Chapter 4.1, Problem 111E

To determine

Find the linear speed of the satellite.

Answer to Problem 111E

  $\text{7}\text{.262}\text{km}\text{/}\text{sec }$

Explanation of Solution

Given:

The radius of the circular orbit is 1250 km and the time taken to complete one revolution in 110 minutes.

Concept Used:

Linear velocity is the change of position of the body with respect to the time when it is travelling along the straight path whereas the angular velocity is the change of position of the body with respect to the time when it is travelling along the circular path.

The formula for linear speed: $\text{ν}\text{=}\text{r}·\text{ω}$

Where $\text{r = radius = 1250 km + 6378 km = 7628 km }$

  $\text{ω}$ = angular speed (radians per time) = $\frac{1\text{rev}}{\text{Time}}=\frac{2\text{π}}{\text{T}}$

Calculation:

Given the time = 110 minutes.

Use the formula for linear speed: $\text{ν}\text{=}\text{r}·\text{ω}$

Where $\text{r = radius = 1250 km + 6378 km = 7628 km }$

  $\text{ω}$ = angular speed (radians per time) = $\frac{1\text{rev}}{\text{Time}}=\frac{2\text{π}}{\text{T}}$

  $=\frac{1\text{rev}}{110\text{minutes}}=\frac{2\text{π}\text{rad}}{110\text{minutes}}$

From the given angular speed, we can find the linear speed by using the formula $\text{ν}\text{=}\text{r}·\text{ω}$

  $\text{ν}\text{=}\text{r}·\text{ω}=7628·\frac{2\text{π}}{110\text{minutes}}\text{km/min}\text{=}7628·\frac{2·3.1416}{110·60}\text{km/sec}\approx \text{7}\text{.262}\text{km}\text{/}\text{sec }$

Linear speed = $\text{7}\text{.262}\text{km}\text{/}\text{sec }$

Thus, the linear speed is $\text{7}\text{.262}\text{km}\text{/}\text{sec }$