Integrating Shell Collapse Velocity Equation for Free-Fall Time Calculation

In summary, the equation for the velocity of a collapsing shell of a collapsing star is derived to be \frac{dr}{dt}=-[GM(\frac{1}{2}-\frac{1}{R})]1/2. To find the free-fall time, the equation needs to be integrated. The hint is to manipulate the equation by substituting x=r/R and then using the standard integral \int[\frac{x}{1-x}]1/2 dx=\pi/2. However, there may be an error in the right-hand side of the equation as it should be a function of r.
  • #1
TheTourist
25
0
I have derived the following equation for the velocity of a collapsing shell of a collapsing star.
[tex]\frac{dr}{dt}[/tex]=-[GM([tex]\frac{1}{2}[/tex]-[tex]\frac{1}{R}[/tex])]1/2​

I now need to integrate this to find the free-fall time, and the hint is to manipulate the equation into a form where you can use a standard integral: substitute x=r/R and then use the standard integral [tex]\int[/tex][[tex]\frac{x}{1-x}][/tex]1/2 dx=[tex]\pi[/tex]/2

I am completely stuck on how to do this
 
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  • #2
Hi TheTourist! :smile:

(have an integral: ∫ and a square-root: √ and a pi: π :wink:)

hmm … that RHS can't possibly be correct :redface: …

shouldn't it be a function of r? :smile:​
 

1. What is the Integrating Shell Collapse Velocity Equation?

The Integrating Shell Collapse Velocity Equation is a mathematical formula that calculates the velocity at which an object will collapse due to its own weight.

2. Why is the Integrating Shell Collapse Velocity Equation important?

This equation is important for calculating the free-fall time of objects, which is crucial for understanding the behavior of objects in free-fall situations, such as during a building collapse or a meteor impact.

3. How is the Integrating Shell Collapse Velocity Equation derived?

The equation is derived from the principles of conservation of energy and momentum, as well as the concept of gravitational potential energy. It takes into account the mass, radius, and density of the object to calculate its collapse velocity.

4. Can the Integrating Shell Collapse Velocity Equation be used for all objects?

No, the equation is specifically designed for objects that collapse due to their own weight, such as buildings, rocks, or planets. It cannot be used for objects that are propelled or accelerated by external forces.

5. What are some practical applications of the Integrating Shell Collapse Velocity Equation?

The equation has many practical applications in engineering and physics, such as predicting the collapse of structures, analyzing the impact of meteorites, and designing safety measures for free-fall situations. It can also be used in simulations and experiments to study the behavior of collapsing objects.

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