Question

question 27 of 29
for a satellite to orbit earth at a constant distance, its centrifugal acceleration must be equal and opposite earths gravitational acceleration. if a satellite is to orbit at a constant distance from earth at a circular radius of 7,000,000 m, what is the required velocity of the satellite? (assume the acceleration due to earths gravity is 8.2 m/s² at this altitude.)

a. 8239 m/s
b. 7043 m/s
c. 6818 m/s
d. 7576 m/s

Explanation:

Step1: Recall centripetal - acceleration formula

The centripetal acceleration formula is $a_c=\frac{v^{2}}{r}$, where $a_c$ is the centripetal acceleration, $v$ is the velocity, and $r$ is the radius of the circular path. In the case of a satellite orbiting Earth, the centripetal acceleration is provided by the gravitational acceleration, so $a_c = g$.

Step2: Rearrange the formula for velocity

We have $g=\frac{v^{2}}{r}$. Rearranging for $v$, we get $v=\sqrt{gr}$.

Step3: Substitute the given values

Given $g = 8.2\ m/s^{2}$ and $r=7000000\ m$. Then $v=\sqrt{8.2\times7000000}$.
$v=\sqrt{57400000}\approx7576\ m/s$.

Answer:

D. 7576 m/s