Understanding Energy & Momentum in Motion

[Lapidus]

Could someone give me some deeper insight why we need to have momentum and energy to describe motion? Kinetic energy is only the square root of momentum divided by two times the mass of the particle. Why the need for this extra entity energy?

I know that energy is a scalar and momentum is a vector, and that momentum generates space symmetry and energy generates time symmetry. But I would like to have some further insights why energy is connected to time and momentum to space.

And why dV/dx = dp/dt ?

Why is the space derivative of the (potential) energy equal to the time derivative?

Again, I would like to have some clever insights and deep-down ideas on how energy, momentum, time and space are related.

thank you

 

[Robrex]

Hi,

Im very new to all this and have only just started to cover it in class but this might be helpful or it might be way to basic, haha

why we need to have momentum and energy to describe motion

Conservation of momentum inside a system doesn't have any bearing on the outside momentum (for example a plane flying along with people moving around inside) but the energy inside the system can change the total amount of energy that system has. So if the plane was to crash and everyone ran forward it would increase the energy the plane has on impact.

Hope this is more or less correct and helps. Also interested to hear feedback on it!