Motion In One Dimension

1. Motion In One Dimension PLATO AND ARISTOTLE GALILEO GALILEI LEANING TOWER OF PISA

2. A B C -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 Distance, Position and Displacement Distance (d) Distance is the total length of a path traveled by an object. Distance is always positive, even if an object reverses its direction. Position (x or y) Position is the location of an object relative to an origin. Position can be positive or negative. 1. What is the distance traveled if an object starts at point C, moves to A, then to B? 12 units Displacement (∆x or ∆y) Displacement is the change in position of an object. 2. What are the positions of objects located at points A, B, and C relative to the origin? -5, -2, +4 units 3. What is the displacement of an object that starts at point C and moves to point B? -2 – (+4) = -6 units 4. What is the displacement of an object that starts at point A, then moves to point C and then moves to point B? Displacement can be positive or negative. -2 – (-5) = +3 units

3. Distance and Position Graphs Distance vs. Time Position vs. Time d (m) x (m) positive constant positive velocity t (s) constant speed negative constant negative velocity t (s) Distance graphs show how far an object travels. Speed is determined from the slope of the graph, which can only be positive. Position graphs show how far, and in which direction, an object travels. Velocity (speed with direction) is determined from the slope of the graph. Notice that these graphs show constant speed. (How do you know?)

4. 200 200 150 150 100 100 50 50 0 0 10 20 10 20 30 40 30 40 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 meters in 10 seconds, and then walks slowly back to the starting blocks in 30 seconds. What is the sprinter’s average speed and average velocity for the entire time? x (m) d (m) slope = speed slope = velocity t (s) t (s)

5. Instantaneous Speed and Velocity Instantaneous speed is the speed of an object at an exact moment in time. Instantaneous velocity includes direction too. 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. negative tangent slope = negative velocity x (m) d (m) slope of tangent = speed t (s) t (s)

6. 30 20 10 0 2 4 6 8 Velocity and Displacement (Honors) 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 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. 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.

7. The Physics of Acceleration “Acceleration is how quickly how fast changes” PAUL HEWITT, CITY COLLEGE, S.F. “how fast” means velocity “how fast changes” means change in velocity “how quickly” mean how much time elapses Acceleration is defined as the rate at which an object’s velocity changes. Metric (SI) units Acceleration has units of meters per second per second, or m/s/s, or m/s2. Acceleration is considered as a rate of a rate. Why?

8. Types of Acceleration Velocity vs. Time Velocity vs. Time v (m/s) v (m/s) slope = average acceleration slope = acceleration slope = instantaneous acceleration t (s) t (s) Varying Acceleration Constant Acceleration 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. (Compare to, but DO NOT confuse with average and instantaneous velocity on a position vs. time graph.) Constant acceleration is the slope of the line for a velocity vs. time graph. (Compare to, but DO NOT confuse with constant velocity on a position vs. time graph.)

9. 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

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

11. v v v v v v v v t t t t t t t t Direction of Velocity and Acceleration constant positive vel. constant negative vel. speeding up from rest speeding up from rest speeding up speeding up slowing down slowing down click for applet

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

13. Freefall Acceleration Aristole wrongly assumed that an object falls at a rate proportional to its weight. Galileo proved that 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. click for video Kinematic equations of freefall acceleration: Since the acceleration is constant, velocity is proportional to time. Latitude, altitude, geology affect g.

14. A Velocity Analogy Compare constant velocity (uniform motion) to making money doing a job. Say you baby sit for $10/hour. It’s easy to graph earnings as a function of time. Compare the graph of earnings versus time to position versus time. The slope of the earnings graph is wage. Slope of the position graph is velocity. slope = wage = $10//hr slope = velocity = 10 m/s

15. A Velocity Analogy Compare constant acceleration to getting raises while doing a job. Maybe you baby sit for $10/hour, but now you get regular wages increases. Compare the graph of earnings versus time to position versus time. Slope of a secant line is the average wage (compare with average velocity.) Slope of a tangent line is the instantaneous wage (compare with instant. velocity.)