A particle is released from rest at point O at time t= 0 and it falls vertically that its acceleration at is given g-kv,where k is positive constant, g is gravity,v is instantaneous velocity at time t.Express v in terms of g,k and t?

1 Answer
Barry H.
Mar 31, 2018

#V=g/k[1-e^[-kt]]#

Explanation:

Acceleration pf particle, #[dv]/[dt]=g-kv,#.ie,

#[dv]/[dt]+kv=g#. This is a linear equation..............#[1]#

Need to multiply by both sides of .....#[1]# by the integrating factor, given by #e^[int[ft]dt# =#e^[kt]#, since we are integrating with respect to time.

Therefore, #ve^[kt]=ginte^[kt]dt+C# = #[g/ke^[-kt]+C]............#[2]#.#...... Therefore #v=g/k+Ce^[-kt]#.

Since the particle falls from rest, #v=0 # when # t=0,#

so #0=g/k+C#,....... therefore# C=-g/k#.

And so #v=g/k[1-e^[-kt]]#, ...[.as #e^[-kt]# approaches #0# as #t # approaches infinity, the velocity approaches a fixed or terminal value #g/k#, as #t# increases]

Hope this was helpful..

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