Motion is Relative We always judge motion by comparing a moving object to something else. The “something else” is called a frame of reference.

Motion is Relative During the last slide: –You didn't move at all relative to your neighbor –You moved about 20 kilometers due to the earth's rotation –You moved about 1000 kilometers due to the earth's motion through space

Frame of Reference for 1-D Motion It’s like a number line It has an origin There is a positive direction that’s defined And a negative direction on the other side

Distance and Displacement Distance: How far you travel (in some time interval) –We’ll use the symbol “d” Displacement: How far away you are from where you started (in some time interval)

Distance and Displacement Example of the difference: I run around a 400-meter track in 60 seconds. Distance traveled during those 60 seconds: 400 m Displacement: 0 m (I ended up back where I started)

Displacement Displacement is the change in position It is not the same as distance traveled It has a direction; in one dimension, we can tell the direction by the sign (+/-)

Rate Rate: how a quantity changes over time. Mathematically: rate = quantity/time Ex: hot dogs/minute (“hot dogs per minute”

Speed Speed is the rate of changing distance Speed is distance per unit time How much time ? How far?

Velocity Velocity is slightly different from speed: we use displacement instead of distance, and direction matters (more about that later) We’ll use “v” for either speed or velocity– pay attention to the context Unit: m/s (meters per second)

Average vs. Instantaneous Average speed is the velocity over an extended period of time (like the previous example) Ave. speed = total distance total time Instantaneous velocity is the velocity at an instant: same equation, but time interval would be a tiny, tiny number

Average vs. Instantaneous I’m driving to work 4 miles away (about 8400 m) I stop for a doughnut I get to work in 30 minutes (1800 s) Average speed = d = 4.67 m/s t When I was getting a doughnut, my instantaneous speed was 0 m/s When I was driving on George Mason, my instantaneous speed was 30 mph (13.4 m/s)

Graphing Motion Position vs. Time Position is same at every time (d = 0) So velocity = 0 Position t Stationary objects

Graphing Motion Position changes same amount every interval If it moves 2m in 1 st second, it will move 2m every second Position t Objects with constant velocity

Graphing Motion The slope is the change in position/ change in time That’s the velocity! KEY FINDING: Slope of position/time graph is the velocity Negative slope: object is moving in the negative direction Position t Objects with constant velocity Change in position Change in time

Average vs. Instantaneous Velocity Slope at any point is the instantaneous velocity Average velocity would be the total displacement divided by total time Position Here slope is 10, so v = 10m/s Here slope is 0, so v = 0 m/s Ave. velocity would be 20/4 = 5 m/s

Interpreting Graphs What’s going on here? Position t Starts in a positive position Moves forward with constant speed Stops for a while Goes backward with constant speed (constant negative velocity) Goes forward with constant speed (slowly) to the origin (x = 0)

Graphing Velocity vs. Time For constant velocity (could be sitting still, could be moving), velocity doesn’t change Graph is just a flat line Position t Velocity t Case 1: No Motion Case 2: Positive Constant Velocity REMEMBER: This is just the slope of the position/time graph! Case 1: No Motion Case 2: Positive Constant Velocity

Area under the curve Question: What does the area under the Velocity vs. Time graph tell you? Velocity (m/s) Time (s) Answer: velocity x time = distance (By “Area under the curve”, we mean area between the curve and the horizontal axis)

Area under the curve It works for changing velocity, too! Velocity (m/s) Time (s) What is the total displacement? Area of the triangle: ½ * 4 * 4 = 8 meters

Careful! Velocity has a direction (in this case, plus or minus) Velocity (m/s) Time (s) If the curve is below the axis, count the area as negative Here, d = -2m + 2m = 0 This triangle: -2m This triangle: 2m

What’s happening here? Getting faster and faster Slope increases, therefore… Velocity increases Position t Velocity t We call it Acceleration

Acceleration Notes Acceleration is any change in speed or direction. Acceleration occurs when an object speeds up, slows down (or changes direction– we’ll see this later)

Acceleration Notes Uniform (or constant) acceleration: when an object accelerates at a constant rate over a period of time. Acceleration = change in velocity/time interval Velocity t

Constant Acceleration Note: In this class, every motion can be broken down to a constant acceleration Velocity t Constant accel NOT Constant accel

Acceleration Notes Mathematically: a = Δv = “change in velocity” v = final velocity v o = initial velocity Units: (m/s) or m s s 2 Δv = v -v o t t

Acceleration Notes Example: A car starts out traveling at 10 m/s and accelerates to 19 m/s in a time of 3 seconds. What is the acceleration of the car? a = v –v o = 19 m/s – 10 m/s = 3 m/s 2 t3s The car accelerates at 3 m/s 2.

Finding Acceleration on a Velocity Graph For linear change in velocity, acceleration is the slope of the velocity graph Speed t Slope = accel = 0 Negative slope, so neg. acceleration (sometimes called “deceleration” Positive slope, so positive acceleration

Average Speed If the speed is changing linearly (constant acceleration) Average speed is just the average of the initial and final speeds v ave = v + v o 2 t V VoVo V ave

Average Speed: Careful! If I accelerate uniformly from 10 to 20 mph (miles per hour), what’s my average speed? Constant acceleration: ½ * ( ) = 15 mph If I drive 10 mph for 10 miles and 20 mph for 10 miles, what’s my average speed? 13.3 mph! Not constant acceleration, so not 15 mph!!!