Q126 / 180 · COMEDK 2024 (Evening Shift) · Physics
A satellite is revolving around the earth in a circular orbit with kinetic energy of $1.69 \times 10^{10} \mathrm{~J}$. The additional kinetic energy required for just escaping into the outer space is
- A$3.38 \times 10^{10} \mathrm{~J}$
- Correct answerB$1.69 \times 10^{10} \mathrm{~J}$
- C$1.35 \times 10^{10} \mathrm{~J}$
- D$0.89 \times 10^{10} \mathrm{~J}$
Solution
For a satellite in a circular orbit of radius $r$ around the earth of mass $M$, the gravitational potential energy is $U = -\dfrac{GMm}{r}$ and the kinetic energy is $K = \dfrac{GMm}{2r}$.
The total mechanical energy of the satellite in the orbit is $E = K + U = \dfrac{GMm}{2r} - \dfrac{GMm}{r} = -\dfrac{GMm}{2r} = -K$.
Given $K = 1.69 \times 10^{10} \text{ J}$, the total energy is $E = -1.69 \times 10^{10} \text{ J}$.
To escape into outer space, the final total energy of the satellite must be at least $0$.
The required additional energy $\Delta E$ is given by $E_{final} - E_{initial} = 0 - E = -(-K) = K$.
Therefore, the additional kinetic energy required is equal to the current kinetic energy, which is $1.69 \times 10^{10} \text{ J}$.
Answer: $1.69 \times 10^{10} \mathrm{~J}$
The total mechanical energy of the satellite in the orbit is $E = K + U = \dfrac{GMm}{2r} - \dfrac{GMm}{r} = -\dfrac{GMm}{2r} = -K$.
Given $K = 1.69 \times 10^{10} \text{ J}$, the total energy is $E = -1.69 \times 10^{10} \text{ J}$.
To escape into outer space, the final total energy of the satellite must be at least $0$.
The required additional energy $\Delta E$ is given by $E_{final} - E_{initial} = 0 - E = -(-K) = K$.
Therefore, the additional kinetic energy required is equal to the current kinetic energy, which is $1.69 \times 10^{10} \text{ J}$.
Answer: $1.69 \times 10^{10} \mathrm{~J}$
Hence, the correct answer is option: 'B'.