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JEE Main 2021 (27 Jul, Shift 1) · Escape Velocity

Escape Velocity question from JEE Main 2021 (27 Jul, Shift 1)

Q54JEE Main 2021 · 27 JUL · SHIFT 1Difficult

Escape Velocity

Suppose two planets (spherical in shape) in radii R and 2R, but mass M and 9M respectively have a centre to centre separation 8 R as shown in the figure. A satellite of mass 'm' is projected from the surface of the planet of mass 'M' directly towards the centre of the second planet. The minimum speed 'v' required for the satellite to reach the surface of the second planet is a7GMR\sqrt{\frac{a}{7}\frac{GM}{R}} then the value of 'a' is ____________.

[Given: The two planets are fixed in their position]

Figure

Numerical answer type. Work it out, then open the answer.

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Answer: 4

  1. Neutral point at xx from MM: Mx2=9M(8R−x)2⇒8R−x=3x⇒x=2R\dfrac{M}{x^2} = \dfrac{9M}{(8R-x)^2} \Rightarrow 8R - x = 3x \Rightarrow x = 2R.
  2. Satellite must just reach the neutral point; beyond it the bigger planet pulls it.
  3. At surface of MM: V1=−GMR−9GM7R=−16GM7RV_1 = -\dfrac{GM}{R} - \dfrac{9GM}{7R} = -\dfrac{16GM}{7R}; at neutral point: VN=−GM2R−9GM6R=−14GM7RV_N = -\dfrac{GM}{2R} - \dfrac{9GM}{6R} = -\dfrac{14GM}{7R}.
  4. 12v2=VN−V1=2GM7R⇒v=47GMR\tfrac12 v^2 = V_N - V_1 = \dfrac{2GM}{7R} \Rightarrow v = \sqrt{\dfrac{4}{7}\dfrac{GM}{R}}
  5. So a=4a = 4.
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