Dynamics - velocity from polar coordinates

In summary, the conversation discusses a scenario where a car is driving straight towards a point at a constant speed and an observer at a different location tracks the car's speed using a radar gun. The question is asked about the speed recorded by the observer. The conversation also mentions the use of polar coordinates and the trigonometric identities. The person involved in the conversation shares their progress in solving the problem, including finding the angle and length of certain lines, but is unsure about how to apply the equation. The final answer is revealed to be the cosine of a certain angle, but the person is still unsure about their approach.
  • #1
Reverend Lee
5
0
Car B is driving straight toward the point O at a constant speed v. An observer, located at A, tracts the car with a radar gun. What is the speed |r(dot)B/A| that the observer at A records? --I've attached a crude version of the example picture. By the way, the angle of the line from the origin to the car is 45 degrees.
2. Homework Equations : v = r(dot)*er + r*theta(dot)*e(sub theta); also the trig identities
Because I'm not too particularly familiar with polar coordinates, I haven't managed to get very far. I found the angle between the x-axis and line AB was 63.4 degrees, the length rb/a is 0.224km, and the angle between the line from the origin to B and the line AB is 18.4 degrees. What I did after that was place line AB to the orgin, and extended er in AB's direction from B and e(sub theta) perpendictular from AB at B. I don't know how to go from there. The answer is provided as 0.949v. Am I on the right track, and if so, how do I apply the equation?
 

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  • #2
I figured out the answer -- its the cosine of 18.4 degrees, but I'm still not entirely sure about how to get the answer, my approach was trial and error.
 
  • #3


Yes, you are on the right track. To apply the equation, you need to first calculate the velocity vector in polar coordinates. This can be done by using the equation v = r(dot)*er + r*theta(dot)*e(sub theta), where r(dot) is the radial velocity and theta(dot) is the angular velocity.

In this case, since the car is driving straight towards the origin at a constant speed v, the radial velocity is simply v and the angular velocity is 0. Therefore, the velocity vector in polar coordinates is v*er.

Next, we need to find the velocity of the car as observed by the observer at A. This can be done by calculating the dot product of the velocity vector and the unit vector pointing from A to B (which is the same as the unit vector pointing from the origin to B since A is located on the x-axis). This can be written as |r(dot)B/A| = v*er dot (cos(45°)*er + sin(45°)*e(sub theta)).

Simplifying this, we get |r(dot)B/A| = v*cos(45°) = 0.949v.

Therefore, the speed recorded by the observer at A is 0.949 times the speed at which the car is driving towards the origin. This makes sense because the observer is located at an angle of 45° with respect to the direction of motion of the car, so the recorded speed will be less than the actual speed.
 

1. What are polar coordinates and how are they used to calculate velocity?

Polar coordinates are a way of representing a point in two-dimensional space using a distance from the origin and an angle from a reference axis. In dynamics, velocity can be calculated using polar coordinates by taking the derivative of the polar position vector with respect to time.

2. How is velocity from polar coordinates different from velocity in Cartesian coordinates?

Velocity in polar coordinates is represented by a magnitude and direction, while velocity in Cartesian coordinates is represented by a combination of x and y components. The two velocity vectors can be converted to each other using trigonometric functions.

3. Can polar coordinates be used to represent velocity in three-dimensional space?

No, polar coordinates can only represent points in two-dimensional space. In order to represent velocity in three-dimensional space, cylindrical or spherical coordinates must be used.

4. How does the direction of velocity change as the polar angle changes?

The direction of velocity changes as the polar angle changes due to the changing direction of the polar position vector. As the polar angle increases, the direction of velocity also changes accordingly.

5. Can velocity from polar coordinates be negative?

Yes, velocity from polar coordinates can be negative. This indicates that the object is moving in the opposite direction of the vector's direction. The magnitude of the velocity will still be positive.

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