Can you feel the gravity?

Classical Mechanics Level 3

Two bodies of masses ${ M }_{ 1 }$ and ${ M }_{ 2 }$ are initially at rest at infinite distance apart. They are then allowed to move towards each other under mutual gravitational attraction. Their relative velocity of approach at a separation distance $r$ between them is $\text{\_\_\_\_\_\_\_\_\_\_}$ .

{ (\frac { 2G({ M }_{ 2 }-{ M }_{ 1 }) }{ r } })^{ \frac { 1 }{ 2 } }

None of these choices

{ (\frac { 2G\sqrt { { M }_{ 2 }{ M }_{ 1 } } }{ r } })^{ \frac { 1 }{ 2 } }

{ (\frac { 2G({ M }_{ 2 }+{ M }_{ 1 }) }{ r } })^{ \frac { 1 }{ 2 } }

1 solution

Andrew Yates
Feb 20, 2016

This is a two-body problem. We can turn it into a one-body problem using the reduced mass $\mu$ . We know that $\delta K = - \delta U$ from conservation of energy, so all we have to do is use find $v$ .

$-\big( -G \frac{M_1 M_2}{r} - 0 \big) = \frac{1}{2} \mu v^2 \\ G \frac{M_1 M_2}{r} = \frac{1}{2} \Big( \frac{M_1 M_2}{M_1 + M_2} \Big) v^2$

Just solve for $v$ .

Can you feel the gravity?

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