It feels to me that you lack a good intuition for how kinetic energy, momentum conservation,, Newton laws, and Galileo’s relativity all play together and this is causing you confusion.
Relativity says that there is no objective notion of “being still”. We can’t objectively distinguish between being still and moving at constant velocity (same speed, same direction).
2nd Newton law: Force is equal to mass times acceleration: F=ma.
3nd Newton law: All forces in the nature exhibit the property that if object A acts on B with a force F, then B acts on A with exactly opposite force.
1st Newton law is boring, it just says that if F = 0, then a = 0.
Force and Energy are tied together through the concept of work which says that change in objects energy = work received by that object. And formula for work is W=F⋅Δs, notice the dot product here! Force perpendicular to movement doesn’t make any impact on energy.
Putting all of this together, it’s useful for you to do following exercises:
Try to derive the formula for kinetic energy.
Try to derive the formula for potential energy in the gravitational field of Earth near the surface (neglecting change of gravitational force with height).
Try to derive the law of conservation of momentum.
Once you do the three above, I have a tricky paradox problem for you to solve:
Imagine two cars driving on the road with the speed v. Suddenly the first car accelerates to double the speed, thus travelling 2v. This naturally required consumption of energy in the form of fuel. The change in kinetic energy was: ΔE=12m(2v)2−12mv2=32mv2 and this should somehow correspond to the amount of the fuel consumed.
However, from the point of the view of the second driver, the car was starting still and accelerated to the velocity v, thus the change in energy is simply ΔE=12mv2.
But, this is paradox. It’s not possible that the car would spend 3 times less fuel from the point of view of the other car than from the point of view of the observer standing still on the ground.
It feels to me that you lack a good intuition for how kinetic energy, momentum conservation,, Newton laws, and Galileo’s relativity all play together and this is causing you confusion.
Relativity says that there is no objective notion of “being still”. We can’t objectively distinguish between being still and moving at constant velocity (same speed, same direction).
2nd Newton law: Force is equal to mass times acceleration: F=ma.
3nd Newton law: All forces in the nature exhibit the property that if object A acts on B with a force F, then B acts on A with exactly opposite force.
1st Newton law is boring, it just says that if F = 0, then a = 0.
Force and Energy are tied together through the concept of work which says that change in objects energy = work received by that object. And formula for work is W=F⋅Δs, notice the dot product here! Force perpendicular to movement doesn’t make any impact on energy.
Putting all of this together, it’s useful for you to do following exercises:
Try to derive the formula for kinetic energy.
Try to derive the formula for potential energy in the gravitational field of Earth near the surface (neglecting change of gravitational force with height).
Try to derive the law of conservation of momentum.
Once you do the three above, I have a tricky paradox problem for you to solve:
Imagine two cars driving on the road with the speed v. Suddenly the first car accelerates to double the speed, thus travelling 2v. This naturally required consumption of energy in the form of fuel. The change in kinetic energy was: ΔE=12m(2v)2−12mv2=32mv2 and this should somehow correspond to the amount of the fuel consumed.
However, from the point of the view of the second driver, the car was starting still and accelerated to the velocity v, thus the change in energy is simply ΔE=12mv2.
But, this is paradox. It’s not possible that the car would spend 3 times less fuel from the point of view of the other car than from the point of view of the observer standing still on the ground.
Can you explain this paradox?
Unfortunately I already came across that paradox a day or two ago on Stack Exchange. It’s a good one though!
Yeah, my numerical skill is poor, so I try to understand things via visualization and analogies. It’s more reliable in some cases, less in others.