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 degrees/h.

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Hi, everyone in this practice problem, we're being asked to calculate the angular velocity of the cargo aircraft in degrees per hour with respect to the earth's surface. So we will have two different aircraft carriers located on a line around the middle of the earth, halfway between the north and the south poles. A distance of 12 50 kilometer separates the two aircraft carriers and the travel time for a cargo aircraft between the two carriers is 3.5 hours. The distance from the Earth Center to a point located on the line around the middle of the planet and on its surface is 63, km. And in this case, we are being asked to calculate the angular velocity of the cargo aircraft in degrees per hour with respect to the earth's surface. The options given are a 0.54 degrees per hour. B 0.99 degrees per hour, C 3.2 degrees per hour and D 10 degrees per hour. So I'm going to start us off with creating a figure of our problem statement. So the figure this, that I'm gonna draw will show the top view of the plane passing through the line around the middle of the earth. So I am going to draw this circle right here, which is representing the earth. And I'm going to draw the two um aircraft carriers, which is represented by these two SPS right here. And essentially I'm going to draw uh the two line connecting the aircraft to the center of the earth which will give, give us the radius R of 63, km just like. So, so I am going to represent the angle in between to be a delta theater. And I'm going to say that the first aircraft is located at theater one and T one and the second one is the two and having the time of T two. All right. So the last information that we have is the distance separating the two aircraft carriers, which is going to be this line right here, Which is separating the two carriers as will be km. Awesome. So now that we've defined our problems or define our system, we know also that the A LA S is 12 50 kilometer and the radius R is 63 78 kilometer, we wanna find what the delta theta is. So using the arc length formula, we can find that delta theta, which is essentially theta two minus theta one is going to equals two S divided by R. So that will mean that delta delta theta is going to equals to 12 50 kilometer Divided by 63, km. I'm gonna cross out the two units there and that will actually equals to 0.1960 radiance. So I'm going to convert the radiance into degrees. So delta theta equals to 0. radiance multiplying that by 180 degrees divided that by pi radiant, that will give us the degree value of delta theta which is 11.23 degrees. Also, we know that the flight time takes 3.5 hours. So 3.5 hours is essentially T two minus T one equals 3.5 hours. Therefore, we can actually calculate the average angular velocity of the cargo aircraft, which is represented by this omega right here to be delta theta Divided by T 2 -1 1 and the L equals to B 11.23 degrees divided by 3.5 hours, which will give us the omega value or the average angular velocity value of the cargo aircraft to be 3. degrees per hour. So with the angular velocity or the average angular velocity of the cargo aircraft being 3.2°/h, the answer to this practice problem is going to correspond to option C. So option C is going to be the answer to this practice problem with the angular speed or the angular velocity of the cargo aircraft averages at 3.2 degrees per hour. So that will be all for this particular practice problem. If you guys still have any sort of foot fusion, please make sure to check out our other lesson videos on similar topics and they'll be all for this one. Thank you.

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Key Concepts

Angular Velocity

Circumference of the Earth

Linear vs. Angular Motion