Question 54: Gravitation

Question

Given below are two statements : one is labelled as
Assertion A
and the other is labelled as
Reason
$\mathbf{R}$
Assertion A :
The kinetic energy needed to project a body of mass $m$ from earth surface to infinity is $\frac{1}{2} \mathrm{mgR}$, 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

Accepted Answer

Let's analyze both the Assertion and the Reason step by step.
Calculation of the Required Kinetic Energy:
The gravitational potential energy of a body of mass $$m$$ at a distance $$r$$ from the center of the Earth is given by:
$$U(r) = -\frac{GMm}{r},$$
where $$G$$ is the gravitational constant and $$M$$ is the mass of the Earth.
At the Earth's surface (where $$r = R$$), the potential energy is:
$$U(R) = -\frac{GMm}{R}.$$
At infinity, the potential energy is defined as:
$$U(\infty) = 0.$$
The energy needed to move the body from the EarthÃ¢ÂÂs surface to infinity is the change in potential energy:
$$\Delta U = U(\infty) - U(R) = 0 - \Bigg(-\frac{GMm}{R}\Bigg) = \frac{GMm}{R}.$$
Using the relation $$g = \frac{GM}{R^2},$$ we find:
$$\frac{GMm}{R} = mgR.$$
So, the required kinetic energy to just reach infinity (with zero speed at infinity) is:
$$	ext{Kinetic Energy} = mgR.$$
The Assertion A states that the required kinetic energy is $$\frac{1}{2}mgR$$, which is only half of the actual value.
Verification of the Reason:
The Reason states: "The maximum potential energy of a body is zero when it is projected to infinity from earth surface."
By convention in gravitational problems, the potential energy at infinity is taken to be zero. This is a standard and correct statement.
Conclusion:
Assertion A is false because it underestimates the needed energy; the correct kinetic energy should be $$mgR$$.
Reason R is true since the gravitational potential energy is indeed defined to be zero at infinity.
Therefore, the correct answer is:
Option A: $$\mathbf{A}$$ is false but $$\mathbf{R}$$ is true.

Answer

$\mathbf{A}$ is false but $\mathbf{R}$ is true

Answer

Both $\mathbf{A}$ and $\mathbf{R}$ are true and $\mathbf{R}$ is the correct explanation of $\mathbf{A}$

Answer

$\mathbf{A}$ is true but $\mathbf{R}$ is false

Answer

Both $\mathbf{A}$ and $\mathbf{R}$ are true but $\mathbf{R}$ is NOT the correct explanation of $\mathbf{A}$

Gravitation

Choose the correct answer: