Kinetic Energy Not Necessarily Equal w/ Same Momentum

[rmarkatos]

Conceptual Question

If two objects have the same momentum do they necessarily have the same kinetic energy? Give your reason.

I would say no basically from just looking at formulas.

P=mv for momentum

K.E.= 1/2mv^2 for kinetic energy

 

[radou]

Of course momentum is not the same as kinetic energy. But momentum and kinetic energy both depend on two variables; the mass and the velocity. Also, note that momentum is a vector quantity, while kinetic energy is a scalar quantity.

Edit: look at this example. Let $$\vec{p}_{1} = m \vec{v}_{1}$$, and $$\vec{p}_{2} = m \vec{v}_{2}$$, where $$\vec{v}_{1} = 3 \vec{i}$$, $$\vec{v}_{2} = 3 \vec{j}$$. The momentums are not the same (the magnitudes are, but the directions are not), and the kinetic energies are the same.

Or, you can look at s non-vector example (assume both momentums are in the same direction): let the momentum be equal, so that m1v1 = m2v2. For example, you can take m1 = 3, and v1 = 4, and m2 = 4, and v2 = 3. Now try to calculate the kinetic energies and see in what relation they are.

 

[kyle8921]

While both formulas include mass and velocity, the way they are calculated is different. Momentum is calculated by multiplying the mass and velocity, while kinetic energy is calculated by multiplying the mass, velocity, and an additional factor of 1/2. This means that even if two objects have the same momentum, their kinetic energy will not necessarily be equal because the additional factor of 1/2 can make a significant difference in the final calculation. Therefore, two objects with the same momentum can have different kinetic energies.