Question

If the distance between the earth and the sun were
quarter its present value, then the number of days in
an year would be approximately
(1) 127 days
(2) 111 days
3)92 days
(4) 46 days​

Answer 1

Answer:

According to Kepler’s law, we have  

T² ∝ a³

Where

T = Earth years

a = astronomical units (1a = one earth year)

We are given that the distance of the earth and the sum were quarter its present value i.e., a₂/a₁ = ¼.

Therefore, based on Kepler's law mentioned above, we can write the eq. as,

T₁² / a₁³ = T₂² / a₂³

⇒ T₂² = [a₂/a₁]³ * T₁²

⇒ T₂² = [1/4]³ * T₁²

⇒ T₂² = [1/64] * T₁²

Here T₁ = 1 year

⇒ T₂² = [1/64] * (1)

⇒ T₂ = √[(1/64) * 1]  

⇒ T₂ = [1/8] * 365 …… [since 1 year = 365 days]

⇒ T₂ = 45.62 days ≈ 46 days

Thus, the number of days in a year would be approximately is 46 days.