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Gravitation question

2025 · 4 Apr · Shift 1 · Q65
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Gravitation question

2025 · 4 Apr · Shift 1 · Q65

JEE MainPhysicsGravitationMCQ+4 / −1
Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R\mathbf{R}R Assertion A : The kinetic energy needed to project a body of mass mmm from earth surface to infinity is 12mgR\frac{1}{2} \mathrm{mgR}21​mgR, where R is the radius of earth. Reason R : The maximum potential energy of a body is zero when it is projected to infinity from earth surface. In the light of the above statements, choose the correct answer from the options given below
  1. A
    A\mathbf{A}A is false but R\mathbf{R}R is true
  2. B
    Both A\mathbf{A}A and R\mathbf{R}R are true and R\mathbf{R}R is the correct explanation of A\mathbf{A}A
  3. C
    A\mathbf{A}A is true but R\mathbf{R}R is false
  4. D
    Both A\mathbf{A}A and R\mathbf{R}R are true but R\mathbf{R}R is NOT the correct explanation of A\mathbf{A}A
View written solutionFree

Correct answer: A

  1. Check Assertion A

The minimum kinetic energy required to send a body of mass mmm from the Earth's surface to infinity is equal to the increase in gravitational potential energy.

Gravitational potential energy at distance rrr from Earth's center is U(r)=−GMmrU(r)=-\frac{GMm}{r}U(r)=−rGMm​

At Earth's surface, r=Rr=Rr=R: U(R)=−GMmRU(R)=-\frac{GMm}{R}U(R)=−RGMm​

At infinity: U(∞)=0U(\infty)=0U(∞)=0

So, the required energy is ΔU=U(∞)−U(R)=0−(−GMmR)=GMmR\Delta U=U(\infty)-U(R)=0-\left(-\frac{GMm}{R}\right)=\frac{GMm}{R}ΔU=U(∞)−U(R)=0−(−RGMm​)=RGMm​

Using g=GMR2g=\dfrac{GM}{R^2}g=R2GM​, GMmR=mgR\frac{GMm}{R}=mgRRGMm​=mgR

Hence, the required kinetic energy is K=mgRK=mgRK=mgR not 12mgR\dfrac{1}{2}mgR21​mgR.

So, Assertion A is false.

  1. Check Reason R

The gravitational potential energy is usually taken as zero at infinity: U(∞)=0U(\infty)=0U(∞)=0

Since for all finite distances from Earth, U(r)=−GMmr<0U(r)=-\frac{GMm}{r}<0U(r)=−rGMm​<0

its maximum value is indeed at infinity, and that value is zero.

So, Reason R is true.

  1. Choose the correct option
  • Assertion A: False
  • Reason R: True

Therefore, the correct option is: A\boxed{\text{A}}A​

  1. Comparison with stored answer

Stored correct answer: A\text{A}A

My derived answer: A\text{A}A

They match.

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