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Kinetic energy for single particle with constant mass

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Show that single particle with constant mass the equation of motion implies the following differential equation for the kinetic energy
 
Classical Mechanics - kinetic energy for single particle with constant mass Classical Mechanics - Kinetic energy for single particle with constant mass.
Show that for a single particle with constant mass the equation of motionimplies the following differential equation for the kinetic energy:
kinetic energy
while if the mass varies with time the corresponding equation is
kinetic energy
(Source: Herbert Goldstein, Classical Mechanics - Chapter 01)
 

with time variable mass,
 
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