it gained’t swing too noteworthy. Now for the homework. Estimate the positioning of the heart of mass and behold how noteworthy it would hump towards the front while you place the engine aid in.

6. Does air resistance topic?

Oh, you manufacture no longer need from now on homework questions? Too rotten.

When the automobile is falling, my earlier evaluation assumed that the ideal force acting on it used to be gravity. Is that legit? Clearly it be no longer fully comely, nonetheless it would possibly maybe possibly possibly be OK. Because the automobile falls, it strikes thru the air. Because it has to push the air out of the manner, the air pushes aid on the automobile. Right here’s the essence of the air traipse force. It be a force in the wrong manner from the price, and it ought to in overall be modeled with the next equation:

Illustration: Rhett Allain

In this mannequin, ρ is the density of the air, A is the inferior-sectional home, C is a traipse coefficient that is dependent upon the form, and unnecessary to direct v is velocity.

Within the event you fancy to must essentially mannequin the motion of a falling object with air traipse, things can glean titillating. Since the automobile will swap velocity, and the air traipse force is dependent upon the price, you can not use easy assumptions fancy we did earlier than. Truly, the steady manner to resolve for the motion of one thing with air traipse is to spoil it into small time steps and use a numerical calculation. Right here is an instance of that.

Nonetheless I’m stunning definite we can ignore the air traipse force here. Right here’s why: The listed height of the tower is 45 meters. Since the air traipse force is in the wrong manner from the gravitational force, a first-rate air traipse would elevate the falling time. Using a long time (while ignoring the air traipse fancy I did earlier than) would give a calculated tower height increased than 45 meters. I did no longer fetch that, so I manufacture no longer yelp the air resistance matters. Nonetheless it is probably going you’ll maybe just restful restful mannequin it.

7. What’s the personality of science and engineering?

Ha! This will maybe just restful aid you busy for a while. Truly, this is never always a homework question, nonetheless it be potentially the steady half of the video. Right here’s what Mark Rober says:

“It be this loop of designing one thing in CAD and then analyzing it to behold if it be steady ample, and then you positively take a look at it to take a look at your solutions. Using pc programs to study designs permits us to manufacture noteworthy more complicated programs than earlier than, when pc programs weren’t as powerful.”

“This belief that we can understand and predict the enviornment spherical us the utilization of math and equations is what first made me fall in savor with science, when I took excessive college physics.”

Yes. It be all about fashions.

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