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It took me quite a long time to click my gears in place and even then I'm not sure it's completely correct.

The problem is that I need to understand these concepts (physics concepts; not just these two) with intuition, not only mathematical representations. So $(E_k = \frac{1}{2} mv^2)$ and $(P_l = mv)$ don't tell me much.

Hence: Here's how I've been viewing them:

  1. Linear momentum is the moving version of inertia; how much it could resist change in its non-zero velocity.
  2. Kinetic energy is how much a moving object could influence other objects upon contact.

So $P_l$ is how much force an object need/can take while $E_k$ is how much it can give. All for moving objects.

Am I correct in this view?

PS. I'm aware of the similar questions already posted. No, they don't address what I need.

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So $P_l$ is how much force an object need/can take while $E_k$ is how much it can give. All for moving objects.

Momentum & Energy are not forces, I believe I understand what you intended but its important not to mince words. Momentum and Energy are what they are defined to be mathematically and nothing more. Along those lines most of the intuition you will need about these equations comes directly from these equations and conservation laws.

As the velocity of an object increases how does its momentum change? Well you know $P_l=mv$ so momentum must increase linearly. How does its $E_k$ change? Again go back to the equation...$\frac{1}{2}mv^2$ ...it increases as a square of the velocity. Then you know that classically, total energy is conserved and that for inelastic collisions momentum is conserved... this tells you how momentum and energy behave when objects interact.

But actually, one could write a book on classical mechanics (actually they have, and they're everywhere) and then you could read all of them but that won't give you the intuition that solving problems provides. My suggestion: solve problems, then solve more problems and then find problems you can't solve and get frustrated, get really frustrated and think. That bit where you get frustrated is the most valuable, as this is how you develop intuition. Intuition requires context and context comes from experience.

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I never said they were forces but rather how much force they need done on them for an effect to show. Namely, why a 1000 kg train at 1 m/s is better than a 1 kg ball at 1000 m/s although both would take the same force and time to stop them although the ball would have a much more destructive effect on any object it contacts because its KE is much higher (1:1,000). –  Noein Apr 30 '12 at 19:00
    
Objects that aren't accelerating don't apply a force, so both of those objects are applying 0 force until they come in to contact with some that has some inertia. Destructive properties is ambiguous, it depends on several things like how the energy is dispersed in the collision. For example if the ball is made of rubber maybe it will heat up and deform greatly upon collision which would absorb a lot of its energy. Meanwhile if the train is completely inelastic then when it collides all of its energy will be transferred kinetically. Otherwise though I agree with your last statement. –  acadien May 2 '12 at 21:36
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