Motion Variables and Models Or “We’ve got to keep on moving” California Content Standards: 1a Students know how to solve problems that involve constant speed and average speed.

Variables that measure how far? Distance Displacement Absolute position change w/o regard to direction Scalar quantity Measured in meters, feet, miles D or s Position change w/ regard to direction Vector quantity Same units as distance Dx = Xf – Xi Activities: Define distance and displacement Define former as scalar quantity and later as vector quantity due to it taking into account the direction of the motion. State the units for the distance variables Write equation for displacement Discuss the track example and how distance and displacement values differ. Discuss meaning of positive and negative distances. By convention we usually consider moving to a right as positive and moving towards the left as negative. These values tell nothing about where the origin is. Retest question Positive means moves to right, negative to left

Variables that measure how fast? Speed Velocity Distance moved/time to move distance w/o regard to direction Scalar Measured in m/s, miles/hr, ft/s Displacement /time interval w/ regard to direction Vector Same units as speed Vavg =(Xf – Xi) / (tf –ti) =Dx/Dt Activites: Write definitions of speed and velocity, state which is a scalar and which is a vector value and list some units of each. Write the mathematical equations for speed and velocity. Point out difference between total distance covered and displacement. Discuss the pacing professor question and how the velocity and speed calculations will differ. Discuss how both quantities can have instantaneous and average values. The former needs to be inferred from a graph or calculated using calculus, the later simply can be calculated using the formula or finding the slope of a x-t graph. Discuss how again positive, by convention, means moving to the right and negative means moving towards the left. These are independent of where the origin is. Positive means moving to right, negative to left

Instantaneous vs. Average Quantities Instantaneous quantities occur at a single clock reading. Only one speed Often slope of tangent line to curve on graph Average quantities occur over several clock readings or a interval of time. Object may have several speeds during interval Often slope of chord on graph Examples: Average speed and position, Instantaneous speed and position Question: Distinguish between instantaneous and average quantities of motion. Give examples of each. Activities: For physics E point out that instantaneous quantities are often the slope of a tangent line on a graph (calculus problem 1) while average quantities are often slope of chord between two points on graph.

Variables Measuring How fast, fast changes Acceleration How fast an object speeds up, slows down, or changes direction Measured in m/s/s, m/s2, km/hr/s etc. A=(vf – vi) / (tf – ti) = Dv / Dt Positive means direction is to right, negative to left NOT that acceleration is increasing or decreasing Instantaneous Acceleration is acceleration at one moment in time Activities: Write definition for acceleration, state that it is a vector quantity that includes direction and list some of its units. Write the mathematical equation for acceleration. Point out the final – initial pattern on the time and distance intervals. Differentiate acceleration from velocity by discussing how accelerations can be felt while velocities cannot such as speeding up or slowing down a car during an illegal drag race or sliding inside your car and elbowing your sibling as the car goes around a turn. Contrast this to being in a train traveling a constant speed and not being able to tell the difference between if you are moving or the countryside is moving. Discuss how positive and negative values of acceleration again only refer to direction NOT whether the speed is increasing or decreasing. For the later you need to know the direction of the velocity. Go through example of car w/ velocity and acceleration in same direction and in opposite directions and note motion changes.

Motion Models Object moves equal distance each second Dx=v Dt Constant Velocity Constant Acceleration Object moves increasing or decreasing distances each second Fab 4 Equations Dx=v0 Dt +1/2 aDt2 Vf=V0+aDt Vf2 = V02 +2aDx Dx=((Vf + V0) /2) Dt X-t graph is parabola V-t graph is line Examples: car speeding up or slowing down Object moves equal distance each second Dx=v Dt X-t graph is line, slope is velocity V-t graph is horizontal line Example: car moving at one speed

Motion Graph Interpretations For X-t graphs Slope over time interval is the average velocity Slope of tangent line is the instantaneous velocity For V-t graphs Slope over time interval is the average acceleration (slope of chord of graph) Slope of tangent line is instantaneous acceleration Area under curve is displacement (can be + or -) Examples

Lecture 3 Question 1 A runner takes 4 laps around a 400 m oval track. What are the distance and displacement the runner runs? 1600 m, 1600 m 0 m, 1600 m 1600 m, 0 m 0 m, 0 m

Lecture 3 Question 2 A confused physics professor paces 20 m in one direction in 5 seconds and then 10 m in the opposite direction in 10 seconds. What is the magnitude of the average velocity of the professor? 1 m/s 2 m/s 3 m/s 4 m/s

Lecture 3 Question 3 What is the acceleration of a car that goes from 20 mph to 100 mph in 4 seconds? 5 mph/s 20 mph/s 25 mph/s 30 mph /s

Lecture 3 Question 4 Which of the following describes the graphs of a car with a constant acceleration? V-t and X-t graphs are straight lines V-t and X-t graphs are curves V-t graph is straight line, X-t graph is a parabola V-t graph is horizontal line, X-t graph is straight line

Lecture 3 Question 5 What do the slope and area under a velocity time graph represent? Acceleration and velocity Velocity and displacement Acceleration and displacement Displacement and velocity