Two identical balls of equal mass m₁ and m₂ are placed at rest at the top of separate hills. How do the velocities v₁ and v₂ of the balls compare, measured after each has rolled down to the bottom of its respective hill? (Note: Assume the absence of friction and negligible rotational kinetic energy.) C)v1=2⋅v2 D)v1=4⋅v2

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11ecba2d_1325_0372_a3bf_b5fdc9fa616e_MD0008_00 In mechanical systems, the total energy (U) of an object is quantified by the sum of kinetic energy (KE) and potential energy (PE): 11ecba2d_1325_0373_a3bf_8bb6bf8b9759_MD0008_00 KE describes the energy of translational or rotational motion. Specifically, translational kinetic energy refers to the energy associated with the motion of an object through three-dimensional space and relates to the mass (m) and velocity (v) of an object: 11ecba2d_1325_2a84_a3bf_3108d42293d7_MD0008_00 PE describes the stored energy associated with an object's configuration (eg, a compressed spring) or relative position. Gravitational potential energy describes the energy an object possesses when placed at some distance above the Earth's surface, and it relates to the object mass (m), gravitational constant (g = 10 m/s²), and height (h): 11ecba2d_1325_2a85_a3bf_df87675aff33_MD0008_00 Conservation of energy requires that an object's total energy observed between two points (A and B) remains the same, but contributions from potential energy may convert to kinetic energy (or vice versa) within mechanical systems: 11ecba2d_1325_2a86_a3bf_357e50550825_MD0008_00 In this question, each of two balls of equal mass (m₁ and m₂) is at rest at the top of a hill, each with a different height (h and 2h). Based on the conservation of energy, all the potential energy stored within the balls at the top of the hills is converted into kinetic energy as the balls reach the bottom of the hills (ie, the earth's surface) such that: 11ecba2d_1325_2a87_a3bf_672c40aafc52_MD0008_00 The velocity of each ball may be determined by rearranging and canceling terms within the equations above: 11ecba2d_1325_2a88_a3bf_b90febbea148_MD0008_00 Consequently, the ratio between the velocity of each ball is: 11ecba2d_1325_5199_a3bf_8933ef1eb02c_MD0008_00 11ecba2d_1325_519a_a3bf_f3cff9a3a7a3_MD0008_00 Therefore, placing a ball on a hill at half the half the height of an equal ball results in the first ball having 70% of the velocity of the second ball upon reaching the bottom of the hill (Choice B). Educational objective: Translational kinetic energy is the energy of motion whereas gravitational potential energy describes the energy of position. Due to the conservation of energy, in the absence of friction, the total energy of an object along a path remains constant, but contributions from potential energy may convert to kinetic energy (or vice versa).

Two Identical Balls of Equal Mass M1 and M2 Are

Two Identical Balls of Equal Mass M₁ and M₂ Are