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 • 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 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 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 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 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 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 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 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 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