A blue care is travelling along at 70 units, and a red car (exact same make and model) is catching up to it going 100. When they're both right beside each other a bend in the road reveals an obstacle blocking both lanes, so both cars brake at the same intensity and deceleration.
The blue care stops right before the obstacle. Since the red car was going at a faster speed, and braked at the same rate, it doesn't managae to stop: but what speed is it going when it hits the obstacle?
The blue car, using ½mv², shed (~70²=) 4900 units of energy (we'll hand wave away the constants). So the red car, which had (100²=) 10000 units of kinetic energy to start, also shed 4900 units, which means it had 5100 units of energy when it collided, and so was going (√5100~) 71.
> The blue car, using ½mv², shed (~70²=) 4900 units of energy (we'll hand wave away the constants). So the red car, which had (100²=) 10000 units of kinetic energy to start, also shed 4900 units, which means it had 5100 units of energy when it collided, and so was going (√5100~) 71
But if the cars produce downforce this is no longer true because you brake harder (more friction available) at higher speeds!
This is how F1 cars pull 4G when breaking. Some custom cars (like one of Ken Block’s last monsters or the Valkyre) use active aero braking to even greater effect.
Physics is an endless source of frustration to me. It feels like a mix of random tricks, most of which I don’t understand.
I find math and compsci reasonably understandable, can read research papers in both fields ( and have published papers) etc. There’s something specific about physics I don’t get but I’ve never been able to figure out what. The main symptom is that most cause -> consequence in such demonstrations , which are seemingly obvious to everyone, make no sense to me.
Am I the only one ? Are there good resources to learn it?
More than twenty years ago, I quit a program that taught math/cs/physics (the notorious French "classes préparatoires") ~almost precisely over this: I felt like I was being taught physics like it was an axiomatic system where the tricks should not be questioned, they just work so "shut up and calculate" (and you don't even need to be doing quantum mechanics for that).
I just felt like we never got to the heart of the matter of why the models work and how to approach developing them, it was all about learning a bag of tricks.
Meanwhile, math and CS being a lot more axiomatic by nature, they also made a lot more sense to me.
That being said, that specificity of physics, the unbridgeable gap between reality and the models we build to describe it, in retrospect, is what makes it more interesting to me today (it's not just a "closed" system in the sense that math is — of course the relationship between math and physics is itself fascinating but that's yet another topic), but I still feel like I haven't found the right pedagogical approach to make it fit my mindset.
The world just is, regardless of what we think about it. Physics is our best attempt so far to understand and predict it at a low level, but it will always be incomplete.
Maths (and especially compsci!) are constructions by and for humans.
Is it any wonder it is as you describe? It would be odd if it was any other way.
My point is precisely that I was often taught physics as if it was mathematics, where there is in fact a profound ontological difference between the two.
What's the problem exactly? Could you not follow the example in the text?
The standard text to build understanding in physics is University Physics by Sears & Zemansky.
It's worth remembering you're quite far from the ground in physics, and it's mostly taught with "neat" cases that give insight into physics. I.e. the thought experiment to show why kinetic energy must scale quadratically with velocity is carefully designed to show that conclusion. You shouldn't expect to have come up with it off the cuff.
It seems that we're exact opposites! But if mathematics is your thing, it might be interesting for you to explore trying to learn things from a lagrangian perspective first?
Not sure if it'll help you with gaining an intuitive understanding, but at least it'll be interesting!
Lagrangian / Hamiltonian mechanics, the principle of least action, always seemed neat, in L&L and other places I encountered it, until I tried doing exactly what you're saying: gaining an intuitive understanding. At that point it just never made sense to me and seemed like a gratuitous deus ex machina that happens to work beautifully but for no apparent reason. You won't be surprised to learn I dropped out of my STEM program shortly after, though I keep a keen interest in the topic.
Actually, it is momentum, sorta. Galilean 3D momentum isn't conserved under special relativity. The energy-momentum four-vector, however, is, under all lorentz-transformed frames.
So in some sense energy is momentum in the time direction (though it's not a Euclidean 4D space, so beware of assumptions). For an object at rest, this becomes its E=mc² equivalence. Kinetic energy is just a straightforward "rotation" of the frame.
Mikes' answer is the most intuitive, but he rephrases the question in a possibly non intuitive way.
Odd that nobody mentioned power, which scales linearly with speed. Of course if you add linear increasing amounts of power to the system the energy will increase quadratically.
Power scaling linearly is more intuitive because doubling your speed requires twice the power to maintain the same force, why does it require twice the power? because you have half the time to power it.
Fun little anecdote:
A blue care is travelling along at 70 units, and a red car (exact same make and model) is catching up to it going 100. When they're both right beside each other a bend in the road reveals an obstacle blocking both lanes, so both cars brake at the same intensity and deceleration.
The blue care stops right before the obstacle. Since the red car was going at a faster speed, and braked at the same rate, it doesn't managae to stop: but what speed is it going when it hits the obstacle?
The blue car, using ½mv², shed (~70²=) 4900 units of energy (we'll hand wave away the constants). So the red car, which had (100²=) 10000 units of kinetic energy to start, also shed 4900 units, which means it had 5100 units of energy when it collided, and so was going (√5100~) 71.
* Numberphile: https://www.youtube.com/watch?v=i3D7XYQExt0
> The blue car, using ½mv², shed (~70²=) 4900 units of energy (we'll hand wave away the constants). So the red car, which had (100²=) 10000 units of kinetic energy to start, also shed 4900 units, which means it had 5100 units of energy when it collided, and so was going (√5100~) 71
But if the cars produce downforce this is no longer true because you brake harder (more friction available) at higher speeds!
This is how F1 cars pull 4G when breaking. Some custom cars (like one of Ken Block’s last monsters or the Valkyre) use active aero braking to even greater effect.
But what if the cars are spherical cows?
Cows can't roll that fast.
Cool anecdote!
Couldn’t help but notice you misspelled car twice but only when talking about the blue car..
heh, thats a fun little experiment.
Physics is an endless source of frustration to me. It feels like a mix of random tricks, most of which I don’t understand.
I find math and compsci reasonably understandable, can read research papers in both fields ( and have published papers) etc. There’s something specific about physics I don’t get but I’ve never been able to figure out what. The main symptom is that most cause -> consequence in such demonstrations , which are seemingly obvious to everyone, make no sense to me.
Am I the only one ? Are there good resources to learn it?
More than twenty years ago, I quit a program that taught math/cs/physics (the notorious French "classes préparatoires") ~almost precisely over this: I felt like I was being taught physics like it was an axiomatic system where the tricks should not be questioned, they just work so "shut up and calculate" (and you don't even need to be doing quantum mechanics for that).
I just felt like we never got to the heart of the matter of why the models work and how to approach developing them, it was all about learning a bag of tricks.
Meanwhile, math and CS being a lot more axiomatic by nature, they also made a lot more sense to me.
That being said, that specificity of physics, the unbridgeable gap between reality and the models we build to describe it, in retrospect, is what makes it more interesting to me today (it's not just a "closed" system in the sense that math is — of course the relationship between math and physics is itself fascinating but that's yet another topic), but I still feel like I haven't found the right pedagogical approach to make it fit my mindset.
The world just is, regardless of what we think about it. Physics is our best attempt so far to understand and predict it at a low level, but it will always be incomplete.
Maths (and especially compsci!) are constructions by and for humans.
Is it any wonder it is as you describe? It would be odd if it was any other way.
My point is precisely that I was often taught physics as if it was mathematics, where there is in fact a profound ontological difference between the two.
Also, physics (the discipline) is also a construction by and for humans.
What's the problem exactly? Could you not follow the example in the text?
The standard text to build understanding in physics is University Physics by Sears & Zemansky.
It's worth remembering you're quite far from the ground in physics, and it's mostly taught with "neat" cases that give insight into physics. I.e. the thought experiment to show why kinetic energy must scale quadratically with velocity is carefully designed to show that conclusion. You shouldn't expect to have come up with it off the cuff.
It seems that we're exact opposites! But if mathematics is your thing, it might be interesting for you to explore trying to learn things from a lagrangian perspective first?
Not sure if it'll help you with gaining an intuitive understanding, but at least it'll be interesting!
https://en.wikipedia.org/wiki/Lagrangian_mechanics
Lagrangian / Hamiltonian mechanics, the principle of least action, always seemed neat, in L&L and other places I encountered it, until I tried doing exactly what you're saying: gaining an intuitive understanding. At that point it just never made sense to me and seemed like a gratuitous deus ex machina that happens to work beautifully but for no apparent reason. You won't be surprised to learn I dropped out of my STEM program shortly after, though I keep a keen interest in the topic.
Because it's not momentum. ;p
F=ma (Force equals mass times acceleration)
W=Fd (work equals force multiplied by distance)
V^2=2ad (velocity squared equals two times acceleration times distance)
So W = Fd = ma(v^2/2a)
Finally: W=1/2mv^2 (work equals 1/2 mass times velocity squared)
So this explains why car crashes can be so dramatic, as a doubling of speed results in 4x the kinetic energy.
Actually, it is momentum, sorta. Galilean 3D momentum isn't conserved under special relativity. The energy-momentum four-vector, however, is, under all lorentz-transformed frames.
So in some sense energy is momentum in the time direction (though it's not a Euclidean 4D space, so beware of assumptions). For an object at rest, this becomes its E=mc² equivalence. Kinetic energy is just a straightforward "rotation" of the frame.
P=mv (momentum equals mass times velocity)
This is linear.
Mikes' answer is the most intuitive, but he rephrases the question in a possibly non intuitive way.
Odd that nobody mentioned power, which scales linearly with speed. Of course if you add linear increasing amounts of power to the system the energy will increase quadratically.
Power scaling linearly is more intuitive because doubling your speed requires twice the power to maintain the same force, why does it require twice the power? because you have half the time to power it.