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Chapter 3: Two and three dimensional motion. Prof. Chris Wiebe Prof. Simon Capstick. The Northern Lights (charged particles accelerating in our atmosphere, giving off light). Chapter Three: 2D and 3D motion.

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chapter 3 two and three dimensional motion

Chapter 3: Two and three dimensional motion

Prof. Chris Wiebe

Prof. Simon Capstick

The Northern Lights

(charged particles accelerating in our atmosphere, giving off light)

chapter three 2d and 3d motion
Chapter Three: 2D and 3D motion
  • Before we can talk about motion in 2D and 3D, we need to talk about vectors and scalars.
  • Scalar: a number with units (examples: mass, speed)
  • Vector: a mathematical quantity with both magnitude and direction.
  • Example: a plane can travel with a speed of 300 km/h. This is a scalar. If I state it’s velocity as being 300 km/h North, then it is a vector.
  • In this course, you must be familiar with how to express vectors in i, j, k components (components along the x, y or z direction).
chapter three 2d and 3d motion1
Chapter Three: 2D and 3D motion
  • We can define unit vectors i, j, k to each have a length of one unit in the directions of x, y and z respectively.
  • With this definition, we can represent any 3D vector as:
  • A = Ax i + Ay j + Az k
  • The magnitude of this number is:
  • A = (Ax2 + Ay2 + Az2)1/2, and this is a scalar.

We will need this to represent position, velocity, and acceleration as vectors in 3D

chapter three 2d and 3d motion2
Chapter Three: 2D and 3D motion
  • How do we use unit vectors?
  • Say that we have a vector in the x-y plane with components Ax = 5 m and Ay = 3 m.
  • We can then write A as: A = (5 m)i + (3 m)j
  • Now it is easy to add or subtract vectors – add or subtract the components!
  • If I had a vector B which is Bx= 2 m and By = 1 m, I can add: A + B = (Ax +Bx)i + (Ay + By)j = (5 m + 2 m)i + (3 m + 1 m)j = (7 m)i + (4 m)j
  • Exercise: Calculate B – A = (-3 m)i + (-2m)j
  • I can also multiply by a scalar: 2A = (2*5 m)i + (2* 3 m)j = (10 m)i + (6 m)j. Note that this just changes the length of the vector (only changes direction if multiplied by a negative scalar).
chapter three 2d and 3d motion3
Chapter Three: 2D and 3D motion
  • So, when adding or subtracting vectors, you can do it geometrically… but as we just demonstrated it is a little easier to break up into components.
  • How can we find components of any arbitrary vector?
  • We can use trigonometry!

Using geometry to add/subtract vectors

chapter three 2d and 3d motion4
Chapter Three: 2D and 3D motion
  • Let’s say that we were given a vector A with it’s length and direction (some angle θ from the x-axis, for example)
  • How can we find the components A = Axi + Ayj?
  • Use trigonometry:
  • Ax = A cos θ
  • Ay = A sin θ
  • We can then add or subtract vectors using these components (for example, C = A + B = (Ax + Bx)i + (Ay + By)j).
chapter three 2d and 3d motion5
Chapter Three: 2D and 3D motion
  • Now we can go back and redefine position, velocity, and acceleration as vectors.
  • Example: A position vector r = x i + y j + z k, and a displacement vector Δr = rf – ri (final pos. – init. pos.)
  • The length of a position vector r = (x2+y2+z2)1/2
  • Velocity vectors: vav = Δr/Δt
  • (Note: vav is still a vector even though we divide by the scalar Δt)
  • Instantaneous velocity:
chapter three 2d and 3d motion6
Chapter Three: 2D and 3D motion

Average velocity

  • So, v(t) is now defined as the limit of Δr/Δt as Δt →0 (or the slope of the tangent line)
  • Another way of expressing this:
  • Magnitude: v = (vx2 + vy2 + vz2)1/2
  • Example of 2D motion: relative motion (ie. boat crossing a river, plane traveling against the wind, etc.) - problem 3-63
  • Adding velocity vectors works only for low speeds – as you approach the speed of light, you need relativity!

Instantaneous velocity

chapter three 2d and 3d motion7
Chapter Three: 2D and 3D motion
  • Acceleration!
  • Average acceleration vector: aav = Δv/Δt
  • Instantaneous acceleration vector:
  • Everything becomes a little more confusing with vectors!
  • For example: the velocity vector is always pointing in the direction of a particle’s motion.
  • The acceleration vector can point in a different direction (it represents the rate ofchange of the velocity vector).
chapter three 2d and 3d motion8
Chapter Three: 2D and 3D motion
  • Velocity and acceleration vectors for a particle moving down a winding path

Particle is speeding up

at point 4

Particle slows

down at pt. 1

Particle is turning

at pts. 2 and 3