2.2 Motion In Two Dimensions Flashcards
What is the diagonal distance from one corner to the opposite corner of a soccer field?
150 m
What are the two components into which a vector can be separated?
- x component
- y component
What is the x component of the diagonal vector in the soccer field example?
120 m
What is the y component of the diagonal vector in the soccer field example?
90 m
What must a vector have?
- magnitude
- direction
What are the two methods commonly used to show direction for vector quantities in two dimensions?
- polar coordinates method
- navigator method
True or False: A vector can exist without a direction.
False
Fill in the blank: The diagonal motion vector can be resolved into two _______.
[perpendicular parts]
In the soccer field example, the sideline is considered the _______ axis.
[x]
In the soccer field example, the goal line is considered the _______ axis.
[y]
What is the bearing method for measuring direction?
Angles are measured clockwise from north, from 0° to 360°.
East is 90°, south is 180°, and west is 270°.
How do sailors create their sailing plans using technology?
By using digitized maps and global positioning satellites to place vectors on a desired path on screen with a mouse.
The computer calculates total distance, identifies directions, and estimates time for the trip.
In the polar coordinates method, where is the positive x-axis located?
At 0°.
Angles are measured counterclockwise from this axis.
What is the total degrees in one complete rotation?
360°.
In the polar coordinates method, how is a vector of 5 m [240°] oriented in the Cartesian plane?
Located in quadrant III, rotated 240° counterclockwise from the positive x-axis.
What does the navigator method use to indicate vector direction?
Compass bearings: north [N], south [S], east [E], and west [W].
How is the displacement vector 5 m [60° S of W] drawn using the navigator method?
Start at west and move 60° towards south.
What determines the method used for stating vector directions?
The type of problem and context of the question.
If a question involves a boat sailing [30° N of W], which method should be used?
The navigator method.
If a plane has a heading of 135°, which method should be used?
The polar coordinates method.
What does non-collinear mean?
Not along a straight line.
What is the difference between collinear and non-collinear vectors?
Collinear vectors lie along the same straight line, while non-collinear vectors lie along different lines.
In vector addition, what does the plus sign (+) indicate?
Connect the vectors tip to tail while keeping track of their directions.
How do you determine the magnitude and direction of the sum of non-collinear vectors?
Use an accurately drawn scale vector diagram.
What is the formula for finding displacement when vectors are not collinear?
Ad = d₁ + d₂.
What is required to find displacement in two dimensions?
You must add the two position vectors.
Why can you not simply subtract your initial position from your final position when dealing with non-collinear vectors?
Because the vectors are not collinear.
What must you do to find the resultant vector in non-collinear vector addition?
Draw the vectors to scale, connect them tip to tail, and measure the magnitude of the resultant.
True or False: The resultant vector is a scalar quantity.
False.
How can you find the direction of the resultant vector?
Using a protractor.
Fill in the blank: In vector addition, you must connect the vectors _______.
tip to tail.
What is the total distance walked if one walks north 40 m and then west 30 m?
70 m if they were collinear, but not the resultant for non-collinear vectors.
What is the angle between vector R and the x-axis denoted by?
Theta (θ)
Theta represents the angle in trigonometric calculations involving vectors.
What trigonometric functions can be used to determine the components of a vector?
Sine, cosine, and tangent
These functions relate the angles and sides of a right triangle.
How do you calculate the x component of vector R?
R_x = R cos θ
This formula uses the cosine function to find the adjacent side of the triangle.
How do you calculate the y component of vector R?
R_y = R sin θ
This formula uses the sine function to find the opposite side of the triangle.
What are the respective components of a vector R with a magnitude of 10 km/h and an angle of 37°?
- R_x = 8.0 km/h
- R_y = 6.0 km/h
These values are calculated using the cosine and sine functions, respectively.
In the navigator method, which direction corresponds to the positive x direction?
East (E)
This establishes a standard way to represent directions in a Cartesian plane.
In the navigator method, which direction corresponds to the positive y direction?
North (N)
This aligns with the conventional representation of vertical movement in the Cartesian plane.
What does the Pythagorean theorem state about a right triangle?
a² + b² = c²
Here, ‘c’ is the hypotenuse, and ‘a’ and ‘b’ are the other two sides.
What is the relationship between the sides of a right triangle and the trigonometric functions?
sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent
These definitions are critical for solving problems involving right triangles.
Fill in the blank: The x component of vector R is the same as the ______ component.
East component
This highlights the relationship between vector components and cardinal directions.
Fill in the blank: The y component of vector R is the same as the ______ component.
North component
This reaffirms the directional conventions used in vector analysis.
What are the components used to describe a two-dimensional vector?
x and y components
The x and y components are perpendicular to each other.
In vector addition, what does a change in one component affect?
It does not affect the other component
This is due to the perpendicular nature of the x and y components.
What theorem is used to find the magnitude of a resultant vector formed by two components?
Pythagorean theorem
This theorem applies as the components form a right triangle with the resultant vector.
Given R₁ = 9.0 m and R₂ = 12.0 m, what is the magnitude of the resultant vector R?
15 m
Calculated using R = √(R₁² + R₂²).
How do you determine the angle of the resultant vector R using the tangent function?
tan(θ) = opposite/adjacent
In this case, opposite = R₂ and adjacent = R₁.
What is the angle of the resultant vector R if R₁ = 9.0 m and R₂ = 12.0 m?
53.1°
This angle is calculated using θ = tan⁻¹(1.33).
Using the polar coordinates method, what is the direction of the resultant vector R?
53.1°
This direction is relative to the east direction.
Using the navigator method, how is the direction of the resultant vector R expressed?
53.1° N of E
This notation indicates the angle is measured north of the east direction.
Fill in the blank: The x and y components of a vector are _______ to each other.
perpendicular
True or False: A change in the x component of a vector will affect the y component.
False
What can be tracked using vectors in a lacrosse game?
The movement of the players and the ball
Vectors help in analyzing the motion and interactions during the game.
What type of angles should be used to simplify calculations for finding components?
Acute (<90°) angles
Acute angles are easier to work with in vector calculations.
How do you determine the acute angle when given an obtuse angle?
Subtract the obtuse angle from 180°
For example, 180° - 155° = 25°.
What is the formula to find the acute angle when given an angle greater than 180°?
Subtract 180 from the angle
For example, 240° - 180° = 60°.
What does most vector motion involve?
Adding non-collinear vectors
Non-collinear vectors are not aligned along the same line.
What are the four steps for adding non-collinear vectors algebraically?
- Determine x and y components of each vector.
- Add all components in the x direction and y direction.
- Use the Pythagorean theorem to find the magnitude of the resultant vector.
- Use trigonometric ratios to find the angle of the resultant vector.
These steps provide a systematic approach to solving vector addition problems.
In a lacrosse game, if player A passes the ball 12.0 m to player B at an angle of 30°, what is the first step to analyze this displacement?
Determine the x and y components of the vector
Resolving the vector into components is crucial for further calculations.
If player B relays the ball to player C, 9.0 m away at an angle of 155°, what is significant about the vectors involved?
The two vectors are not at right angles to each other
This requires the use of non-standard methods for vector addition.
What is the first calculation needed when resolving displacement vectors?
Resolve each displacement vector into its components
This involves calculating the x and y components for each vector.
What directions are designated as positive when calculating components?
Up and right
This is a common convention in vector analysis.
What is the displacement of a cross-country skier who travelled 15.0 m [220°] and then 25.0 m [335°]?
To determine the displacement, resolve each vector into its x and y components.
How do you resolve a vector into its x component?
Use the formula R_x = R cos θ.
How do you resolve a vector into its y component?
Use the formula R_y = R sin θ.
What angles are designated as positive in the analysis?
Up and to the right are designated as positive.
What is the first vector the skier travels?
15.0 m [220°]
What is the second vector the skier travels?
25.0 m [335°]
In the context of vector resolution, what does θ represent?
θ represents the angle of the vector.
Fill in the blank: The skier’s first displacement vector is _____ m [220°].
15.0
Fill in the blank: The skier’s second displacement vector is _____ m [335°].
25.0
True or False: The skier’s displacement vectors can be resolved into components before calculating the total displacement.
True
How do you find the angle using the tangent function?
Use the formula: tan(angle) = opposite/adjacent
Refer to Figure 2.40 for visual representation.
What is the tangent value calculated in the example?
1.810
This value is derived from the ratio of opposite to adjacent sides.
What is the angle corresponding to tan(1.810)?
619°
This angle is calculated using the arctangent function.
What is the significance of the angle 8 in relation to the x-axis?
It lies below the positive x-axis
Refer to Figure 2.41 for the angle’s position.
Using the polar coordinates method, what is the angle found?
299°
This angle represents the skier’s direction in polar coordinates.
What is the displacement of the cross-country skier?
23.1 m
The displacement is measured at an angle of 299°.
What is the first step in solving a two-dimensional motion problem?
Split the motion into two one-dimensional problems by using the vectors’ x and y components.
How do you combine the results from the x and y components?
Add the x and y components separately.
What theorem is used to find the magnitude of the resultant vector?
Pythagorean theorem.
Which function is used to find the angle of the resultant vector?
Tangent function.
Fill in the blank: To find the magnitude of the resultant, use the _______.
Pythagorean theorem.
True or False: You can directly add the two-dimensional vectors without breaking them into components.
False.
What are the two components used to describe motion in two dimensions?
- x component
- y component
