Answer
See explanation. In a nutshell, the sum of two vectors can remain unchanged, while the sum of their squares changes.
Work Step by Step
Momentum is a vector, and kinetic energy is a non-negative scalar. The sum of two vectors can remain unchanged while the sum of their squares changes.
To cite a concrete example: consider 2 identical objects, one moving east and one moving west, at the same speed. Because they are moving, the system momentum is zero, but the system’s kinetic energy is positive.
Let them make a head-on collision and stick together. They must be at rest (because now the momentum of this single object is zero). The final kinetic energy is zero. The momentum did not change, but the kinetic energy did.
In a nutshell, the sum of two vectors can remain unchanged, while the sum of their squares changes.
