1. Motion In One Dimension PLATO AND ARISTOTLEGALILEO GALILEILEANING TOWER OF PISA

2. Distance vs. Time for Toy Car (0-5 sec.) Distance (cm) Time (s) 0 200 400 600 800 1000 1.02.03.04.05.0 Constant speed is the slope of the (best fit) line for a distance vs. time graph. Remember, the standard metric unit for length is the meter! Graphing Constant Speed

3. Distance vs. Time for Toy Car (0-0.5 sec.) Distance (mm) Time (s) 0 100 200 300 400 500 0.10.20.30.40.5 Average speed is the slope of a secant line for a distance vs. time graph. 600 (0.13, 0) (0.5, 350) Instantaneous speed is the slope of a tangent line for a distance vs. time graph. click for applet Graphing Average and Instantaneous Speed

4. Distance, Position and Displacement Distance (d) The length of a path traveled by an object. It is never negative, even if an object reverses its direction. Position (x or y) The location of an object relative to an origin. It can be either positive or negative. Displacement (∆x or ∆y) The change in position of an object. Also can be positive or negative. 1.What is the distance traveled if an object starts at point C, moves to A, then to B? 2.What is the displacement of an object that starts at point C and moves to point B? 3.What is the displacement of an object that starts at point A, then moves to point C and then moves to point B? One dimensional motion Two dimensional motion -6-5-4-3-20+1+2+3+4+5+6 CAB x (m) 3.What is the distance traveled and the displacement of the bicycle that starts at point A, then moves to point B, and ends at point C?

5. Distance and Position Graphs Distance vs. TimePosition vs. Time d (m) t (s) x (m) t (s) Distance graphs show how far an object travels. Speed is determined from the slope of the graph, which cannot be negative. Position graphs show initial position, displacement, velocity (magnitude and direction). That’s why position graphs are better! CAR B: constant positive velocity CAR C: constant negative velocity Remember, all these graphs show constant speed. (How do you know?) positive negative

6. Average Speed vs. Average Velocity Average speed is the distance traveled divided by time elapsed. Average velocity is displacement divided by time elapsed. Example: A sprinter runs 100 m in 10 s, jogs 50 m further in 10 s, and then walks back to the finish line in 20 seconds. What is the sprinter’s average speed and average velocity for the entire time? 200 150 100 50 0 4030 2010 d (m) t (s) 200 150 100 50 0 40302010 x (m) t (s) slope = ave. speed slope = ave. velocity

7. Instantaneous Speed and Velocity Instantaneous speed is the how fast an object moves at an exact moment in time. Instantaneous velocity has speed and direction. Instantaneous speed (or velocity) is found graphically from the slope of a tangent line at any point on a distance (or position) vs. time graph. slope of tangent = instantaneous speed d (m) t (s) x (m) t (s) slope of tangent = instantaneous velocity sign of slope = sign of velocity Honors:

8. The Physics of Acceleration “Acceleration is how quickly how fast changes” PAUL HEWITT, CITY COLLEGE, S.F. “how fast” “how fast changes” “how quickly” Acceleration is defined as the rate at which an object’s velocity changes. means velocity means change in velocity mean how much time elapses Acceleration is considered as a rate of a rate. Why? Acceleration has units of meters per second per second, or m/s/s, or m/s 2. Metric (SI) units

9. Types of Acceleration Constant acceleration is the slope of a velocity vs. time graph. (Sound familiar?! Compare to, but DO NOT confuse with constant velocity on a position vs. time graph.) Average acceleration is the slope of a secant line for a velocity vs. time graph. Instantaneous acceleration is the slope of a tangent line for a velocity vs. time graph. (Again, compare to, but DO NOT confuse with average and instantaneous velocity on a position vs. time graph.) Constant Acceleration Velocity vs. Time v (m/s) t (s) Velocity vs. Time v (m/s) t (s) Varying Acceleration slope = acceleration slope = average acceleration slope = instantaneous acceleration

10. Velocity and Displacement (Honors) A velocity graph can be used to determine the displacement (change in position) of an object. The area of the velocity graph equals the object’s displacement. Velocity vs. Time v (m/s) t (s) area = displacement = (.5)(3 s)(30 m/s) + (4 s)(30 m/s) + (.5)(1 s)(30 m/s) = 180 m 30 20 10 0 8642 For a non-linear velocity graph, the area can be determined by adding up infinitely many pieces each of infinitely small area, resulting in a finite total area! This process is now known as integration, and the function is called an integral.

11. An Acceleration Analogy Compare the graph of wage versus time to a velocity versus time graph. The slope of the wage graph is “wage change rate”. Slope of the velocity graph is acceleration. What is the slope for each graph, including units? In this case the “wage change rate” is constant. The graph is linear because the rate at which the wage changes is itself unchanging (constant)! The analogy helps distinguish velocity from acceleration because it is clear that wage and “wage change rate” (acceleration) are different. slope = “wage change rate” = $1//hr/month slope = acceleration = 1 m/s/s

12. An Acceleration Analogy Can a person have a high wage, but a low “wage change rate”? Earnings, Wage, and “Wage Change Rate”Position, Velocity, and Acceleration Can an object have a high velocity, but a low acceleration? Can a person have a positive wage, but a negative “wage change rate”? Can an object have a positive velocity, but a negative acceleration? Can a person have zero wage, but still have “wage change rate”? Can an object have zero velocity, but still have acceleration? Can a person have a low wage, but a high “wage change rate”? Can an object have a low velocity, but a high acceleration? Making good hourly money, but getting very small raises over time. Moving fast, but only getting a little faster over time. Making little per hour, but getting very large raises quickly over time. Moving slowly, but getting a lot faster quickly over time. Making money, but getting cuts in wage over time. Moving forward, but slowing down over time. Making no money (internship?), but eventually working for money. At rest for an instant, but then immediately beginning to move.

13. Direction of Velocity and Acceleration vivi amotion +0 –0 0+ 0– ++ –– +– –+ constant positive vel. constant negative vel. speeding up from rest speeding up slowing down v t v t v t v t v t v t v t v t speeding up click for applet Velocity vs. Time

14. Kinematic Equations of Motion Assuming constant acceleration, several equations can be derived and used to solve motion problems algebraically. Velocity vs. Time (Constant Acceleration) v (m/s) t (s) Slope equals acceleration Area equals displacement Eliminate time Eliminate final velocity vfvf vivi t

15. Freefall Acceleration Aristole wrongly assumed that an object falls at a rate proportional to its weight. Galileo assumed all objects freefall (in a vacuum, no air resistance) at the same rate. An inclined plane reduced the effect of gravity, showing that the displacement of an object is proportional to the square of time. Kinematic equations of freefall acceleration: Since the acceleration is constant, velocity is proportional to time. Locationg Equator9.780 Honolulu9.789 Denver9.796 San Francisco9.800 Munich9.807 Leningrad9.819 North Pole9.832 click for video Latitude, altitude, geology affect g.