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Question

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 V20V2=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 V20V2=2gh1+h2R
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Solution

The correct options are
A Speed V at a height h is given by V20V2=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
12mV2oGMmR=GMmR+h+12mV2
12(V2oV2)=GMRGMR+h
12(V2oV2)=GM(1R1R+h)
12(V2oV2)=GMRhR+h
12(V2oV2)=ghRR+h
V2oV2=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=1.2341Rh=0.234h=4.26R;
Option C is correct.

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