QuestionJuly 19, 2025

The Earth travels around the sun at a distance of 1.5times 10^11 m. The mass of the sun is 1.99times 10^30 kg. What is the Earth's orbital speed?What is the Earth's orbital acceleration? square

The Earth travels around the sun at a distance of 1.5times 10^11 m. The mass of the sun is 1.99times 10^30 kg. What is the Earth's orbital speed?What is the Earth's orbital acceleration? square
The Earth travels around the sun at a distance of 1.5times 10^11 m. The mass of the sun is 1.99times 10^30 kg. What is the Earth's orbital speed?What is the Earth's orbital acceleration? square

Solution
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Answer

Orbital Speed: 29,783 \, \text{m/s}; Orbital Acceleration: 0.00593 \, \text{m/s}^2 Explanation 1. Calculate the gravitational force Use **F = \frac{G \cdot M \cdot m}{r^2}** where G = 6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2, M = 1.99 \times 10^{30} \, \text{kg}, m is Earth's mass (cancels out later), and r = 1.5 \times 10^{11} \, \text{m}. 2. Equate gravitational force to centripetal force Set **\frac{G \cdot M \cdot m}{r^2} = \frac{m \cdot v^2}{r}**. Simplify to find v: v = \sqrt{\frac{G \cdot M}{r}}. 3. Calculate orbital speed Substitute values: v = \sqrt{\frac{6.674 \times 10^{-11} \cdot 1.99 \times 10^{30}}{1.5 \times 10^{11}}}. 4. Calculate orbital acceleration Use **a = \frac{v^2}{r}** with v from Step 3.

Explanation

1. Calculate the gravitational force<br /> Use **$F = \frac{G \cdot M \cdot m}{r^2}$** where $G = 6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2$, $M = 1.99 \times 10^{30} \, \text{kg}$, $m$ is Earth's mass (cancels out later), and $r = 1.5 \times 10^{11} \, \text{m}$.<br /><br />2. Equate gravitational force to centripetal force<br /> Set **$\frac{G \cdot M \cdot m}{r^2} = \frac{m \cdot v^2}{r}$**. Simplify to find $v$: $v = \sqrt{\frac{G \cdot M}{r}}$.<br /><br />3. Calculate orbital speed<br /> Substitute values: $v = \sqrt{\frac{6.674 \times 10^{-11} \cdot 1.99 \times 10^{30}}{1.5 \times 10^{11}}}$.<br /><br />4. Calculate orbital acceleration<br /> Use **$a = \frac{v^2}{r}$** with $v$ from Step 3.
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