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It's easiest to visualize in terms of conversion from potential energy. We know intuitively that a ball atop a 20ft ladder has twice the potential energy of a
by cubic_earth 3mo ago
It's easiest to visualize in terms of conversion from potential energy.
We know intuitively that a ball atop a 20ft ladder has twice the potential energy of a ball atop a 10ft ladder. And we also know when they fall, by the time they reach the ground and all the potential energy has been converted to kinetic energy, the previously higher ball will have twice the kinetic energy too.
But a twice higher ball won't have even close to twice the speed at impact. So let's look at why not.
The force of gravity is a constant force that causes constant acceleration in free fall regardless of speed. (Ignoring air resistance, inverse sq considerations, etc.)
Suppose it takes 1 second for the ball on the 10ft ladder to hit the ground with kinetic energy of 10 and a speed of 100. Again, gravity as a constant acceleration force is speed increase per time... not speed per distance. In the ladder example, it took 1 full second for gravity to accelerate the object to speed 100.
Now think about the 20ft ladder: the ball is dropped. How much kinetic energy and speed does the ball have after it has fallen 10 feet (but still has 10 left to go)? Well it has the same exact amount as the other ball did after falling 10 feet for a duration of 1 second: kinetic energy of 10 and speed of 100.
Now the crux: thinking about when the final 10 feet of the fall look like. We know for sure the ball still has 10 ft of potential energy to covert into kinetic, and that that will happen as it falls. But what of the impact speed? Since the current velocity of the ball as it enters the last 10 feet is already 100, we know it will spend less time transiting this distance than it did the first half where it started at off at speed 0. Since gravity imparts speed in free fall as a function of time - consequently less speed will be imparted over the second 10 foot interval. That concept is enough to prove the relationship isn't linear.
If you do the actual calculation or tests, you will see one ball needs to be dropped from 4x the hight of another to hit the ground at 2x the speed, but yet with still 4x the kinetic energy.
- hunter2_ 3mo agoBrilliant. For those wanting more numbers [0], the ball on the 10ft ladder hits the ground at (I'll stick with imperial units) 17.296 MPH, the ball on the 20ft ladder hits the ground at 24.46 MPH or 41.42% faster, and the ball on the 40ft ladder hits the ground at 34.59 MPH or 100% faster. [0] https://www.omnicalculator.com/physics/free-fall https://www.omnicalculator.com/physics/free-fall
- card_zero 3mo agoNice. Nitpick: in the middle paragraph you put "speed 10" instead of 100.
- cubic_earth 3mo agoFixed. Thanks.
- nlawalker 3mo ago> We know intuitively that a ball atop a 20ft ladder has twice the potential energy of a ball atop a 10ft ladder. What makes this intuitive? The foundation of the asker’s question is that it seems intuitive that kinetic energy would increase linearly with speed, but that turns out to be wrong.
- hunter2_ 3mo agoThat's a good question, and I suppose the mgh formula isn't a suitable answer, so my answer would be something like: if you lift an object to some height, and then you repeat that action (lifting it from there to twice the height), you've done twice the work, and doing twice the work requires twice the caloric intake.
- SilasX 3mo agoOkay but that depends on the intuitions the question is trying to justify, which makes it circular. We also know, for example, that the body uses more than twice as much energy to do twice as much work (because of fatigue on the muscles or whatever the right term is here). In fact it takes positive energy just told a weight at a fixed height, doing zero mechanical work! So you’re actually appealing to even weaker intuition than the one the question is trying to ground!
- hibernator149 3mo agoAll intuitions are wrong, but some are usefull. You have to do the experiment, discover the formula, and then adapt your intuitions accordingly.
- SilasX 3mo agoWhat point that I made are you responding to? I was disputing someone’s appeal to a specific intuition for being an unhelpful one to use here. So I obviously get the concept of some intuitions being useful. Did you see the comment being responded to?
- 3mo ago
- NaiveBayesian 3mo agoI agree that this feels intuitive, that potential energy should increase linearly with height. But in the end, it's all up to the units/quantities we choose to measure, no? If we, say, decided to measure "Squenergy" in Sqoules, with 1Sq² = 1J, then suddenly, squenergy does increase linearly with speed! The formula for kinetic Squenergy becomes sqrt(m/2)v. Of course this complicates other stuff, like potential Squenergy becoming sqrt(MgH), it not being additive, etc.
- Matumio 3mo agoNo, it's not an arbitrary choice. You can convert any energy form into thermal energy. When you boil water with an immersion heater, what happens if you use two heaters? Right, they add up. But not if you think in Squenergy. Or you can measure the effect (in heat) that a ball impact has. In Squenergy two balls don't add up add up, so it's less natural to think that way.
- feoren 3mo agoThe difference is that Squenergy is not conserved.
- once-in-a-while 3mo ago[dead]
- PunchyHamster 3mo ago> We know intuitively that a ball atop a 20ft ladder has twice the potential energy of a ball atop a 10ft ladder. ...no ? dropping something 10 times from 1ft is nowhere near energetic/damaging as once from 10tf
- namdnay 3mo agolifting something 10 times 1 foot is exactly the same as lifting it 10 feet :)
- TeMPOraL 3mo ago10ft vs 20ft is factor of 2 difference in height, 1ft vs 10ft is factor of 10. Also how it translates to difference in damage depends on how elastic is the collision. If the balls are made of rubber, things get much less dramatic, and energy exchange ratios more obvious.
- cubic_earth 3mo agoDropping something from 1 foot has 1/10th the kinetic energy compared to the same thing dropped from 10 feet. Damage is a very poor proxy for energy because there are all kinds of variables and thresholds and structural consideration in determining damage. A cup might not break at all in a 1 foot fall and might shatter when dropped from 10. The outcome is binary, which isn't useful as a scale of input energy.
- dragonwriter 3mo agoThat might seem an attractive intuition based on observing damage and having an understandable intuition of how damage should relate to energy, its wrong. Energy is force × distance. For gravitational potential (and, equivalently, the kinetic energy acquired when that is tranformed into kinetic energy by a fall), the force is the force of gravitational attraction between the body and the object it is falling toward (the object’s weight) and the distance is the distance of the (actual or potential, as appropriate) fall. The nonlinearity you observe in damage is because damage is a complicated outcome that depends on a lot more than aggregate energy involved.
- SideburnsOfDoom 3mo ago
- acchow 3mo ago> We know intuitively that a ball atop a 20ft ladder has twice the potential energy of a ball atop a 10ft ladder. This assumes that energy increases linearly with distance (given constant force over that distance