Two bodies of masses m1 and m2 have equal kinetic energies....

Two bodies of masses $m_{1}$ and $m_{2}$ have equal kinetic energies. If $P_{1}$ and $P_{2}$ are their respective momentum, what is ratio of $\mathrm{P}_{2}: \mathrm{P}_{1}$ ? (A) $\mathrm{m}_{1}: \mathrm{m}_{2}$ (B) $\sqrt{\mathrm{m}}_{2} / \sqrt{\mathrm{m}_{1}}$ (C) $\sqrt{m_{1}}: \sqrt{m_{2}}$ (D) $\mathrm{m}_{1}^{2}: \mathrm{m}_{2}^{2}$

Two bodies of masses $m_{1}$ and $m_{2}$ have equal kinetic energies. If $P_{1}$ and $P_{2}$ are their respective momentum, what is ratio of $\mathrm{P}_{2}: \mathrm{P}_{1}$ ?
(A) $\mathrm{m}_{1}: \mathrm{m}_{2}$
(B) $\sqrt{\mathrm{m}}_{2} / \sqrt{\mathrm{m}_{1}}$
(C) $\sqrt{m_{1}}: \sqrt{m_{2}}$
(D) $\mathrm{m}_{1}^{2}: \mathrm{m}_{2}^{2}$

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Key Concepts

Relationship between Kinetic Energy and Momentum

In classical mechanics, kinetic energy and momentum are related through the equation K = p²/(2m), which expresses kinetic energy in terms of momentum and mass. This relationship enables comparisons between the momenta of different objects that possess the same kinetic energy.

Square Root Dependence in Momentum Calculations

When two objects have the same kinetic energy, their momenta are related to the square root of their masses. This is derived from the energy-momentum relationship, indicating that momentum is proportional to the square root of mass for a fixed kinetic energy.

Kinetic Energy

Kinetic energy is the energy an object possesses by virtue of its motion. It is typically expressed as one-half of the product of mass and the square of velocity, and it plays a central role in dynamics by quantifying the energy due to movement.

Momentum

Momentum is a measure of the quantity of motion an object has, calculated as the product of its mass and velocity. It is a vector quantity and is conserved in closed systems, making it a fundamental concept in analyzing physical interactions.

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