the two concepts are both based on the same variables, mass and velocity. What is the necessity of keeping both? Why we use momentum to analyse collision but not Ek? What's special about momentum, except it looks more convenient (without a spuare)
But look at what's happening to velocity in the second equation - it is being squared and divided by two.
Ultimately, momentum and kinetic energy are simply quantities we can calculate from on object's mass and velocity. Scientists do these calculations because they have some neat properties, namely conservation in a wide variety of circumstances.
These quantities are intimately related. Since momentum is given by
p = mv
and kinetic energy is given by
K = (1/2)mv^2
We can write kinetic energy as
K = (p^2)/2m
(with m=mass, v=velocity, p=momentum, and K=kinetic energy)
Calculate this yourself to make sure it works out.
So, it's easy to convert an object's momentum into it's kinetic energy and vice versa. But what do they mean?
To give my idea, I'm going to use the term "energy" (in quotes) in the much looser, everyday sense of the word (sort of a mashed together momentum-energy--how much would it hurt to be hit by this sort of thing). I hope this won't muddle the illustration but I can't think of a better word (maybe "oomph"). Anyway, this may take a while because energy and momentum are ultimately abstract concepts that are useful in and of themselves, regardless of interpretation. The paragraph that follows is the direct route to an answer; what comes after is the scenic route.
Direct: In my mind, momentum carries kinetic energy from one place to another.
Scenic: Imagine you and a friend are standing on the edge of a lake and you are teaching your friend how to skip stones. You say to hold the stone this way and throw it this hard. He understands everything except, when he throws his stone, he throws it straight down into the water, make a huge splash and soaking you both.
You decide to demonstrate. You take your stone and sling it low over the water at a shallow angle, watching it softly touch the water surface with little splash and sail upwards again. The stone skips several times and disappears into the opposite shore.
Now, if both you and your friend threw your stones at the same velocity, and thus the same magnitude of momentum and kinetic energy, why the different results? It goes back to the fact, mentioned by others, that momentum is a vector--specifying magnitude and direction--and kinetic energy is a scalar--specifying only magnitude.
When your friend threw his stone straight down, the "energy" of the stone was directed straight down (the momentum vector was directed straight down) towards the water. The "energy" of the stone (kinetic energy) had nowhere to go but into the water. The water reacts to this "energy" by splashing.
When you threw your stone, the "energy" was directed mostly parallel to the surface of the water with a small portion heading downwards (the direction of the momentum). When the stone hit the water, only that small portion of "energy" went down into the water, causing small splashes. The rest of the "energy" carried on forward towards the next skip. Because this stone has the same kinetic energy as your friend's stone, all of these little splashes put together would be the same size as your friend's big splash.
Kinetic energy is how much energy a moving body can carry. Momentum determines how that energy is delivered.
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