The time period of a satellite of Earth is 24 hours. If the separ… — Solution | JEE Physics

MCQEasyJEE 2023Kepler's Laws of Planetary Motion

The time period of a satellite of Earth is $24 \, \text{hours}$ . If the separation between the Earth and the satellite is decreased to one-fourth of the previous value, then its new time period will become:

A
$4 \, \text{hours}$
B
$6 \, \text{hours}$
C
$12 \, \text{hours}$
D
$3 \, \text{hours}$

Answer

Correct answer:D

Step-by-step solution

Standard Method

Given: The initial time period is $T_1 = 24 \, \text{hours}$ and the new separation is $R_2 = \frac{R_1}{4}$ .

Find: The new time period $T_2$ .

Using Kepler's third law:

T^2 \propto R^3

So,

\left(\frac{T_2}{T_1}\right)^2 = \left(\frac{R_2}{R_1}\right)^3

Substitute $R_2 = \frac{R_1}{4}$ and $T_1 = 24 \, \text{hours}$ :

\left(\frac{T_2}{24}\right)^2 = \left(\frac{1}{4}\right)^3 = \frac{1}{64}

Taking square root,

\frac{T_2}{24} = \frac{1}{8}

Therefore,

T_2 = 24 \times \frac{1}{8} = 3 \, \text{hours}

The correct option is D. The solution lists option C, but the worked calculation clearly gives $3 \, \text{hours}$ .

Direct Ratio Method

Given: $T \propto R^{3/2}$ from Kepler's third law.

If the radius becomes $\frac{1}{4}$ of the original, then the time period becomes:

\frac{T_2}{T_1} = \left(\frac{1}{4}\right)^{3/2} = \frac{1}{8}

Hence,

T_2 = 24 \times \frac{1}{8} = 3 \, \text{hours}

Therefore, the new time period is $3 \, \text{hours}$ , so the correct option is D.

Common mistakes

Using $T \propto R$ instead of Kepler's law is incorrect. The orbital time period varies as $R^{3/2}$ , not directly with $R$ . Always start from $T^2 \propto R^3$ .
Reducing the radius to one-fourth and assuming the time period also becomes one-fourth is wrong. Because the relation is not linear, you must apply the exponent $\frac{3}{2}$ or use $T^2 \propto R^3$ .
Mapping the answer from the listed correct option without checking the working can cause an error here. The solution calculation gives $3 \, \text{hours}$ , so the defensible answer is option D, not the displayed option label in the solution.

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