A rocket of constant mass starts vertically upwards with speed V0. Choose the correct options: (R is the radius of the earth).
A
Speed V at a height h is given by V20−V2=2gh1+hR
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B
The maximum height reached by the rocket fired with a speed of 90% of escape velocity is 2.26R.
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C
The maximum height reached by the rocket fired with a speed of 90% of escape velocity is 4.26R.
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D
Speed V at a height h is given by V20−V2=2gh1+h2R
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Solution
The correct options are A Speed V at a height h is given by V20−V2=2gh1+hR C The maximum height reached by the rocket fired with a speed of 90% of escape velocity is 4.26R. To find the height h reached by the rocket, we conserve energy between the starting point and at the height h 12mV2o−GMmR=−GMmR+h+12mV2 ⇒12(V2o−V2)=GMR−GMR+h ⇒12(V2o−V2)=GM(1R−1R+h) ⇒12(V2o−V2)=GMRhR+h ⇒12(V2o−V2)=ghRR+h ⇒V2o−V2=2gh1+hR
Option A is correct.
For the maximum height reached, V in the above equation would be zero. Hence with an initial velocity 0.9Ve; (0.9×√2gR)2=2gh1+hR ⇒0.81×2gR=2gRRh+1 ⇒1+Rh=10.81 ⇒Rh=1.234−1⇒Rh=0.234⇒h=4.26R;