Physics 131 9/9/15 Vectors Each point in space is specified by three numbers: (x, y, z), and a position vector– an arrow showing the displacement from the origin to that position. Vectors add like successive displacements or algebraically by Prof. E. F. Redish

Physics 131 Notation 9/9/15 We specify the directions we are talking about by drawing two little arrows of unit length in two perpendicular directions. “x” and “y” are called the coordinates and can be positive or negative. Note that if x is negative, it means is a vector pointing in the direction opposite to Prof. E. F. Redish

Velocity Average velocity is defined by Physics 131 9/9/15 Velocity Average velocity is defined by Instantaneous velocity is what we get when we consider a very small time interval (compared to times we care about) Note: an average velocity goes with a time interval. Note: an instantaneous velocity goes with a specific time. Prof. E. F. Redish

Vectors (2-D coordinates) Physics 131 9/9/15 Vectors (2-D coordinates) We have 2 directions to specify. We must Choose a reference point (origin) Pick 2 perpendicular axes (x and y) Choose a scale We specify our x and y directions by drawing little arrows of unit length in their positive direction. A position vector (displacement from the origin) is written x and y have units, do not. Prof. E. F. Redish

1D Velocity Velocity is the rate of change of position Average velocity = (how far did you go?)/(how long did it take you?) Instantaneous velocity = same (but for short Δt)

Graphing velocity: Figuring it out from the position Physics 131 9/9/15 Graphing velocity: Figuring it out from the position Slope = <v> You can figure out the velocity graph from the position graph using Δx Δt v (m/s) 9/9/15 Prof. E. F. Redish

Physics 131 9/9/15 Position to velocity Ratio of change in position that takes place to the (small) time interval Difference of two positions at two (close) times 9/9/15 Prof. E. F. Redish

Recall: Vectors (2-D coordinates) Physics 131 9/9/15 Recall: Vectors (2-D coordinates) We have 2 directions to specify. We must Choose a reference point (origin) Pick 2 perpendicular axes (x and y) Choose a scale We specify our x and y directions by drawing little arrows of unit length in their positive direction. A position vector (displacement from the origin) is written x and y have units, do not. Prof. E. F. Redish

Vector velocity and speed A displacement – a change in position – has a direction. This means velocity = displacement/time interval has one too. We define speed as the magnitude of velocity. (No vector on this. Why?)

Physics 131 9/9/15 Velocity to position change in position that takes place in a small time interval sum (“Σ”) in the changes in position over many small time intervals 9/9/15 Prof. E. F. Redish

What have we learned? Representations and consistency Physics 131 9/9/15 What have we learned? Representations and consistency Visualizing where an object is  a position graph at different times Visualizing how fast an object is moving  a velocity graph at different times Position graph  velocity graph Velocity graph  position graph slopes areas Prof. E. F. Redish

Vectors If a position vector (displacement from the origin) lies in each of the four indicated quadrants, what can you say about the vector representation of that displacement, x and y are both positive x and y are both negative x is positive and y is negative x is negative and y is positive y D A x C B