Question

Assuming that the free-fall acceleration of the surface of a collapsing cloud remains constant during the entire collapse, derive an expression for the free-fall time. Show that your answer differs from Eq. (12.26) only by a term of order unity.

Answer 1

To derive the free-fall time for a collapsing cloud with constant acceleration, use the equation t = √(2R/g), which represents the time it takes for an object to fall from rest from a height R under constant gravitational acceleration g. This equation will differ from Eq. (12.26) only by a term of order unity.

Step-by-step explanation:

To derive an expression for the free-fall time of a collapsing cloud with constant acceleration, we can use the standard kinematic equation for one-dimensional motion under constant acceleration, which is:

s = ut + ½ at²

Where s is the displacement, u is the initial velocity, a is the acceleration, and t is the time. For an object starting from rest (u=0) and falling under the influence of gravity (a = g), the displacement s would equal the radius R of the cloud just before it begins to collapse. The equation can be rewritten as:

R = ½ g t²

Solving for t gives us:

t = √(2R/g)

This is the free-fall time for an object falling from rest from a height R under constant gravitational acceleration g. Equation (12.26) in the provided context might contain additional factors or constants that slightly modify the result, but the derived free-fall time will differ only by a term of order unity, assuming the modifications are negligible or equate to a constant of proportionality close to one.