Chapter 3: Two-Dimensional Kinematics

Introduction

  • Most motions in nature follow a curved path rather than a straight line.
  • One-dimensional kinematics (such as those from Ch. 2) can easily be extended to two- and three-dimensional kinematics.

3-1 Kinematics in Two Dimensions

  • Vector: a quantity with both magnitude (length) and direction (angle).
  • Vectors can be used to describe the path of objects in two dimensions
    • Example: one vector may show the horizontal component of motion, while another shows the vertical component.
    • Two perpendicular vectors form a right angle. If we wish to find the straight-line path between the start and end points of these vectors, we can use the Pythagorean theorem.
      • a2 + b2 = c2
      • a and b are the magnitudes of the perpendicular vectors, while c is the magnitude of the hypotenuse.
    • The horizontal and vertical components of motion are independent of each other, and neither affects the other

3-2 Vector Addition and Subtraction: Graphical Methods

  • Given a vector V, its magnitude is represented by the variable in italics V and its direction is represented by θ.
  • Adding vectors graphically: place all vectors head to tail, and draw the resultant vector.
  • Subtracting vector graphically: subtracting a vector is like adding its negation. Negate the vectors that will be subtracted, then add all the vectors together by placing them head to tail and drawing the resultant vector.
  • Vectors are often separated into their perpendicular component vectors to be analyzed.

3-3 Vector Addition and Subtraction: Analytical Methods

  • Trigonometry is used to determine the magnitudes and directions of vectors
  • Say there exists a vector, vector A, with direction θ. It is composed of two perpendicular vectors added together: its x-component (Ax) and its y-component (Ay).
    • Magnitude of Ax: Ax = A cos θ
    • Magnitude of Ay: Ay = A sin θ
    • Magnitude of A: A2 = Ax2 + Ay2
    • Direction θ: θ = tan-1(Ax / Ay)

3-4 Projectile Motion

  • Objects thrown into the air are called projectiles, and their path is called the trajectory. Assume air resistance is negligible for now.
  • Recall that motions along perpendicular axes are independent of each other.
    • This allows us to analyze motion by splitting it into two motions: one vertical, one horizontal.
  • Assume that gravity only affects motion in the vertical direction.
    • ay = -g = -9.80 m/s2
    • ax = 0
  • Total displacement of a projectile can be found by using a vector s that's composed of perpendicular x- and y-components. Velocity is v.
    • Horizontal motion
      • x = x0 + vxt
      • vx = v0x = constant velocity
    • Vertical motion
      • y = y0 + 1/2(v0y + vy)t
      • vy = v0y - gt
      • y = y0 + v0y - 1/2(gt)
      • vy2 = v0y2 - 2g(y - y0)
    • Total displacement and velocity (of the original vector s)
      • s2 = x2 + y2
      • θ = tan-1(vy / vx)
      • v2 = vx2 + vy2
      • θv = tan-1(vy / vx)
    • Maximum height of a projectile
      • h = (v0y2) / (2g)

Compare the purple square with its x-component (blue) and y-component(red)



3-5 Addition of Velocities

  • Velocity is a vector, so the same rules of vector addition explained above apply to them.
  • The velocity of an object can be split into its component velocity vectors along the x- and y-axes.
  • Given a velocity vector v with direction θ that is made up of component velocity vectors vx and vy:
    • vx = v cos θ
    • vy = v sin θ
    • v2 = vx2 + vy2
    • θ = tan-1(vy / vx)