Two objects with masses M₁ = 30,000 kg and M₂ = 70,000 kg are initially very far apart and at rest.If due to gravitational attraction the two objects attract each other,calculate their total kinetic energy when they are 100 km apart. A)1.4 $\times$ 10^-3 J B)1.4 $\times$ 10^-11 J C)1.4 $\times$ 10^-6 J D)0.7 $\times$ 10^-3 J E)0.7 $\times$ 10^-6 J

Question

Quizplus · Accepted Answer

The total kinetic energy of the system when the objects are 100 km apart can be calculated using the conservation of energy principle. Initially, when the objects are very far apart, their potential energy is approximately zero, and their kinetic energy is also zero since they are at rest. As they attract each other due to gravity, their potential energy decreases (becomes more negative), and their kinetic energy increases. The change in potential energy is equal to the negative of the change in kinetic energy.The gravitational potential energy when they are $r$ meters apart is given by $$U = -\frac{Gm_1m_2}{r}$$ where $G$ is the gravitational constant ($6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2$), $m_1$ and $m_2$ are the masses of the two objects, and $r$ is the distance between their centers in meters.Given $m_1 = 30,000 \, \text{kg}$, $m_2 = 70,000 \, \text{kg}$, and $r = 100 \, \text{km} = 100,000 \, \text{m}$, we can calculate the potential energy at 100 km apart:$$U = -\frac{(6.674 \times 10^{-11}) \times (30,000) \times (70,000)}{100,000}$$$$U = -\frac{(6.674) \times (21) \times 10^{-11+4+4}}{10^5}$$$$U = -\frac{140.154 \times 10^{-3}}{10}$$$$U = -1.4 \times 10^{-3} \, \text{J}$$Since the total mechanical energy is conserved, the increase in kinetic energy of the system is equal to the decrease in potential energy (ignoring the negative sign, as we're interested in the magnitude of the kinetic energy). Therefore, the total kinetic energy of the system when the objects are 100 km apart is $1.4 \times 10^{-3} \, \text{J}$. However, this calculation seems to have a mistake in interpreting the gravitational potential energy formula's outcome in terms of kinetic energy and the scale of the system's energy. The correct approach involves recognizing that the potential energy calculated directly translates into kinetic energy but considering the vast difference in units and the actual gravitational interaction, the correct magnitude of energy considering the conservation of energy and the correct application of formulas leads to a different interpretation, focusing on the system's total energy change from an initial state to the given state.The correct calculation should consider the total energy conservation from an initial state of zero kinetic and potential energy (at infinite separation) to the final state with significant gravitational potential energy (negative value) and kinetic energy. The error in the explanation above lies in the misinterpretation of the potential energy calculation and its relation to kinetic energy, not accounting for the initial conditions correctly and the conservation of energy principle over the entire interaction. The correct answer involves calculating the change in gravitational potential energy as the objects move from being very far apart to 100 km apart and equating this change to the gain in kinetic energy, using the correct values and understanding of gravitational interactions and energy conservation.

Two Objects with Masses M1 = 30,000 Kg and M2 ×\times×

Two Objects with Masses M₁ = 30,000 Kg and M₂ $\times$