The minimum energy required to launch a satellite of mass <math> <mrow> <mi>m</mi> </mrow> </math> from the surface of earth of mass <math> <mrow> <mi>M</mi> </mrow> </math> and radius <math> <mrow> <mi>R</mi> </mrow> </math> in a circular orbit at an altitude of <math> <mrow> <mn>2</mn> <mi>R</mi> </mrow> </math> from the surface of the earth is:

Apply energy conservation, U i + K i = U f + K f  ⇒ − G M m R + K i = − G M m 3 R + 1 2 m v 2  ⇒ − G M m R + K i = − G M m 3 R + 1 2 × m × G M 3 R  ⇒ K i = − 1 6 G M m R + G M m R  K i = 5 6 G M m R

The minimum energy required to launch a satellite of mass <math> <mrow> <mi>m</mi> </mrow> </math> from the surface of earth of mass <math> <mrow> <mi>M</mi> </mrow> </math> and radius <math> <mrow> <mi>R</mi> </mrow> </math> in a circular orbit at an altitude of <math> <mrow> <mn>2</mn> <mi>R</mi> </mrow> </math> &nbsp;from the surface of the earth is:

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Physics Gravitation Physics Class 11th Gravitation

The minimum energy required to launch a satellite of mass $m$ from the surface of earth of mass $M$ and radius $R$ in a circular orbit at an altitude of $2 R$ from the surface of the earth is:

Option 1 -

$\frac{5 G m M}{6 R}$

Option 2 -

$\frac{2 G m M}{3 R}$

Option 3 -

$\frac{G m M}{2 R}$

Option 4 -

$\frac{G m M}{3 R}$

2 Views | Posted 2 months ago

Asked by Shiksha User

1 Answer

Answered by

alok kumar singh | Contributor-Level 10

2 months ago

Correct Option - 1

Detailed Solution:

Apply energy conservation,

$U_{i} + K_{i} = U_{f} + K_{f}$

$\Rightarrow - \frac{G M m}{R} + K_{i} = - \frac{G M m}{3 R} + \frac{1}{2} m v^{2}$

$\Rightarrow - \frac{G M m}{R} + K_{i} = - \frac{G M m}{3 R} + \frac{1}{2} \times m \times \frac{G M}{3 R}$

$\Rightarrow K_{i} = - \frac{1}{6} \frac{G M m}{R} + \frac{G M m}{R}$

$K_{i} = \frac{5}{6} \frac{G M m}{R}$

Apply energy conservation, <math> <mrow> <msub> <mrow> <mi>U</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>K</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>U</mi> </mrow> <mrow> <mi>f</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>K</mi> </mrow> <mrow> <mi>f</mi> </mrow> </msub> </mrow> </math>  <math> <mrow> <mo>⇒</mo> <mo>−</mo> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> <mi>m</mi> </mrow> <mrow> <mi>R</mi> </mrow> </mfrac> <mo>+</mo> <msub> <mrow> <mi>K</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> <mi>m</mi> </mrow> <mrow> <mn>3</mn> <mi>R</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mi>m</mi> <msup> <mrow> <mi>v</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </math>  <math> <mrow> <mo>⇒</mo> <mo>−</mo> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> <mi>m</mi> </mrow> <mrow> <mi>R</mi> </mrow> </mfrac> <mo>+</mo> <msub> <mrow> <mi>K</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> <mi>m</mi> </mrow> <mrow> <mn>3</mn> <mi>R</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>×</mo> <mi>m</mi> <mo>×</mo> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <mrow> <mn>3</mn> <mi>R</mi> </mrow> </mfrac> </mrow> </math>  <math> <mrow> <mo>⇒</mo> <msub> <mrow> <mi>K</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>6</mn> </mrow> </mfrac> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> <mi>m</mi> </mrow> <mrow> <mi>R</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> <mi>m</mi> </mrow> <mrow> <mi>R</mi> </mrow> </mfrac> </mrow> </math>  <math> <mrow> <msub> <mrow> <mi>K</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mn>5</mn> </mrow> <mrow> <mn>6</mn> </mrow> </mfrac> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> <mi>m</mi> </mrow> <mrow> <mi>R</mi> </mrow> </mfrac> </mrow> </math>

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The minimum energy required to launch a satellite of mass $m$ from the surface of earth of mass $M$ and radius $R$ in a circular orbit at an altitude of $2 R$ from the surface of the earth is:

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The minimum energy required to launch a satellite of mass m from the surface of earth of mass M and radius R in a circular orbit at an altitude of 2 R from the surface of the earth is:

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The minimum energy required to launch a satellite of mass $m$ from the surface of earth of mass $M$ and radius $R$ in a circular orbit at an altitude of $2 R$ from the surface of the earth is: