Physics Q29 – JEE-MAIN 2025

An object is kept at rest at a distance of

3 R

above the earth's surface where

R

is earth's radius. The minimum speed with which it must be projected so that it does not return to earth is : (Assume

M = mass of earth

G = Universal gravitational constant

)

Clue 1 of 3: Escape Velocity Concept

To escape a gravitational field, an object must be projected with a minimum speed such that its total mechanical energy becomes non-negative.

🥷

Ninja Strategy•Conceptual Understanding of Escape Velocity

Recognize that escape velocity decreases with increasing distance from the center of the gravitational body. This allows for the elimination of options that are too high or have an incorrect radial dependence.

🧠 Full Solution Path

Step 1: Determine Initial Position✦ Active

The object is at a distance $3 R$ above the earth's surface. Therefore, its initial distance from the center of the earth is:

r = R + 3 R = 4 R

Step 2: Apply Conservation of Energy○ Expand

For the object to escape the gravitational field, its total mechanical energy must be zero at infinity. Using the principle of conservation of mechanical energy:

K_{i n i t i a l} + U_{i n i t i a l} = K_{f i n a l} + U_{f i n a l}

Where $K_{i n i t i a l} = \frac{1}{2} m v^{2}$ (kinetic energy at projection), $U_{i n i t i a l} = - \frac{G M m}{r}$ (gravitational potential energy), and $K_{f i n a l} = 0$ , $U_{f i n a l} = 0$ at infinity for minimum escape speed.

\frac{1}{2} m v^{2} - \frac{G M m}{r} = 0 + 0

💡 Teacher's Secret Hint

Remember that gravitational potential energy is negative and zero at infinity.

Step 3: Calculate the Escape Speed○ Expand

From the energy conservation equation, we can solve for the minimum projection speed $v$ :

\frac{1}{2} m v^{2} = \frac{G M m}{r} ⟹ v^{2} = \frac{2 G M}{r} ⟹ v = \sqrt{\frac{2 G M}{r}}

Substitute the initial distance $r = 4 R$ :

v = \sqrt{\frac{2 G M}{4 R}} = \sqrt{\frac{G M}{2 R}}

💡 Teacher's Secret Hint

Ensure correct substitution of the distance from the center of the earth, not just from the surface.

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Physics Question 29 – JEE-MAIN 2025

Clue 1 of 3: Escape Velocity Concept

🧠 Full Solution Path