Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

physics.stackexchange.com

Post reply on HN
Print view View original

151–160 of 231 posts

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#151

Post by perilunar »

6510 wrote: post #118
AlexandrB wrote: post #32

Earlier quoted context omitted.

There's a great Australian traffic safety ad that makes this same point: https://www.youtube.com/watch?v=7x7c0qNGbv0

Nice bit of camera trickery. He says "both drivers react and a moment later they break", but the cars are still side by side. It (apparently) takes drivers 1.5 seconds to respond, the 5 km/h speed difference cuts the distance by 2 meter. Which apparently is a big deal. Rough estimate breaking distance: 5 km/h = 0.13 meter 30 km/h = 4.5 meter 60 km/h = 14 to 18 meter 65 km/h = 21 to 24 meter The +5 km/h adds 6 to 7 me…

"Breaking distance": how much shorter the car is after the impact.

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#152

Post by jijijijij »

> The previous answers all restate the problem as "Work is force dot/times distance". But this is not really satisfying, because you could then ask "Why is work force dot distance?" and the mystery is the same.

...

> But now look at this in a train which is moving along with one of the balls before the collision. In this frame of reference, the first ball starts out stopped, the second ball hits it at 2v, and the two-ball stuck system ends up moving with velocity v.

That's still just pushing the problem elsewhere. Intuitively, why does the two-ball system end up with a velocity of 1v?

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#153

Post by GlibMonkeyDeath »

For me, the most intuitive explanation is that:

Force = change in momentum with time

Energy = Force x distance

Now consider how much energy can be dissipated by a tiny change in momentum over a small distance dx, when we are at a given velocity v:

dE = Fdx = (dp/dt)dx = m(dv/dt)dx = mdv(dx/dt) = mv*dv

The intuition is that in order to apply a force through some distance, I have to change the velocity of an object by dv. But, the distance I just traveled also depends on the current velocity v. That's why the total energy available isn't just simply proportional to velocity - every time we change v, the amount of force available goes down, too.

Summing all the little bits of energy dE over our velocity changes dv, from the starting velocity down to zero, and we get the formula for kinetic energy.

BTW, the intuition here really starts from the idea that force = momentum change with time. The definition of "force", "momentum", and "energy" can be maddeningly circular, even if we have clear mathematical representations and a common world we experience.

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#154

Post by electricwallaby »

It helps to re-frame the premise.

An object which has a constant force applied will have it's distance increase quadratically with respect to time.

Energy is force times distance. Intuition: the energy it takes to lift an object up is proportional to the height you lift it to.

So if you apply a constant force, you get a constant acceleration which leads to a quadratically increasing distance.

If you accept that energy is force times distance, the energy required to move the object in this scenario increases quadratically.

This means that if you apply a force F for 1 second, the amount of energy that is imparted by that force depends on how fast the object is already going. The energy required to apply a force to an already fast moving object is much higher. Intuition: you have to expend all the energy required to get up to the moving object's speed before you can start applying a force. So there's a cost to even get in the game

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#155

Post by tcoff91 »

hyperhello wrote: post #46

I didn’t think this was that weird. When you double your speed you are also going to be going twice as far in the same time, not just twice as fast, and they both have the effect of work.

To me the simplest way to understand it is through calculus. Kinetic energy is the integral of momentum so You go from p = mv to k = 1/2mv^2

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#156

Post by Yajirobe »

abetusk wrote: post #98

Ron Maimon uses an argument that relies purely on symmetry, which circumvents the standard explanations, including many in this thread. In some sense, this is the simplified version of Noether's theorem (as far as I understand it). As an aside, I believe Ron Maimon's account was suspended after he challenged the character of someone who was soliciting votes for a moderator position. Ron Maimon's stance was that if so…

It wasn't a permanent ban. He will be unbanned at Mar 18, 2292 at 16:28

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#158

Post by trinari »

freehorse wrote: post #117
nlawalker wrote: post #61

Earlier quoted context omitted.

> 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.

The effort to move a piece of furniture from 1st to 2nd floor is the same as the effort to move it from the 2nd to the 3rd. We have good intuition for this by our experience, which derives a linear relationship. The effort to move a piece of furniture up two floors is double the effort of moving it up one floor (ie you have to put the same effort twice, assuming enough rest). I would not say we have the same intuitio…

Can someone help me understand the following?

Getting up from a seat als walking a couple steps feels that same at home and in a flying airplane (or does it?). But the base speed is 0 in the former and several hundred mph in the latter case

Re: Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

#160

Post by Tepix »

trinari wrote: post #158
freehorse wrote: post #117

Earlier quoted context omitted.

The effort to move a piece of furniture from 1st to 2nd floor is the same as the effort to move it from the 2nd to the 3rd. We have good intuition for this by our experience, which derives a linear relationship. The effort to move a piece of furniture up two floors is double the effort of moving it up one floor (ie you have to put the same effort twice, assuming enough rest). I would not say we have the same intuitio…

Can someone help me understand the following? Getting up from a seat als walking a couple steps feels that same at home and in a flying airplane (or does it?). But the base speed is 0 in the former and several hundred mph in the latter case

I guess Galileo came up with it first:

https://en.wikipedia.org/wiki/Galilean_invariance

Post reply on HN