When a rocket leaves a launch pad the forces become unbalanced.
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regardless friction and energy losses,
according to second Newton's law,
F_reaction = dP/dt
F_reaction = m dv/dt + v dm/dt
Let u = velocity of gas burst and v is velocity of rocket,
F_action = -(u + v) dm/dt
negative means in opposite direction to action force,
and consider for the third Newton's law,
F_reaction = -F_action
m dv/dt + v dm/dt = (u + v) dm/dt
m dv/dt + v dm/dt = u dm/dt + v dm/dt
m dv/dt = u dm/dt
dm/m = dv
∫ dm/m = ∫ dv
v = ln (m) + C
applying the initial condition to determine an arbitrary constant C
according to second Newton's law,
F_reaction = dP/dt
F_reaction = m dv/dt + v dm/dt
Let u = velocity of gas burst and v is velocity of rocket,
F_action = -(u + v) dm/dt
negative means in opposite direction to action force,
and consider for the third Newton's law,
F_reaction = -F_action
m dv/dt + v dm/dt = (u + v) dm/dt
m dv/dt + v dm/dt = u dm/dt + v dm/dt
m dv/dt = u dm/dt
dm/m = dv
∫ dm/m = ∫ dv
v = ln (m) + C
applying the initial condition to determine an arbitrary constant C
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Newton's first law of motion states that "Every body remains in a state of constant velocity unless it is acted upon by an external unbalanced force." In this case, the acceleration of gravity multiplied by the mass of the rocket is balanced by the force of the launch pad in the opposite direction from the force of gravity. When the rocket fuel is ignited, the mass of the ignited fuel is enough to unbalance those forces prior to ignition. The rocket then accelerates off the launch pad.