Problem 4 - A Complete Resource Book in Physics for JEE Main

Question

The orbital speed of Jupiter is (A) greater than the orbital speed of earth. (B) lesser than the orbital speed of earth. (C) equal to the orbital speed of earth. (D) Zero.

Step-by-Step Solution

Answer

The orbital speed of Jupiter is (B) lesser than the orbital speed of Earth.

Step 1: Find the orbital speed formula

The orbital speed (v) of a celestial body can be determined using the following formula: \(v = \sqrt{\frac{GM}{r}}\) where: - v = orbital speed - G = gravitational constant (\(6.674 \times 10^{-11} Nm^2/kg^2\)) - M = mass of the central body (the Sun in our case) - r = distance between the celestial body and the central body (orbital distance) This formula will be useful for determining and comparing the orbital speeds of both Jupiter and Earth.

Step 2: Find the orbital distances of Jupiter and Earth

The distance between the Sun and Jupiter is approximately 778 million kilometers, and the distance between the Sun and Earth is approximately 150 million kilometers. For the purpose of calculating orbital speeds, we will use these values: - Jupiter's orbital distance (rJ) = \(778 \times 10^6 km\) - Earth's orbital distance (rE) = \(150 \times 10^6 km\)

Step 3: Calculate the orbital speeds

Using the orbital speed formula and the values of the orbital distances, we can calculate Jupiter's and Earth's orbital speeds: - Jupiter's orbital speed: \(v_J = \sqrt{\frac{GM}{r_J}}\) - Earth's orbital speed: \(v_E = \sqrt{\frac{GM}{r_E}}\) Keep in mind that the mass of the Sun (M) and the gravitational constant (G) are the same for both celestial bodies.

Step 4: Compare the orbital speeds

We don't need the exact values of orbital speeds for determining the answer. We can just compare the distances in the denominators of the formula. Since the orbital distance of Jupiter is greater than Earth's (rJ > rE), and they divide the same value (GM), the result will be smaller for the planet with a greater distance. Therefore: - Jupiter's orbital speed: \(v_J < v_E\)

Step 5: Choose the correct option

Based on our comparison, Jupiter's orbital speed is lesser than the Earth's orbital speed. Therefore, the correct option is: (B) lesser than the orbital speed of Earth.