Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R

Q: Why doesn't the mass of the escaping object appear in the formula?

The escape velocity is derived by equating the kinetic energy of the object to its gravitational potential energy. Since both of these energies are directly proportional to the mass of the object, the mass cancels out on both sides of the equation.

Q: Can a smaller planet have a higher escape velocity than a larger one?

Absolutely! Escape velocity depends on the ratio of mass to radius ($M/R$), not just the size. A very small but incredibly dense planet (like a neutron star) can have a much higher escape velocity than a large, low-density gas giant.

Q: What if the Reason statement said $M_A/R_A = M_B/R_B$?

If the Reason statement had said that the ratio of mass to radius must be the same, then it would have been the correct mathematical explanation for the Assertion, and the answer would have been option (c).

Q: Does escape velocity depend on the angle at which the object is launched?

No, escape velocity is a scalar quantity derived from energy conservation. As long as the object is not pointed directly at the ground or another obstacle, the minimum speed required to escape remains the same regardless of the launch angle.

Q: Is it possible for two planets with different masses to have the same escape velocity?

Yes, as long as their radii scale proportionally with their masses. For example, if Planet B has twice the mass of Planet A, it must also have twice the radius for their escape velocities to be equal.

Question

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.

**Assertion (A)** The escape velocities of planet A and B are same. But A and B are of unequal mass.

**Reason (R)** The product of their mass and radius must be same, $M_1R_1 = M_2R_2$

In the light of the above statements, choose the most appropriate answer from the options given below.

Options:

(A) Both A and R are correct but R is not the correct explanation of (a)

(B) A is correct but R is not correct.

(C) Both A and R are correct and R is the correct explanation of (a)

(D) A is not correct but R is correct.

Sigma Percentile · Accepted Answer

### The Cosmic Jailbreak: Understanding Escape Velocity

Imagine you are standing on a planet and you throw a ball straight up into the sky. Usually, the planet's gravity pulls it back down. But what if you could throw it so incredibly fast that it never comes back? That magical speed is called the **escape velocity**.

Mathematically, the escape velocity $v_e$ of a planet is determined by its mass $M$ and its radius $R$, and is given by the famous equation:

$$
v_e = \sqrt{\frac{2GM}{R}}
$$

Here, $G$ is the universal gravitational constant. Notice carefully that the escape velocity depends on the *ratio* of the planet's mass to its radius, not just how heavy or how large it is independently.

### Analyzing the Assertion

The problem presents us with an interesting scenario: two planets, A and B, have different masses but exactly the same escape velocity. Let's translate this assertion into a mathematical equation:

$$
v_A = v_B
$$

By substituting our escape velocity formula for both planets, we get a raw setup that looks like this:

$$
\sqrt{\frac{2GM_A}{R_A}} = \sqrt{\frac{2GM_B}{R_B}}
$$

### The Mathematical Verdict

To make sense of this, we need to strip away the square roots. By squaring both sides and canceling out the common $2G$ term, the equation simplifies beautifully:

$$
\frac{M_A}{R_A} = \frac{M_B}{R_B}
$$

This tells us that for two planets to have the same escape velocity, the ratio of their mass to their radius must be identical. If we cross-multiply this relation, we get $M_A R_B = M_B R_A$.

Now, let's look at the Reason (R) provided in the question. It claims that the *product* of their mass and radius must be the same, i.e., $M_A R_A = M_B R_B$. This directly contradicts our derived mathematical truth!

Therefore, we can confidently conclude that while the Assertion (A) is a perfectly valid physical possibility, the Reason (R) is fundamentally incorrect.

Select Answer:

Visualized Solution

\text{Visualizing the Planets}

\text{Escape Velocity Formula}

\text{Equating Escape Velocities}

\text{Substituting the Formula}

\text{Simplifying the Relation}

\text{Cross-Multiplying}

\text{Evaluating the Reason}

\text{Final Conclusion}

The Sigma Insight: Escape Speed and Motion of Satellites

The Cosmic Jailbreak

Analyzing the Assertion

The Mathematical Verdict

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