Question

If the size and shape of two objects A & B are the same, but object B has a smaller mass: (a) for which object would the beta factor be larger? Justify your answer. (b) Which object would have a larger terminal velocity? Justify your answer. (c) Using a-b: if you dropped styrofoam ball and a steel ball of the same size/shape off a tall building, which would land land first, discounting air resistance? How about including air resistance? beta= 0.006.

Answer 1

(a) Beta factor is nothing but the velocity of an object with respect to velocity of light.

say, beta = v/c

Obviously an object having lower mass will move fastly as its acceleration will be larger.note that this is not the case of free fall.

dv/dt = a , dv = adt --------------(1)

hence velocity can be calculated by integration.

(b) terminal velocity is the maximum velocity achievable. From eq. (1), it is clear that when terminal velocity is achieved acceleration becomes zero afterwards. terminal velocity for a more massive object say A is achieved at a later time and is greater than B.

Explanation- As the mass of B is less so it will reach its terminal velocity sooner and as terminal velocity is achieved object no longer accelerates so A accelerates for a longer time ,

as dv = adt , so terminal velocity of A will be larger than B.

(c) density of styrofoam ball is less than that of a steel ball

mass= density * volume

so mass of steel ball will be more .steel ball is object A of our (b) part and styrofoam is object B of less mass.As A was having higher terminal velocity so it will reach faster than on the ground than B.

This is also clarified from our daily experiences that a steel ball reaches faster than a styrofoam ball .

If we include air resistance then terminal velocities of both will be reduced little bit as it acts opposite to gravity(gravity acts downward and air resistance acts upward) so ,they both balls will hit the ground at a later time.

Comment regarding any query as it is a very interesting concept .