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# If two objects have equal kinetic energies, do they necessarily have the same momentum? Defend your answer.?

by shawn s
Member since:
May 01, 2008
Total points:
582 (Level 2)

They do not necessarily have the same momentum
The quantity of momentum of an object is given by the mass multiplied by the velocity while the quantity of kinetic energy is given by one-half the mass multiplied by the square of the velocity.
So its : m*v for momentum and 0.5 *m* v^2 for kinetic energy. Only one example is needed to prove that these are not necessarily the same. (leaving units off for simplicity of typing)
Object 1 m= 8 v=2 momentum = 8*2 =16 kinetic energy = 0.5 * 8 * 4 = 16
Object 2 m= 2 v=4 momentum = 2*4 =8 kinetic energy = 0.5 * 2 * 16 = 16
While this answers the question there is more to it.
Turn the question around and ask " When will you have a case when two objects have the same value of Kinetic energy AND have the same quantity of momentum?" Try and make up some concrete examples like the one above where both object's momentum values are the same AND the object's kinetic energy values are the same. You will quickly get the feel of how the v^2 versus the v is making it difficult. In fact there will be only two cases where you can make it work: 1) when the masses are the same AND the velocities are the same which is an obvious case
2) when the velocities are both zero which is also an obvious case. You can prove algebraically that these are the only cases if you care to pursue it (Lets' not consider the case of both masses being zero which is fine in relativity but not so meaningful in classical physics)
So in general you will NOT see the value of the momentum and kinetic energy coming out the same unless it is one of these obvious cases.
BUT WAIT there is one more point! The way this question is worded means you really should take the VECTOR nature of momentum into account. So even if the masses are the same and the speed is the same, the momentum is NOT necessarily the same. Two objects of equal mass going at the same speed but in opposite directions, as in an explosion, have the same kinetic energy, and the same quantity for the size of the momentum, but they do not have the "same momentum" as the question specifies.
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Thanks a million to everyone who answered! <3

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• by Paulie92
Member since:
April 12, 2010
Total points:
303 (Level 2)
Nope. Since kinetic energy is a scalar quantity and momentum is a vector quantity, direction is the important key to determine this question.

Kinetic energy=(1/2) x mass x speed^2
Momentum=mass x velocity

Assume that both objects have the same mass, since they have the same kinetic energies, they will have the same speed. However, if they are travelling in different directions, they will have different momentum as the direction of velocity (Which is part of the calculation of momentum) will be different.

For example, if the two objects travel in opposite direction, one will have velocity, v, while the other will have velocity of -v. so the momentum of one will be negative while the other will be positive.

Of course, if we assume that both objects have different mass, their momentum may be different since the value of velocity not be the same anymore.
• by ?
Member since:
April 02, 2010
Total points:
1,161 (Level 3)
nop they don't because, the velocity of the momentum is stable but in kinetic energy, it's powered by two.
check its formula.