Question

Two identical satellites are at the heights $R$ and $7R$ from the earth's surface. ($R$ = Radius of the earth). Then which of the following statement is correct:

• A. Ratio of total energy of both is $4$
• B. Ratio of total energy of both is $5$
• C. Ratio of potential energy of both is $9$
• D. Ratio of kinetic energy of both is $6$

Answer 1

The correct option is A

Ratio of total energy of both is $4$

Step 1: Calculation of energy of the satellites

The distance from the center of the earth to the satellites can be depicted by using the following diagram.

The height of the first satellite from the center of the earth is $2R$.

The height of the second satellite from the center of the earth is $8R$.

Step 2: Calculation of ratio potential energy

The potential energy of the satellite can be calculated as follows:

$p.e.=\frac{-GMm}{h}$

Where, $p.e.$ is the potential energy, $G$ is the gravitational constant, $M$ is the mass of the earth, $m$ is the mass of the satellite, and $h$ is the height of the satellite.

Since the satellites are identical, the masses will be the same.

Hence, the potential energy will depend upon the height only, which is inversely proportional.

$$\begin{gathered} \frac{{\left(p.e.\right)}_1}{{\left(p.e.\right)}_2}=\frac{h_2}{h_1} \\ \frac{{\left(p.e.\right)}_1}{{\left(p.e.\right)}_2}=\frac{8R}{2R} \\ \frac{{\left(p.e.\right)}_1}{{\left(p.e.\right)}_2}=4 \end{gathered}$$

Hence, option (C) is incorrect.

Step 3: Calculation of ratio kinetic energy

The kinetic energy can be calculated as follows:

$k.e.=-\frac{1}{2}p.e.$

Here, $k.e.$ is the kinetic energy.

Calculation of ratio of kinetic energies

$$\begin{gathered} \frac{{\left(k.e.\right)}_1}{{\left(k.e.\right)}_2}=\frac{-{\displaystyle \frac{{\left(p.e.\right)}_1}{2}}}{-{\displaystyle \frac{{\left(p.e.\right)}_2}{2}}} \\ \frac{{\left(k.e.\right)}_1}{{\left(k.e.\right)}_2}=\frac{{\left(p.e.\right)}_1}{{\left(p.e.\right)}_2} \\ \frac{{\left(k.e.\right)}_1}{{\left(k.e.\right)}_2}=4 \end{gathered}$$

Hence, option (D) is incorrect.

Step 4: Calculation of total energy

The total energy is the sum of the kinetic energy and the potential energy.

Calculation of ratio of total energy

$$\begin{gathered} \frac{{\left(t.e.\right)}_1}{{\left(t.e.\right)}_2}=\frac{{\left(k.e.\right)}_1+{\left(p.e.\right)}_1}{{\left(k.e.\right)}_2+{\left(p.e.\right)}_2} \\ \frac{{\left(t.e.\right)}_1}{{\left(t.e.\right)}_2}=\frac{{\displaystyle \frac{-{\left(p.e.\right)}_1}{2}}+{\left(p.e.\right)}_1}{{\displaystyle \frac{-{\left(p.e.\right)}_2}{2}}+{\left(p.e.\right)}_2} \\ \frac{{\left(t.e.\right)}_1}{{\left(t.e.\right)}_2}=\frac{{\displaystyle \frac{{\left(p.e.\right)}_1}{2}}}{{\displaystyle \frac{{\left(p.e.\right)}_2}{2}}} \\ \frac{{\left(t.e.\right)}_1}{{\left(t.e.\right)}_2}=4 \end{gathered}$$

Where, $t.e.$is the total energy.

Hence, the ratio of total energy is $4$.

Therefore, option (A) is correct.