What is the angular velocity of an aircraft passing overhead at 900 km/h?

[haxan7]

Homework Statement

An aircraft passes directly over you with a speed of 900 km/h at an altitude of 10,000 m. What is the angular velocity of the aircraft (relative to you) when directly overhead? Three minutes later?

Homework Equations

Where were studying center of mass and Moment of inertia.

The Attempt at a Solution

No idea, how to approach this question, Aircraft is flying straight, will there even by any angular velocity?

 

[ehild]

Draw a straight line from your eye to the aircraft. It changes direction as the aircraft flies over you, does not it? The line encloses an angle alpha with the vertical. Find the expression for this angle in terms of time. The angular velocity is the time derivative of the angle.

ehild

 

[haxan7]

Draw a straight line from your eye to the aircraft. It changes direction as the aircraft flies over you, does not it? The line encloses an angle alpha with the vertical. Find the expression for this angle in terms of time. The angular velocity is the time derivative of the angle.

ehild

Speed = 900km/hour
tan(α)=900t/10000
α=arctan(900t/10000)

Derivative is 900/(10000+81 t^2)
If i put t = 0 and t = 3/60, will i get the correct answer? I don't think so, Help me.

 

[ehild]

Do not mix units. Convert the speed of the aircraft to m/s.

ehild

 

[paranormal]

The angular velocity of an aircraft passing overhead at 900 km/h will be zero. Angular velocity is defined as the rate of change of angular displacement, and since the aircraft is flying straight, there is no change in its angular displacement relative to an observer on the ground. The aircraft's motion can be described in terms of its linear velocity, which is 900 km/h in this case.

To calculate the angular velocity of the aircraft relative to you when it is directly overhead, we can use the equation ω = v/r, where ω is the angular velocity, v is the linear velocity, and r is the distance from the axis of rotation. In this case, the axis of rotation is the observer on the ground, and the distance from the axis of rotation to the aircraft is the altitude of 10,000 m.

Therefore, the angular velocity of the aircraft when directly overhead would be ω = (900 km/h) / (10,000 m) = 0.09 rad/s.

Three minutes later, the aircraft will have moved a distance of (900 km/h) * (3 min) = 45 km from its initial position. To calculate the angular velocity at this point, we can use the same equation but with a different distance from the axis of rotation. The new distance would be the hypotenuse of a right triangle with sides of 10,000 m and 45 km. Using the Pythagorean theorem, we can calculate the hypotenuse to be approximately 45.001 km.

Therefore, the angular velocity of the aircraft three minutes later would be ω = (900 km/h) / (45.001 km) = 0.02 rad/s. This means that the angular velocity decreases as the distance from the observer increases, which makes sense since the aircraft is moving further away from the observer's axis of rotation.