# Two objects have masses of #5 MG# and #7 MG#. How much does the gravitational potential energy between the objects change if the distance between them changes from #90 m# to #2000 m#?

The gravitational potential energy will *decrease* by about

The change in gravitational potential energy will then be:

We have the following information:

Substituting in these values into the above equation:

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The change in gravitational potential energy between the objects is 1.22 × 10^14 Joules.

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The change in gravitational potential energy (( \Delta U )) between two objects can be calculated using the formula:

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

Where:

- ( G ) is the gravitational constant (( 6.674 \times 10^{-11} , \text{N m}^2/\text{kg}^2 ))
- ( m_1 ) and ( m_2 ) are the masses of the two objects in kilograms (1 MG = ( 10^6 ) kg)
- ( r_f ) is the final distance between the objects
- ( r_i ) is the initial distance between the objects

Converting the masses from megagrams (MG) to kilograms:

- 5 MG = ( 5 \times 10^6 ) kg
- 7 MG = ( 7 \times 10^6 ) kg

Given:

- ( m_1 = 5 \times 10^6 ) kg
- ( m_2 = 7 \times 10^6 ) kg
- ( r_i = 90 ) m
- ( r_f = 2000 ) m

Substituting the values into the formula: [ \Delta U = -\left(6.674 \times 10^{-11}\right) \cdot \frac{(5 \times 10^6) \cdot (7 \times 10^6)}{2000} + \left(6.674 \times 10^{-11}\right) \cdot \frac{(5 \times 10^6) \cdot (7 \times 10^6)}{90} ]

Calculating: [ \Delta U = -\left(6.674 \times 10^{-11}\right) \cdot \frac{35 \times 10^{12}}{2000} + \left(6.674 \times 10^{-11}\right) \cdot \frac{35 \times 10^{12}}{90} ] [ \Delta U = -\left(6.674 \times 10^{-11}\right) \cdot 17.5 \times 10^{9} + \left(6.674 \times 10^{-11}\right) \cdot 388.9 \times 10^{9} ]

[ \Delta U = -1.16795 \times 10^{-1} + 2.59383 \times 10^{-1} ] [ \Delta U = 1.42588 \times 10^{-1} , \text{J} ]

Therefore, the change in gravitational potential energy between the objects is ( 1.42588 \times 10^{-1} ) joules (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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