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MCQEasyJEE 2024Kepler's Laws of Planetary Motion

JEE Physics 2024 Question with Solution

A planet takes 200200 days to complete one revolution around the Sun. If the distance of the planet from the Sun is reduced to one-fourth of the original distance, how many days will it take to complete one revolution?

  • A

    2525

  • B

    5050

  • C

    100100

  • D

    2020

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: Original period T1=200 daysT_1 = 200 \, \text{days} and new distance r2=r14r_2 = \frac{r_1}{4}.

Find: The new time period T2T_2.

Use Kepler's Third Law:

T2∝r3T^2 \propto r^3

So,

T22T12=r23r13\frac{T_2^2}{T_1^2} = \frac{r_2^3}{r_1^3}

Substitute r2=r14r_2 = \frac{r_1}{4}:

T22T12=(r14)3r13=164\frac{T_2^2}{T_1^2} = \frac{\left(\frac{r_1}{4}\right)^3}{r_1^3} = \frac{1}{64}

Taking square root,

T2T1=18\frac{T_2}{T_1} = \frac{1}{8}

Hence,

T2=2008=25 daysT_2 = \frac{200}{8} = 25 \, \text{days}

Therefore, the planet will take 25 days25 \, \text{days} to complete one revolution. The correct option is A.

Extracted Working and Discrepancy Note

Given: T1=200 daysT_1 = 200 \, \text{days} and r2=14r1r_2 = \frac{1}{4}r_1.

Find: T2T_2 using the working shown in the solution.

The solution states:

(T1T2)2=(r1r2)3\left(\frac{T_1}{T_2}\right)^2 = \left(\frac{r_1}{r_2}\right)^3

With r2=r14r_2 = \frac{r_1}{4},

(200T2)2=43=64\left(\frac{200}{T_2}\right)^2 = 4^3 = 64

From this,

200T2=8\frac{200}{T_2} = 8

so,

T2=2008=25 daysT_2 = \frac{200}{8} = 25 \, \text{days}

One extracted approach on the page contains inconsistent intermediate arithmetic, including 200T2=32\frac{200}{T_2} = 32 and T2=6.25T_2 = 6.25, but the final stated answer there is still 2525 days. The second approach is internally consistent and confirms the correct answer.

Therefore, the correct answer is 2525 days, that is, option A.

Common mistakes

  • Using direct proportionality T∝r3T \propto r^3 is incorrect because Kepler's Third Law gives T2∝r3T^2 \propto r^3. First square the time period relation, then take the square root at the end.

  • Forgetting to cube the distance ratio is a conceptual error. When the distance becomes r14\frac{r_1}{4}, the ratio in the law involves (14)3\left(\frac{1}{4}\right)^3, not just 14\frac{1}{4}.

  • Reversing the ratio of old and new quantities can lead to the wrong result. Keep the same order on both sides, such as T22T12=r23r13\frac{T_2^2}{T_1^2} = \frac{r_2^3}{r_1^3}, before substituting values.

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