Vector Addition and Subtraction
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Questions and Answers

What is the result of adding two equal and opposite vectors A and -A?

  • A
  • A + B
  • 0 (correct)
  • -A

    • Which of the following correctly expresses the difference of two vectors A and B?

  • A - B = A + (-B) (correct)
  • A - B = (A + B)/2
  • A - B = A - B
  • A - B = A + B

    • Which property is NOT true for the null vector?

  • It results from adding equal and opposite vectors.
  • Its magnitude is zero.
  • It can result from multiplying a vector by zero.
  • It can have a specified direction. (correct)

    • Using the parallelogram method, what do you draw to complete the shape when adding two vectors A and B?

    <p>A line from A's tail parallel to B.</p> Signup and view all the answers

    Which equation correctly illustrates the property that a vector plus the null vector equals the vector itself?

    <p>A + 0 = A</p> Signup and view all the answers

    What does the average acceleration depend on?

    <p>The change in velocity divided by the time interval</p> Signup and view all the answers

    What does the equation A + (-A) equal?

    <p>0</p> Signup and view all the answers

    As the time interval ∆t approaches zero, what happens to the average acceleration?

    <p>It converges to instantaneous acceleration</p> Signup and view all the answers

    If the direction of the change in velocity ∆v changes, what else is affected?

    <p>The direction of acceleration</p> Signup and view all the answers

    Which of the following statements correctly describes the properties of the null vector when a vector is multiplied by zero?

    <p>Multiplying by zero yields a null vector.</p> Signup and view all the answers

    According to the associative law, how is the addition of vectors defined?

    <p>(A + B) + C = A + (B + C)</p> Signup and view all the answers

    In the expression for acceleration in terms of x and y, what do ax and ay represent?

    <p>Rates of change of velocity in the x and y directions</p> Signup and view all the answers

    Which of the following statements best describes the relationship between velocity and acceleration in one dimension?

    <p>They are always along the same straight line</p> Signup and view all the answers

    What does the variable θ represent in the context of the average acceleration?

    <p>The angle relative to the x-axis</p> Signup and view all the answers

    Which mathematical operation is necessary to derive ax and ay?

    <p>Differentiation with respect to time</p> Signup and view all the answers

    What happens to the direction of acceleration as the time interval ∆t decreases?

    <p>It can change based on ∆v</p> Signup and view all the answers

    What is the maximum height attained by the ball?

    <p>10.0 m</p> Signup and view all the answers

    What is the time taken for the ball to return to the same level?

    <p>2.9 s</p> Signup and view all the answers

    What speed did the ball achieve at its maximum height?

    <p>0 m/s</p> Signup and view all the answers

    What factor must be neglected to simplify calculations of range and height?

    <p>Air resistance</p> Signup and view all the answers

    What is the distance from the thrower to the point where the ball returns to the same level?

    <p>69 m</p> Signup and view all the answers

    In uniform circular motion, what does 'uniform' refer to?

    <p>Constant speed</p> Signup and view all the answers

    For an object in uniform circular motion, how is the direction of velocity changing?

    <p>It changes continuously</p> Signup and view all the answers

    What is the total angle of projection used in calculating the height and distance?

    <p>30°</p> Signup and view all the answers

    What is the total velocity vector v(t) of the particle at time t?

    <p>3.0i + 4.0t j + 5.0k</p> Signup and view all the answers

    How is the average acceleration defined in this context?

    <p>It is the change in velocity divided by the time interval.</p> Signup and view all the answers

    What does the notation v_0 represent in the context given?

    <p>The initial velocity of the particle at t = 0.</p> Signup and view all the answers

    What does the acceleration vector a(t) equal at time t?

    <p>4.0j</p> Signup and view all the answers

    Which component of the velocity does not change with time as per the equations given?

    <p>The z-component of the velocity.</p> Signup and view all the answers

    At time t = 1.0 s, what is the direction of the velocity vector v(t)?

    <p>A combination of x and y directions.</p> Signup and view all the answers

    What final form does the expression v = v_0 + at imply about motion?

    <p>The object moves with constant acceleration.</p> Signup and view all the answers

    If the acceleration is constant and in the j direction, what impact does this have on the x component of velocity?

    <p>The x component remains constant.</p> Signup and view all the answers

    What term describes an object that is in flight after being projected?

    <p>Projectile</p> Signup and view all the answers

    Which equation represents the vertical position of a projectile at time t?

    <p>y = (vo sin θo) t − (1/2) g t^2</p> Signup and view all the answers

    What shape describes the path of a projectile?

    <p>Parabolic</p> Signup and view all the answers

    What is the formula for the maximum height of a projectile?

    <p>hm = (vo^2 sin θo) / 2g</p> Signup and view all the answers

    What is the formula for the horizontal range of a projectile?

    <p>R = (vo^2 sin 2θo) / g</p> Signup and view all the answers

    In uniform circular motion, the direction of the acceleration is always towards what?

    <p>The center of the circle</p> Signup and view all the answers

    What is the relationship between linear velocity (v) and angular speed (ω) in circular motion?

    <p>v = ω R</p> Signup and view all the answers

    The path length traversed by an object between two points differs from what?

    <p>Magnitude of displacement</p> Signup and view all the answers

    Study Notes

    Addition of Vectors

    • The addition of vectors is associative, meaning the order in which vectors are added does not affect the resultant vector: (A + B) + C = A + (B + C).
    • The sum of two equal and opposite vectors is a null vector, represented by 0, with zero magnitude and undefined direction : A – A = 0. |0| = 0.

    Subtraction of Vectors

    • The subtraction of vectors can be defined as the sum of the first vector and the negative of the second: A – B = A + (–B).
    • The vector –B is added to vector A to get the resulting vector R2 = (A – B).

    Parallelogram method for Vector Addition

    • In the parallelogram method, two vectors are added by placing their tails at a common origin. A parallelogram is constructed with the vectors as adjacent sides.
    • The diagonal of the parallelogram starting from the common origin represents the resultant vector R.

    Motion in a Plane

    • The instantaneous acceleration of an object in two or three dimensions is the limiting value of the average acceleration as the time interval approaches zero.
    • Velocity and acceleration vectors can have any angle between them from 0° to 180°.

    Motion in a Plane with Constant Acceleration

    • If an object moves in a plane with constant acceleration, the velocity at any time t is given by: v = v0 + at.
    • This equation can also be expressed in terms of components as: vx = vox + axt and vy = voy + ayt.

    Projectile Motion

    • The path of a projectile is parabolic.
    • Equations for the position and velocity of a projectile launched with initial velocity vo at an angle θo with the horizontal are:
      • x = (vo cos θo) t
      • y = (vo sin θo) t − (1/2) g t2
      • vx = vox = vo cos θo
      • vy = vo sin θo − g t
    • The maximum height (hm) attained by a projectile is given by: hm = (vo sin θo)2 / 2g
    • The time taken to reach the maximum height is: tm = vo sin θo / g
    • The horizontal range (R) of a projectile is given by: R = (vo2 sin 2θo) / g

    Uniform Circular Motion

    • The magnitude of the acceleration of an object in uniform circular motion is ac = v2 / R, directed towards the center of the circle.
    • The angular speed (ω) is the rate of change of angular distance and is related to the velocity v by: v = ω R.
    • The acceleration in uniform circular motion is given by: ac = ω2R.
    • If T is the time period of revolution and ν is the frequency, then: ω = 2πν, v = 2πνR, ac = 4π2ν2R

    Points to Ponder

    • Path length and displacement are not generally the same.

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    Related Documents

    Motion In A Plane PDF

    Description

    This quiz covers the key concepts of vector addition and subtraction, including their properties and methods such as the parallelogram method. Understand how vectors interact in motion across two or three dimensions and apply these concepts to solve related problems.

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