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The last of the fundamental forces to evade unification, deep mysteries remain in our understanding of gravity. Get started at the beginning with the Newtonian description of gravity.

What is the approximate gravitational potential energy of the two-particle system, of masses \(5.4\text{ kg}\) and \(2.9 \text{ kg},\) separated by a distance of \(15.0\text{ m}?\)

**Assumptions and details**

- The universal gravitational constant is \(G=6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2\text{/kg}^2.\)

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Consider an \(xy\)-plane in deep space, where two particles **A** and **B** are located. Particle **A** is fixed at the origin and particle **B** is apart from particle **A** with a distance of \(3.6\text{ m}.\) The masses of particles **A** and **B** are \(24\text{ kg}\) and \(12\text{ kg},\) respectively. If particle **B** is released from rest, what is the kinetic energy of **B** when it has moved \(0.1\text{ m}\) toward **A**?

**Assumptions and Details**

- The universal gravitational constant is \(G=6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2\text{/kg}^2.\)

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The mean diameter of Mars is \(6.7\times 10^3\text{ km}\) and its mass is \(6.7\times 10^{23}\text{ kg}.\) What is the approximate escape speed on Mars?

**Assumptions and Details**

- The universal gravitational constant is \(G=6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2\text{/kg}^2.\)

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Four identical particles are forming a square with side length \(12.0\text{ m},\) as shown in the above figure. Then the side length of the square is reduced to \(4.0\text{ m}.\) If the mass of each particles is \(19.0\text{ kg},\) approximately how much gravitational potential energy of the four-particle system is reduced, assuming that the four particles are point particles?

**Assumptions and Details**

- The universal gravitational constant is \(G=6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2\text{/kg}^2 ?\)

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If a \(6000 \text{ kg}\) space rocket is launched vertically from the surface of the Earth with initial energy \(7.0 \times 10^{10} \text{ J},\) what will be the kinetic energy of the space rocket when it is \(1.4 \times 10^7 \text{ m}\) from the center of the Earth, assuming that the mass and the radius of the Earth are \(6.0 \times 10^{24}\text{ kg}\) and \(6.4 \times 10^6 \text{ m},\) respectively?

**Assumptions and Details**

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