Physics Q31 – JEE-MAIN 2026

If a body of mass

1 kg

falls on the earth from infinity, it attains velocity (

v

) and kinetic energy (

k

) on reaching the surface of earth. The values of

v

and

k

respectively are _______. (Take radius of earth to be

6400 km

and

g = 9.8 {m/s}^{2}

)

Clue 1 of 3: Energy Conservation

When a body falls from infinity under the influence of gravity, its total mechanical energy is conserved.

🧠 Full Solution Path

Step 1: Apply Conservation of Mechanical Energy✦ Active

The initial total mechanical energy ( $E_{i}$ ) at infinity is zero, as the potential energy is zero and the initial velocity is assumed to be zero. The final total mechanical energy ( $E_{f}$ ) at the Earth's surface is the sum of kinetic energy ( $K_{f} = \frac{1}{2} m v^{2}$ ) and gravitational potential energy ( $U_{f} = - m g R$ ). By conservation of energy:

E_{i} = E_{f} ⟹ 0 = \frac{1}{2} m v^{2} - m g R

This simplifies to:

\frac{1}{2} m v^{2} = m g R

Step 2: Calculate Velocity (

v

)○ Expand

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

v^{2} = 2 g R ⟹ v = \sqrt{2 g R}

Given $R = 6400 km = 6.4 \times 10^{6} m$ and $g = 9.8 {m/s}^{2}$ . Substitute these values:

v = \sqrt{2 \times 9.8 {m/s}^{2} \times 6.4 \times 10^{6} m} = \sqrt{125.44 \times 10^{6}} m/s \approx 11.2 \times 10^{3} m/s = 11.2 km/s

💡 Teacher's Secret Hint

Ensure all units are consistent (SI units) before calculation.

Step 3: Calculate Kinetic Energy (

k

)○ Expand

The kinetic energy $k$ at the surface is $k = \frac{1}{2} m v^{2}$ . From Step 1, we established that $\frac{1}{2} m v^{2} = m g R$ . Given $m = 1 kg$ :

k = m g R = 1 kg \times 9.8 {m/s}^{2} \times 6.4 \times 10^{6} m = 62.72 \times 10^{6} J = 6.27 \times 10^{7} J

Comparing the calculated values ( $v = 11.2 km/s$ and $k = 6.27 \times 10^{7} J$ ) with the given options, Option 1 matches.

💡 Teacher's Secret Hint

Remember that the kinetic energy gained is equal to the loss in potential energy when falling from infinity.

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Physics Question 31 – JEE-MAIN 2026

Clue 1 of 3: Energy Conservation

🧠 Full Solution Path