Physics Q47 – JEE-MAIN 2025

A satellite of mass

1000 kg

is launched to revolve around the earth in an orbit at a height of

270 km

from the earth's surface. Kinetic energy of the satellite in this orbit is _______

\times 10^{10} J

. (Mass of earth

= 6 \times 10^{24} kg

, Radius of earth

= 6.4 \times 10^{6} m

, Gravitational constant

= 6.67 \times 10^{- 11} {Nm}^{2} {kg}^{- 2}

)

Enter Calculated Numerical Value:

Clue 1 of 3: Orbital Mechanics

For a satellite in a stable circular orbit, the gravitational force provides the necessary centripetal force.

🧠 Full Solution Path

Step 1: Calculate the orbital radius✦ Active

The orbital radius $r$ is the sum of the Earth's radius $R_{e}$ and the height $h$ of the satellite from the surface. Convert all values to meters.

R_{e} = 6.4 \times 10^{6} m

h = 270 km = 270 \times 10^{3} m = 0.27 \times 10^{6} m

r = R_{e} + h = 6.4 \times 10^{6} m + 0.27 \times 10^{6} m = 6.67 \times 10^{6} m

Step 2: Determine the kinetic energy formula for a satellite in orbit○ Expand

For a satellite in a stable circular orbit, the gravitational force provides the centripetal force. Equating the gravitational force $F_{g} = \frac{G M_{e} m}{r^{2}}$ and the centripetal force $F_{c} = \frac{m v^{2}}{r}$ allows us to find the orbital velocity squared $v^{2}$ . The kinetic energy is $K . E . = \frac{1}{2} m v^{2}$ .

\frac{G M_{e} m}{r^{2}} = \frac{m v^{2}}{r} ⟹ v^{2} = \frac{G M_{e}}{r}

K . E . = \frac{1}{2} m v^{2} = \frac{1}{2} m (\frac{G M_{e}}{r}) = \frac{G M_{e} m}{2 r}

Step 3: Substitute values and calculate the kinetic energy○ Expand

Substitute the given values for $G$ , $M_{e}$ , $m$ , and the calculated $r$ into the kinetic energy formula.

K . E . = \frac{(6.67 \times 10^{- 11} {Nm}^{2} {kg}^{- 2}) \times (6 \times 10^{24} kg) \times (1000 kg)}{2 \times (6.67 \times 10^{6} m)}

K . E . = \frac{6 \times 10^{(- 11 + 24 + 3)}}{2 \times 10^{6}} = \frac{6 \times 10^{16}}{2 \times 10^{6}} = 3 \times 10^{10} J

The kinetic energy is $3 \times 10^{10} J$ . Therefore, the value to be filled in the blank is $3$ .

💡 Teacher's Secret Hint

Pay attention to unit conversions and scientific notation for accurate calculation.

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Physics Question 47 – JEE-MAIN 2025

Clue 1 of 3: Orbital Mechanics

🧠 Full Solution Path