Define escape velocity and its expression?

1 Answer
Sunayan S
Feb 2, 2018

ESCAPE VELOCITY:-

The minimum velocity with which a body must be projected vertically upwards in order that it just escape the gravitational field of the earth(specially,not earth but for also other planet) is called Escape velocity.

Let's derive the formula to determine escape velocity...

Let's mass of Earth#="M ";#Radius of Earth#="R"#

  • The attraction force by earth on a object of mass #"m"# from a distance of #x# from the center of the earth is #color(brown)(F=(GMm)/x^2" ";G->#Gravitational constant
  • Against the attraction force,to move the object through the attraction line by a small distance #color(red)(dx#, It need to work #d"W"=(GMm)/x^2 cdot dx#
  • To move the object to infinite, work need,
    #"W"=intd"W"=int_R^oo(GMm)/x^2 cdot dx=GMm[-1/x]_R^oo=(GMm)/R#
  • If escape velocity from earth is #V_("escape")#,then initial kinetic energy of the body is #E_k=1/2mV_("esc")^2#
  • Now, #"W"=E_k#
    #=>(GMm)/R=1/2mV_("esc")^2#
    #=>V_("esc")=sqrt((2GM)/R#
  • For, earth #color(red)([g=(GM)/R^2]#,hence #color(red)(ul(bar(|color(black)(V_("esc")=sqrt(2gR))|#

For earth, #g=9.8ms^-2# and #R=6400xx10^3m#

For earth,the value of escape velocity is #sqrt(2xx9.8xx6400xx10^3)=11200m//s=11.2"km"//s#

Hope it helps..
Thank you...

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