21st sum

( dv/dt ) = - v  

 

dv/v = - dt

 

By integrating above eqn, we get,   ln(v) = -t + C  ..................(1)

 

where C is constant of integration.

 

At t = 0, we have v = 10, hence C = ln(10)

 

Hence eqn.(1) becomes,  ln(v/10) = -t   or    v = 10 e^-t ...............(2)

 

velocity after 2 seconds,   v = 10 e^-2 m/s

 

from eqn.(2), we get, v = dS/dt = 10 e^-t ...................(3)

 

By integrating, S = -10 e^-t + C  .....................(4)

 

where C is constant of integration

 

Let us assume, at t= 0, distance travelled with retrdation is zero

 

Hence using eqn.(4),  0 = -10 + C  or   C = 10

 

Hence eqn.(40 becomes,  S(t) = 10( 1 - e^-t )

 

Hence ditance travelled after 2 seconds,  S = 10 ( 1 - e^-2 ) m