Question

The earth's radius is about 4000 miles. Kampala, the capital of Uganda, and Singapore are both nearly on the equator. The distance between them is 5000 miles. The flight from Kampala to Singapore takes 9.0 hours. What is the plane's angular velocity with respect to the earth's surface? Give your answer in $\%/\text{h}$.

Short answer

The angular velocity of the plane with respect to the earth's surface is ${7.955}^{°}/\text{h}$.

Step-by-step solution

Step 1.  Given information

1. The radius of earth is 4000 miles, the distance between the Kampala and Singapore is 5000 miles, the flight from Kampala to Singapore takes $9.0\text{h}$.

Step 2.

The angular velocity of the plane is,

$\omega =\frac{d\theta }{t}\mathrm{...}\left(1\right)$

Here,$\omega$ is the angular velocity of the time,$d\theta$ is the change in angular position of the plane, and $t$ is the time period.

The angular position of the plane is,

$d\theta =\frac{L}{r}$

Here, $L$is the distance between Kampala and Singapore, and$r$ is the radius of the earth.

Substitute 5000 miles for and 4000 miles for in the above equation to find$d\theta$ .

$d\theta =\frac{5000\text{miles}}{4000\text{miles}}=\left(1.25\text{rad}\times \frac{{180}^{°}}{\pi }\right)={71.619}^{°}$

Substitute${71.619}^{°}$ for$d\theta$ and$9.0\text{h}$ for$t$ in the equation (I) to find$\omega$ .

$\omega =\frac{{71.619}^{°}}{9.0\text{h}}={7.955}^{°}\text{h}$

Therefore, the angular velocity of the plane with respect to the earth's surface is${7.955}^{°}/\text{h}$ .