Cugel wrote: ↑18 Aug 2022, 11:46amThe energy required to get there will go up exponentially with the value of the assisted speed.
It isn't exponential.
The energy needed to increase the kinetic energy goes up as the difference of the square of the speeds.
On the level as speed increases aerodynamic drag becomes the largest force to be overcome to maintain speed. The force of the drag goes up as the square of the speed, the power needed as the cube of the speed.
Yes, yes ... I use the word "exponential" only in that loose sense of increase/decrease in a nonlinear fashion.
...
This is similar to that recent discussion of physical quantities. In informal usage loose wording may not be too important. But when anything important is discussed or when disagreement occurs it becomes necessary to get the wording right. The original post already includes quantitation of both power and speed. It's well into the zone where loose wording won't deliver the required outcome.
Jdsk wrote: ↑18 Aug 2022, 11:52am
It isn't exponential.
The energy needed to increase the kinetic energy goes up as the difference of the square of the speeds.
On the level as speed increases aerodynamic drag becomes the largest force to be overcome to maintain speed. The force of the drag goes up as the square of the speed, the power needed as the cube of the speed.
Jonathan
Beat me to it! Quite so. Kinetic energy = 0.5 x m x (v squared).
Some confusion between kinetic energy and the energy required to overcome drag. Once a set velocity is achieved it is only overcoming drag and friction which require energy (on the flat). Drag goes up as a square of the speed so is exponential - twice the speed needs four times the energy. Kinetic energy is what goes into the brakes when someone pulls out in front of you
This is not, mathematically "exponential", but rather a "polynomial".
Jdsk wrote: ↑18 Aug 2022, 11:52am
It isn't exponential.
The energy needed to increase the kinetic energy goes up as the difference of the square of the speeds.
On the level as speed increases aerodynamic drag becomes the largest force to be overcome to maintain speed. The force of the drag goes up as the square of the speed, the power needed as the cube of the speed.
Beat me to it! Quite so. Kinetic energy = 0.5 x m x (v squared).
Some confusion between kinetic energy and the energy required to overcome drag. Once a set velocity is achieved it is only overcoming drag and friction which require energy (on the flat). Drag goes up as a square of the speed so is exponential - twice the speed needs four times the energy. Kinetic energy is what goes into the brakes when someone pulls out in front of you
That's why I separated the energy required to move to a new speed from the effects of drag. They even have separate paragraphs!
: - )
Chris Jeggo wrote: ↑18 Aug 2022, 11:59am
Beat me to it! Quite so. Kinetic energy = 0.5 x m x (v squared).
Some confusion between kinetic energy and the energy required to overcome drag. Once a set velocity is achieved it is only overcoming drag and friction which require energy (on the flat). Drag goes up as a square of the speed so is exponential - twice the speed needs four times the energy. Kinetic energy is what goes into the brakes when someone pulls out in front of you
This is not, mathematically "exponential", but rather a "polynomial".
Yes.
Jonathan
PS: Congratulations to everyone who passed their Mathematics A level and received their results today. Most popular subject again.
bluespeeder wrote: ↑18 Aug 2022, 1:51pmSome confusion between kinetic energy and the energy required to overcome drag. Once a set velocity is achieved it is only overcoming drag and friction which require energy (on the flat). Drag goes up as a square of the speed so is exponential - twice the speed needs four times the energy. Kinetic energy is what goes into the brakes when someone pulls out in front of you
Apart from it not being exponential...
Wording that describes energy being needed to overcome drag is confusing. A force is needed to counter drag, and that force goes up as the square of the speed. And power is needed to counter drag, and that power goes up as the cube of the speed.
bluespeeder wrote: ↑18 Aug 2022, 1:51pm
Some confusion between kinetic energy and the energy required to overcome drag. Once a set velocity is achieved it is only overcoming drag and friction which require energy (on the flat). Drag goes up as a square of the speed so is exponential - twice the speed needs four times the energy. Kinetic energy is what goes into the brakes when someone pulls out in front of you
This is not, mathematically "exponential", but rather a "polynomial".
Yes.
Jonathan
PS: Congratulations to everyone who passed their Mathematics A level and received their results today. Most popular subject again.
Two things learnt today!
I've just got my head around the fact the power goes up as a cube - double the speed, four times the drag over twice the distance per second (I think)
bluespeeder wrote: ↑18 Aug 2022, 2:29pmTwo things learnt today!
I've just got my head around the fact the power goes up as a cube - double the speed, four times the drag over twice the distance per second (I think)
Force to overcome the drag goes up as the square of speed, so x4 for double the speed. Power to overcome the drag goes up as the cube of the speed, so x8 for double the speed.
if anyone's wondering where the cubing comes into it... the squaring comes in with force being proportional to the square of speed, and power = force x velocity, and velocity is like speed so power is proportional to the cube of speed because of that extra multiplication by velocity.
bluespeeder wrote: ↑18 Aug 2022, 1:51pm
Some confusion between kinetic energy and the energy required to overcome drag. Once a set velocity is achieved it is only overcoming drag and friction which require energy (on the flat). Drag goes up as a square of the speed so is exponential - twice the speed needs four times the energy. Kinetic energy is what goes into the brakes when someone pulls out in front of you
This is not, mathematically "exponential", but rather a "polynomial".
Yes.
Jonathan
PS: Congratulations to everyone who passed their Mathematics A level and received their results today. Most popular subject again.
This is not, mathematically "exponential", but rather a "polynomial".
Yes.
Jonathan
PS: Congratulations to everyone who passed their Mathematics A level and received their results today. Most popular subject again.
Two things learnt today!
I've just got my head around the fact the power goes up as a cube - double the speed, four times the drag over twice the distance per second (I think)
Yes, power = force * velocity. Eg to keep a constant speed up a hill of constant gradient (neglecting air resistance) you don't need any more force to do it at a different speed, but you do need more power.
For an exponential, there is a constant doubling (or halving) time. Classic example is Newton's law of cooling where the difference in temperature halves at constant rate
eg put a bottle of water at 25 degrees in a fridge at 5 degrees, temperature difference 20 degrees. if you get to one hour and you're at 15 degrees - half way to 5 degrees - then in another hour you'll lose 5 degrees - half way from 15 to 5 - then 2.5 degrees the next hour, etc.
PS: Congratulations to everyone who passed their Mathematics A level and received their results today. Most popular subject again.
Two things learnt today!
I've just got my head around the fact the power goes up as a cube - double the speed, four times the drag over twice the distance per second (I think)
Yes, power = force * velocity. Eg to keep a constant speed up a hill of constant gradient (neglecting air resistance) you don't need any more force to do it at a different speed, but you do need more power.
For an exponential, there is a constant doubling (or halving) time. Classic example is Newton's law of cooling where the difference in temperature halves at constant rate
eg put a bottle of water at 25 degrees in a fridge at 5 degrees, temperature difference 20 degrees. if you get to one hour and you're at 15 degrees - half way to 5 degrees - then in another hour you'll lose 5 degrees - half way from 15 to 5 - then 2.5 degrees the next hour, etc.
If that makes any sense
I was working in my head with power = energy/time, speed = distance/time, energy = force*distance, your equation cuts out the middle man! I always thought velocity was the vector and speed the scalar with direction not relevant in this case but please correct if I'm wrong.
Does the exponential example mean the water never gets to 5 degrees? as Zeno's paradox
bluespeeder wrote: ↑18 Aug 2022, 3:31pmI was working in my head with power = energy/time, speed = distance/time, energy = force*distance, your equation cuts out the middle man! I always thought velocity was the vector and speed the scalar with direction not relevant in this case but please correct if I'm wrong.
Two things learnt today!
I've just got my head around the fact the power goes up as a cube - double the speed, four times the drag over twice the distance per second (I think)
Yes, power = force * velocity. Eg to keep a constant speed up a hill of constant gradient (neglecting air resistance) you don't need any more force to do it at a different speed, but you do need more power.
For an exponential, there is a constant doubling (or halving) time. Classic example is Newton's law of cooling where the difference in temperature halves at constant rate
eg put a bottle of water at 25 degrees in a fridge at 5 degrees, temperature difference 20 degrees. if you get to one hour and you're at 15 degrees - half way to 5 degrees - then in another hour you'll lose 5 degrees - half way from 15 to 5 - then 2.5 degrees the next hour, etc.
If that makes any sense
I was working in my head with power = energy/time, speed = distance/time, energy = force*distance, your equation cuts out the middle man! I always thought velocity was the vector and speed the scalar with direction not relevant in this case but please correct if I'm wrong.
Does the exponential example mean the water never gets to 5 degrees? as Zeno's paradox
Yes re scalar, I should have said force*speed for power, not velocity.
And yes, in principle the water never gets to 5 degrees. In practice, the fridge temperature isn't absolutely constant anyway, so within a few hours it's as close as can be meaningfully measured.
roubaixtuesday wrote: ↑18 Aug 2022, 2:47pm
Yes, power = force * velocity. Eg to keep a constant speed up a hill of constant gradient (neglecting air resistance) you don't need any more force to do it at a different speed, but you do need more power.
I was working in my head with power = energy/time, speed = distance/time, energy = force*distance, your equation cuts out the middle man! I always thought velocity was the vector and speed the scalar with direction not relevant in this case but please correct if I'm wrong.
Yes re scalar, I should have said force*speed for power, not velocity.
For this purpose power = force x velocity is fine. Of course you have to use the appropriate components of the vectors.
