Motion in One Dimension Mechanics: macroscopic objects Kinematics: describing motion

1. Motion in One Dimension • Mechanics: macroscopic objects • Kinematics: describing motion • Dynamics: studying the causes of motion • Describing Motion • Reference frame: • Definition: [n] a system that uses coordinates to establish position • Synonyms: coordinate system, frame of reference, reference system * • often refers to a physical object • Coordinate System: • A system of assigning numbers to locate a point within a reference frame • Consider a ball thrown straight up • from the view point of the thrower • from the view point of a passenger in a passing automobile * http://www.hyperdictionary.com/dictionary/reference+frame

2. Describing Motion • Displacement: the change in the position of an object • (includes sign, which indicates direction!) • Dx = xf – xi • Example 2.2: You walk along a straight sidewalk for 45m, then turn around and walk 25 m in the opposite direction. You then walk 37 m in the original direction and stop. What is your displacement from your starting point? (hint: combine segments)

3. average speed: the ratio total distance s traveled to the time interval t required for the travel • example: Fall Foliage Road Trip! Two students take a three hour trip to enjoy the fall foliage. In the first two hours they travel 100 km at a constant speed. In the third hour they travel another 80 km. What was their average speed for each hour of the trip? What was their average speed for the entire trip?

4. velocity: rate of change of displacement • indicates speed and direction of travel • average velocity: the ratio of the displacement during a time interval to the length if the time interval • example: What is the average speed and average velocity of a helicopter • if it takes of from the hospital and travels 150 km due east in one hour? • if it takes of from the hospital and travels 150 km due west in one hour? • if it takes of from a position 20 km east of the hospital and travels 50 km due east, then turns around and travels to a spot 80 km west of the hospital, all in one hour?

5. displacement Dx Dt time • Graphical Interpretation of Velocity • Velocity is the slope of the graph of position versus time • may be positive or negative (direction!) • steeper = faster • math animation • http://phys23p.sl.psu.edu/~mrg3/mathanim/tangent.html • http://phys23p.sl.psu.edu/~mrg3/mathanim/tangents.html • motion sensor exercise • Instantaneous Velocity • average velocity equals instantaneous velocity if velocity is constant

6. instantaneous velocity = slope of displacement versus-time graph displacement = area under the curve of velocity-time graph s = s1+s2 = +v1Dt1+v2Dt2 s = vt v s s t t v v v1 v2 t Dt2 ... Dt1 Dt2 t t Dt1 Dt3

7. acceleration: rate of change of velocity • changes in speed as well as direction of travel • average velocity: the ratio of the displacement during a time interval to the length if the time interval • example: A bicyclist starts from rest and increases his velocity at a constant rate until he reaches a speed of 4.0 m/s in 5.0 s. What is his average acceleration?

8. velocity Dv Dt time • Graphical Interpretation of acceleration • Acceleration is the slope of the graph of velocity versus time • may be positive or negative (direction!) • Instantaneous Acceleration • average acceleration equals instantaneous acceleration if acceleration is constant • instantaneous acceleration = slope of velocity versus-time graph • velocity = area under the curve of acceleration-time graph!

9. Motion with Constant Acceleration • constant acceleration slope of velocity-time graph is straight line • displacement is area under velocity-time curve • w/ substitution and some algebra • The “Fab Four” v = v0+at v v0 t

10. Problem Solving Tips • Draw a diagram • Write down all known quantities, identify on diagram. • Write down all unknown quantities, identify on diagram. • Find basic equation which relates each unknown quantity to known quantities. • Solve equation for the desired unknown. • “plug and chug”

11. example: A Boeing 777 airliner, initially at rest, undergoes a constant acceleration of 2.3 m/s2 down the runway for 34 s before it lifts off. How far down the runway does it travel? How fast is it going at lift off? • example: Suppose a child on a go-cart is traveling at 4.0 m/s when she crosses a line 4.0 m from her starting point. She then accelerates at a constant rate of 0.40m/s2 until she crosses a line 40. m from the starting point. How long does it take her to go from the 4.0 m mark to the 40 m mark?

12. example: You are driving your new sports car at a speed of 90 km/h when you suddenly see a dog step into the road 50 m ahead. You hit the brakes hard to get the maximum deceleration of 7.5 m/s2 (i.e. a = - 7.5 m/s2). How far will you go before stopping?

13. Acceleration of Gravity • Historical Perspective: • Attempts to describe motion by Galileo • Experiment and observation as the foundation of science • ramps and freefall • idealizations and simplifications: ignoring air resistance • g = 9.81 m/s2 (9.78 at equator, 9.83 at poles) • = 32 ft/s2 • a = g, acceleration is directed downwards(watch directions in problems!!!)

14. example: On a Free Fall ride at an amusement park, riders are seated in a padded gondola and are taken to the top of a 10-story tower. The gondola is dropped 30 m down a vertical track (which curves at the bottom after which the gondola is slowed to a stop). • How long does it take to fall the 30 m? • What is the speed of the gondola at the bottom?