Solving Convert & Vector Equations: Cartesian, Polar, Push Ball

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In summary, the conversation covered converting polar and rectangular equations, finding ground speed and true bearing using vector addition, and determining combined force magnitude in a push ball game.
  • #1
Apophis
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:confused:
1.Convert the following equation to Cartesian coordinates r sin (angle) = -2.

2.Convert the rectangular equation y = -5x + 4 to a polar equation.

3.An aircraft going from Atlanta, GA to New York, NY on a bearing of S69oE is traveling at a speed of 430 miles per hour. The wind is blowing out of the north to south at a speed of 25 miles per hour. Find the ground speed and the plane's true bearing.

4.Two teams are playing push ball with a large 8 foot diameter ball. One team exerts a force represented by the vector a = 2 i + -5 j , and the other team exerts a force represented by the vector b = -8 i -3 j . Determine the combined force magnitude.
:confused:
 
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  • #2
The translators from polar to cartesian and backwards are:

[tex] x = r cos(\theta) [/tex]

[tex] y = r sin(\theta) [/tex]

And from the above two you can see that

[tex] x^2 + y^2 = r^2 [/tex]

This is enough for #1 and #2

3. Vector addition, draw a vector with heading 69 degrees south of east and label its magnitude 430mph. Then add another vector to it with south heading at 25mph. The total heading and speed will be the sum of the two.

4. Same as #3, draw the two vectors and find the vector sum.
 
  • #3


1. To convert the equation to Cartesian coordinates, we can use the identities x = r cos(angle) and y = r sin(angle). Therefore, the Cartesian equation would be x = -2 cos(angle).

2. To convert the equation to polar coordinates, we can use the identities x = r cos(angle) and y = r sin(angle). Therefore, the polar equation would be r = -5x + 4 or r = -5(r cos(angle)) + 4. This can be simplified to r = -5r cos(angle) + 4 or r(1+5cos(angle)) = 4.

3. To find the ground speed, we can use the formula: ground speed = airspeed + wind speed. Therefore, the ground speed would be 430 + 25 = 455 miles per hour. To find the true bearing, we can use the formula: true bearing = bearing + wind direction. Therefore, the true bearing would be S69oE + 90o = S159oE.

4. To find the combined force magnitude, we can use the formula: magnitude = √(a^2 + b^2). Plugging in the given values, we get magnitude = √(2^2 + (-5)^2) + √((-8)^2 + (-3)^2) = √(4 + 25) + √(64 + 9) = √29 + √73. This cannot be simplified any further without knowing the values of i and j.
 

Related to Solving Convert & Vector Equations: Cartesian, Polar, Push Ball

1. How do I convert Cartesian coordinates to polar coordinates?

To convert Cartesian coordinates (x, y) to polar coordinates (r, θ), use the following equations:
r = √(x2 + y2) and θ = tan-1(y/x).
Make sure to pay attention to the quadrant in which the point lies when determining the angle θ.

2. How do I convert polar coordinates to Cartesian coordinates?

To convert polar coordinates (r, θ) to Cartesian coordinates (x, y), use the following equations:
x = r cos(θ) and y = r sin(θ).
Again, pay attention to the quadrant in which the point lies when determining the values of x and y.

3. What are vector equations?

Vector equations are equations that involve vectors, which are quantities that have both magnitude and direction. These equations can be used to represent physical quantities such as velocity, force, and acceleration.

4. How do I solve vector equations?

To solve vector equations, you can use algebraic methods such as adding, subtracting, and multiplying vectors by scalars. You can also use geometric methods, such as using the head-to-tail method or the parallelogram method, to visualize and manipulate the vectors. It is important to keep track of the direction and magnitude of each vector when solving these equations.

5. How can I use vector equations to solve problems involving motion?

Vector equations can be used to solve problems involving motion by representing the quantities involved (such as velocity and acceleration) as vectors. By manipulating these vectors using algebraic and geometric methods, you can solve for unknown quantities and make predictions about the motion of objects. It is important to keep in mind the physical meaning of each vector in relation to the problem being solved.

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