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How is that If the momentum of the particle is constant with time, its kinetic energy ($E_k$) should also be constant with time a true statement but the converse is false...

When momentum is constant,

\begin{eqnarray} p&=&mv \\ E_k=(mv^2)/2& =& (mv)^2/2m\\ E_k&=&p^2/2m\\ \end{eqnarray} therefore when $p $ is constant, $m$ is a constant. So $E_k$ is also constant...

Isn't it same the other way as well.....when $E_k $ is constant, $m$ anyway a constant therefore $p^2$ is a constant which makes $p$ a constant?

But my note says that when $E_k$ is constant with time, $p$ should also be constant with time is false......so how will the momentum ($p$) vary?

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  • $\begingroup$ NOTE, The opposite of the statement "If A then B" is "If A then NOT B". The statement "If B then A" is the converse of the first, not the opposite. $\endgroup$ – definitelynotbs Jul 13 '19 at 10:47
  • $\begingroup$ Edited it...thank u $\endgroup$ – KillErHawK X Jul 13 '19 at 11:02
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Momentum is a Vector quantity, it is measured with both value and direction. Kinetic Energy is a Scalar quantity, measured only in value, but dependent on momentum.

If momentum is constant this means that we are traveling at a constant speed in a constant direction, our Kinetic Energy is then guaranteed by the equation K=P^2/2m

A constant kinetic energy however, only means that we are traveling at a constant speed; the direction of that speed is not guaranteed. A satellite in a circular orbit, for example, has a constant Kinetic energy but no constant direction, and therefore no constant momentum; while the quantitative value of momentum will be the same everywhere in the orbit, the direction changes, and therefore so does the momentum vector.

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Momentum is a vector, but only its magnitude enters into kinetic energy. So kinetic energy can be constant while the momentum vector varies as long as its magnitude stays constant and only its direction changes (for example like in perfect circular motion).

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