Kinetic velocity independ. of direction?

[razored]

Why is the velocity in the kinetic energy formula independent of the direction? I can't seem to figure out why. This is not a homework problem.

 

[jgens]

Kinetic energy is defined to be the amount - magnitude - of work necessary to accelerate an object to a given speed; hence, only the magnitude is important - regarding kinetic energy.

Mathematically, it's independent because the quantity is squared.

 

[Sisplat]

The kinetic energy formula depends only of v.v (the length of the vector v squared). Obviously the length of a vector doesn't depend on its direction.

It doesn't matter what direction you run in, you use up the same amount of energy*.

* in a very idealized sports hall.

 

[razored]

Mathematically speaking, no problem because it is squared; however, when we throw a ball 45 degrees above horizontal and then 45 degrees below, it just intuitively for me does not make sense that they contain the same kinetic energy. hmm.

 

[Sisplat]

If you did it in the absence of gravity, they would have the same kinetic energy. You would also have trouble defining the term horizontal.

If you do your experiment on Earth the problem is different.

Initially, at the exact moment you throw the two balls, they would have the same kinetic energy.

But after any time has passed, the ball 45 degrees above the horizon will be loosing kinetic energy and gaining gravitational potential energy (calculated with mass*gravity*height in a homogeneous - the same everywhere - gravitational field).

The other ball would gain kinetic energy and loose potential energy.

I hope this helps.

 

[jgens]

Read my definition of kinetic energy.

Definition: Kinetic energy is defined to be the amount - magnitude - of work necessary to accelerate an object to a given velocity. Therefore, kinetic energy is direction independent.

Sisplat also explained a physical scenario that illustrates why kinetic energy is always greater than or equal to zero.

 

[razored]

I seem to comprehend it a bit more now. Thanks a bunch.