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 °/h.

검증된 단계별 안내

Step 1: Understand the relationship between linear velocity and angular velocity. Angular velocity (ω) is related to linear velocity (v) by the formula: ω = v / r, where r is the radius of the circular path.

Step 2: Convert the earth's radius from miles to the same unit as the distance traveled (miles). Since the radius is already given in miles (4000 miles), no conversion is needed.

Step 3: Calculate the plane's linear velocity (v) by dividing the distance traveled (5000 miles) by the time taken (9.0 hours). Use the formula: v = distance / time.

Step 4: Substitute the values of linear velocity (v) and the earth's radius (r) into the formula for angular velocity: ω = v / r. This will give the angular velocity in radians per hour.

Step 5: Convert the angular velocity from radians per hour to degrees per hour. Use the conversion factor: 1 radian = 57.2958 degrees. Multiply the angular velocity in radians per hour by this factor to express the result in degrees per hour.

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주요 개념

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Angular Velocity

Angular velocity is a measure of how quickly an object rotates around a specific point or axis. It is typically expressed in degrees per unit of time, such as degrees per hour. In this context, it describes how fast the plane is moving in relation to the Earth's rotation as it travels from Kampala to Singapore.

Circumference of the Earth

The circumference of the Earth can be calculated using the formula C = 2πr, where r is the radius. Given the Earth's radius of about 4000 miles, the circumference is approximately 25,132 miles. This value is essential for understanding the relationship between linear distance traveled and the angular displacement of the Earth during the flight.

Equatorial Motion

Equatorial motion refers to the movement of objects along the equator, where the Earth's rotational speed is highest. Since both Kampala and Singapore are near the equator, the plane's flight path can be analyzed in terms of how it relates to the Earth's rotation. This concept is crucial for calculating the angular velocity of the plane as it travels between these two points.