Two objects have unequal masses, m1>m2; which object has greater momentum?

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In summary, when two objects have unequal masses and equal kinetic energies, the object with the greater momentum is determined by the equation momentum = constant * sqrt(m). This means that the object with the larger mass will have the larger momentum. Another approach is to use the equation ½m1v1² = ½m2v2² and solve for momentum, which also shows that the object with the larger mass will have the larger momentum. However, a single counterexample can disprove the textbook answer, as shown by the example of m1=4, v1=1, m2=1, v2=2. In this case, the momentum of object 1 is larger than the momentum of object 2,
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Two objects have unequal masses, m1 > m2. If their kinetic energies are equal, which has the greater momentum?

This is the way I approached this problem. I know that,
Momentum = mass*velocity
K = 1/2mv^2


So, I solved for v since it is the only unknown in these two equations.
k = 1/2mv^2
2k/m = v^2
√2k/m = v

And plugged it into the first equation for momentum.
Momentum = m√2k/m
= √m^2*√2k/m
= √m^2 2k/m
= √2k m

However, since 2k is a constant between the two massess I removed them from the equation giving me Momentum = √m

Since m1>m2 I thought it was safe to say that the momentum of object 1 is greater than the momentum of object 2; however, I was wrong. According to the answer the momentum of m2 is greater than m1 so I'm just wonder, why?

Is my math wrong, am I not suppose to join these two equation? If somebody can help I would greatly appreciate it. Thank you for taking the time to review my question.
 
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Fascinating problem and solution!
You have momentum = constant * sqrt(m).
So the one with the bigger m has the bigger momentum.
Makes sense.

Another approach is the say ½m1v1² = ½m2v2²
so m1*v1 = m2*v2²/v1
momentum1 = m2*v2*v2/v1
momentum 1 = momentum2*v2/v1
The first line ½m1v1² = ½m2v2² with m1 > m2 implies v1 < v2
so v2/v1 is larger than 1 and momentum1 must be larger than momentum2.

Your conclusion has been found correct by two methods . . .
A single counterexample can disprove the textbook answer.
Say m1=4, v1 = 1, m2=1, . Then v2 = 2 to make the energies equal.
p1 = 4*1 = 4, p2 = 1*2 = 2.
 

1. What is momentum?

Momentum is a physical quantity that measures the motion of an object. It is calculated by multiplying the mass of an object by its velocity.

2. How is momentum affected by mass?

The greater the mass of an object, the greater its momentum will be. This means that an object with a larger mass will have a greater resistance to changes in its motion.

3. How does momentum differ from velocity?

Velocity describes the speed and direction of an object's motion, while momentum takes into account the mass of the object as well. Two objects can have the same velocity, but different momenta if they have different masses.

4. In the scenario of two objects with unequal masses, m1>m2, which object has greater momentum?

The object with a greater mass (m1) will have a greater momentum. This is because momentum is directly proportional to mass.

5. Can an object with a smaller mass have a greater momentum than an object with a larger mass?

No, this is not possible. Momentum is dependent on both mass and velocity, and since the velocity of the two objects is not specified, we can assume they are moving with the same velocity. In this case, the object with a larger mass will have a greater momentum.

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