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A 5kg mass freely falling and experiencing a retarding force of 5v where v is the speed in m/sec has
Asked by sreekarreddypindi | 30 May, 2022, 12:15: PM
answered-by-expert Expert Answer
Force acting on a freely falling body is written as
 
begin mathsize 14px style m space fraction numerator d v over denominator d t end fraction space equals left parenthesis space m space g space right parenthesis space minus left parenthesis space b space v space right parenthesis end style
where m is mass of body , v is velocity at timt t , dv/dt is acceleration i.e., rate of change of velocity,
g is acceleration due to gravity and b is velocity coefficient of resistive force acting on the body.
 
( units of b is kg/s )
 
Above equation is written as
 
begin mathsize 14px style fraction numerator d v over denominator d t end fraction space equals space g space minus space left parenthesis b divided by m right parenthesis space v end style
 
Let b/m = k . ( Units of k is s-1 )
 
begin mathsize 14px style fraction numerator d v over denominator d t end fraction space equals space g space minus space k space v end style
 
begin mathsize 14px style fraction numerator d v over denominator g space minus space k space v end fraction space equals space d t end style
 
By integrating above expression on both sides with initial condition v = 0 at t = 0, we get
 
ln( g - k v ) - ln(g) = - k t
 
begin mathsize 14px style ln fraction numerator g minus k v over denominator g end fraction space equals space ln open parentheses 1 space minus space k over g v close parentheses space equals space minus k t end style

begin mathsize 14px style open parentheses 1 space minus space k over g v close parentheses space equals space e to the power of negative k t end exponent end style

begin mathsize 14px style v space equals space g over k open parentheses 1 minus e to the power of negative k t end exponent close parentheses end style
In above expression , k = b/m = 5 kg s-1/ 5 kg  = 1 s-1
 
Hence we get velocity v as a function of time as
 
v = g ( 1 - e-t )
 
 
 
Answered by Thiyagarajan K | 30 May, 2022, 02:09: PM
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