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MCQEasyJEE 2025Satellites & Orbital Velocity

JEE Physics 2025 Question with Solution

A satellite is launched into a circular orbit of radius RR around the earth. A second satellite is launched into an orbit of radius 1.03R1.03R. The time period of revolution of the second satellite is larger than the first one approximately by:

  • A

    3%3\%

  • B

    4.5%4.5\%

  • C

    9%9\%

  • D

    2.5%2.5\%

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given: The first satellite moves in a circular orbit of radius RR and the second satellite moves in a circular orbit of radius 1.03R1.03R.

Find: The approximate percentage increase in the time period of the second satellite.

Using Kepler's third law for circular orbits:

T2∝R3T^2 \propto R^3

Hence,

T∝R3/2T \propto R^{3/2}

For the two satellites,

T2T1=(R2R1)3/2=(1.03RR)3/2=(1.03)3/2\frac{T_2}{T_1} = \left(\frac{R_2}{R_1}\right)^{3/2} = \left(\frac{1.03R}{R}\right)^{3/2} = (1.03)^{3/2}

Equivalently,

(T2T1)2=(1.03)3\left(\frac{T_2}{T_1}\right)^2 = (1.03)^3

Now,

(1.03)3≈1.092727(1.03)^3 \approx 1.092727

So,

T2T1=1.092727≈1.045\frac{T_2}{T_1} = \sqrt{1.092727} \approx 1.045

Therefore, the time period of the second satellite is larger by

(1.045−1)×100=4.5%(1.045 - 1) \times 100 = 4.5\%

So, the correct option is B.

Approximation Trick

Given: The orbital radius changes from RR to 1.03R1.03R.

Find: The approximate percentage change in time period.

Since

T∝R3/2T \propto R^{3/2}

a small percentage change gives

ΔTT≈32ΔRR\frac{\Delta T}{T} \approx \frac{3}{2} \frac{\Delta R}{R}

Here,

ΔRR=3%\frac{\Delta R}{R} = 3\%

Therefore,

ΔTT≈32×3%=4.5%\frac{\Delta T}{T} \approx \frac{3}{2} \times 3\% = 4.5\%

Thus, the time period increases approximately by 4.5%4.5\%, so the correct option is B.

Common mistakes

  • Using T∝RT \propto R is incorrect because the time period does not vary linearly with orbital radius. Use Kepler's law, which gives T∝R3/2T \propto R^{3/2} instead.

  • Calculating the ratio of radii correctly but forgetting to subtract 11 at the end is wrong. The question asks for the increase, so convert T2T1≈1.045\frac{T_2}{T_1} \approx 1.045 into percentage increase by using 1.045−11.045 - 1.

  • Taking the increase in radius as the increase in time period is incorrect. A 3%3\% increase in radius does not mean a 3%3\% increase in time period because the dependence is a power law, not direct equality.

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