Why can't an elastic collision be a head on collision? Surely you mean inelastic collision and elastic collision.
In an elastic collision, the energy of the system is conserved, while for an inelastic collsion, energy in not conserved.
To clarify further, you are trying to draw a comparison between two different classifications. It's like asking how an American is different from a white (Caucasian) man. Some Americans are white, and some whites are American, but while one is a racial grouping, the other is a national grouping.
To get to the point :
A collision is "head-on", if the point of contact is the point of intersection of the lines of motion of the center of masses. If one of the bodies is stationary (in some frame), or if both bodies are traveling along the same line, then the collision is head-on if the point of contact lies along this line.
Thus a collision is classified as "head-on" only based on the geometry of the bodies and their motions. A head on collision may be elastic or inelastic.
A collision is elastic if energy is conserved (as well as momentum). A collision between steel balls is far more elastic than a collision between two lumps of putty.
Thus the classification of an "elastic collision" depends only on the nature of the materials that make up the objects, and nothing else. An elastic collision may, or may not be head on.
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