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calculus_jy
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recenty i read the rocket equation, derivation of, however i think i have a slight confusion with signs
suppost initially a rocket has
mass= [tex]M[/tex]
velocity= [tex]\overrightarrow{v}[/tex]
then at a time dt later,
mass of rocket= [tex]M-dM[/tex]
velocity of rocket= [tex]\overrightarrow {v} +d\overrightarrow {v} [/tex]
mass of ejacted gas= [tex]dM[/tex]
velocity of gas= [tex]\overrightarrow{u}[/tex]
using conservation of momentum
[tex]\overrightarrow{v}M=(M-dM)(\overrightarrow{v}+d\overrightarrow{v})+\overrightarrow{u}dM[/tex]
[tex](\overrightarrow{u}-\overrightarrow{v})dM+Md\overrightarrow{v}=0[/tex]
but [tex](\overrightarrow{u}-\overrightarrow{v})[/tex]=velocity of gas relative to rocket
let [tex](\overrightarrow{u}-\overrightarrow{v})=\overrightarrow{U}[/tex]which is a constant
[tex]\overrightarrow{U}dM+Md\overrightarrow{v}=0[/tex]
[tex]-\int_{M_0}^{M}\frac{dM}{M}=\frac{1}{\overrightarrow{U}}\int_{\overrightarrow{v}_0}^{\overrightarrow{v}}d\overrightarrow{v}[/tex]
now [tex]-ln\frac{M}{M_0}=\frac{\overrightarrow{v}-\overrightarrow{v_0}}{\overrightarrow{U}}[/tex]
the problem is , when taking the velocity in the direciton rocket is travelling
[tex]\overrightarrow{U}<0[/tex]
[tex]-ln\frac{M}{M_0}>0[/tex]since [tex]\frac{M}{M_0}<1[/tex]
then
[tex]\overrightarrow{v}-\overrightarrow{v_0}<0[/tex] which is impossibe as the rocket is accelerating?
suppost initially a rocket has
mass= [tex]M[/tex]
velocity= [tex]\overrightarrow{v}[/tex]
then at a time dt later,
mass of rocket= [tex]M-dM[/tex]
velocity of rocket= [tex]\overrightarrow {v} +d\overrightarrow {v} [/tex]
mass of ejacted gas= [tex]dM[/tex]
velocity of gas= [tex]\overrightarrow{u}[/tex]
using conservation of momentum
[tex]\overrightarrow{v}M=(M-dM)(\overrightarrow{v}+d\overrightarrow{v})+\overrightarrow{u}dM[/tex]
[tex](\overrightarrow{u}-\overrightarrow{v})dM+Md\overrightarrow{v}=0[/tex]
but [tex](\overrightarrow{u}-\overrightarrow{v})[/tex]=velocity of gas relative to rocket
let [tex](\overrightarrow{u}-\overrightarrow{v})=\overrightarrow{U}[/tex]which is a constant
[tex]\overrightarrow{U}dM+Md\overrightarrow{v}=0[/tex]
[tex]-\int_{M_0}^{M}\frac{dM}{M}=\frac{1}{\overrightarrow{U}}\int_{\overrightarrow{v}_0}^{\overrightarrow{v}}d\overrightarrow{v}[/tex]
now [tex]-ln\frac{M}{M_0}=\frac{\overrightarrow{v}-\overrightarrow{v_0}}{\overrightarrow{U}}[/tex]
the problem is , when taking the velocity in the direciton rocket is travelling
[tex]\overrightarrow{U}<0[/tex]
[tex]-ln\frac{M}{M_0}>0[/tex]since [tex]\frac{M}{M_0}<1[/tex]
then
[tex]\overrightarrow{v}-\overrightarrow{v_0}<0[/tex] which is impossibe as the rocket is accelerating?
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