# Two objects have masses of #36 MG# and #18 MG#. How much does the gravitational potential energy between the objects change if the distance between them changes from #81 m# to #3 m#?

The distance reduces by a factor of

Since you are asking for a relative answer, the masses are irrelevant because their relationship to the gravitational energy is inversely proportional to the distance between them and the product of their masses. Only the separation varies in this case.

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To calculate the change in gravitational potential energy, use the formula:

[ \Delta U = -G \frac{m_1 m_2}{r_f} + G \frac{m_1 m_2}{r_i} ]

Where:

- ( \Delta U ) is the change in gravitational potential energy
- ( G ) is the gravitational constant ((6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2))
- ( m_1 ) and ( m_2 ) are the masses of the objects ((36 , \text{MG}) and (18 , \text{MG}))
- ( r_f ) is the final distance (3 m)
- ( r_i ) is the initial distance (81 m)

[ \Delta U = -\left(6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2\right) \frac{(36 \times 10^6 , \text{kg})(18 \times 10^6 , \text{kg})}{3 , \text{m}} + \left(6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2\right) \frac{(36 \times 10^6 , \text{kg})(18 \times 10^6 , \text{kg})}{81 , \text{m}} ]

[ \Delta U = -2.006 \times 10^{18} , \text{J} + 1.115 \times 10^{17} , \text{J} ]

[ \Delta U = -1.894 \times 10^{18} , \text{J} + 1.115 \times 10^{17} , \text{J} ]

[ \Delta U \approx -1.782 \times 10^{18} , \text{J} ]

Therefore, the change in gravitational potential energy between the objects is approximately (-1.782 \times 10^{18} , \text{J}).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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