Chapter 2: 1-D Kinematics

Transcription

1 Chapter : 1-D Kinematics Brent Royuk Phys-111 Concordia University Displacement Levels of Formalism The Cartesian axis One dimension: the number line Mathematical definition of displacement: Δx = x f x o delta means change What s the difference between distance and displacement? Example: If you travel from x i = -5 m to x f = -1 m what is your displacement? Vectors and Scalars Vectors and Scalars Vectors are numbers with directions. Often symbolized by arrows Scalars have magnitudes but no direction. Chapter motion is all onedimensional. Most important: +/- refer to direction: left/right or up/down usually 3 4 Average Speed and Velocity Speed: how fast an object s position is changing To measure speed, measure distance and time distance average speed = elapsed time If it s not average speed, what kind of speed is it? MKS unit: m/s How does a m/s compare with a mph? Average Speed and Velocity Vector speed: velocity (more later) average velocity = displacement total travel time A note on notation: Relax. When speed/velocity is uniform, problems are doable. v = x - x o t t o v av = Δx Δt = x f x o t f t o, x = x o + vt and x = vt 5 6 1

2 Example Motion Graphs Take a canoe trip down-river miles at mph. Then return upriver miles at 1 mph. What is your average speed? What is your average velocity? The abscissa and the ordinate Uniform v, positive and negative The slope triangle 7 8 Motion Graphs Instantaneous Velocity The abscissa and the ordinate Uniform v, positive and negative The slope triangle 9 10 Examples A fast sprinter can run the 100-m dash in 10.0 s, and a good marathoner can run the marathon (6mi, 385 yd) in hr 10. a) what are their average speeds? Calculate to three sig-figs. b) if the sprinter could maintain sprint speed during the marathon, how long would it take? A bike travels around a circular track of diameter m at a constant speed in min, 10 sec (130.0 s). What is the speed of the bike? Is the velocity constant? 11 Examples A jet is flying 35m above level ground at a speed of 1300 km/h. Suddenly the plane encounters terrain that gently slopes upward at 4.3 o, an amount difficult to detect. How much time does the pilot have to make a correction before flying into the ground? Two trains, each having a speed of 30 km/h, are headed toward each other on the same track. A bird that can fly 60 km/h flies off the front of one train when they are 60 km apart and flies back and forth between the trains until they crash. How far does the bird fly? Assignment: Homework A 13

3 Ranking Task Ranking Task Acceleration a av = Δv Δt = v v f o t f t o The MKS acceleration unit Everyday accelerations What is deceleration? Change in direction but not magnitude of velocity is still an acceleration e.g. car going around curve at constant speed Jerk, snap, crackle & pop Graphing Acceleration V-t graphs What is the slope? The area under the curve Average vs. instantaneous When acceleration is held constant, we can write a = v v o t and v = v o + at Compare with y = mx + b How can you figure out how far an object travels while accelerating? v av = 1 ( v o + v), so x = x o + 1 ( v o + v)t In practice, we can usually let x o =0 without losing generality, so ( ) x = v o + v t

4 A car accelerates from 1 m/s to 18 m/s over a time of 6.0 s. How far did it travel during this time? A particle had a velocity of 18 m/s and.4 s later its velocity was 30 m/s in the opposite direction. What was its average acceleration during this time? How far did it travel? Assume the brakes on your car are capable of creating a deceleration of 5. m/s. If you are going 85 mph and suddenly see a state trooper, how long will it take to decelerate to 55 mph? How far do you travel during this time? 1 v + vo Start with x = t and v = v o + at Eliminate v to get 1 x = vo t + at the t dependence Note See it on an x-t graph Eliminate t to get v = vo + ax Problem-Solving Basics The Complete Equation List 1. x = vt x= (v o. +v )t v = v o + at x = vo t at v = vo + ax 6. Examine the situation to determine which physical principles are involved. Make a list of what is given or can be inferred from the problem as stated (identify the knowns). Identify exactly what needs to be determined in the problem (identify the unknowns). Find an equation or set of equations that can help you solve the problem. Substitute the knowns along with their units into the appropriate equation and obtain numerical solutions complete with units. Check the answer to see if it is reasonable. Does it make sense? 3 4 Examples If my motorcycle could maintain its 0-60 acceleration for 15 seconds, how far would it have traveled? How fast would it be going? If you brake your Porsche from 5 m/s to 4 m/s over a distance of 10 m, what was your acceleration? The driver of a pickup truck going 1.00x10 km/h applies the brakes, giving the truck a uniform deceleration of 6.50 m/s until it slows to 5.00 x 101 km/h. a) How far did the truck travel while it was braking? b) How much time has elapsed? A subway train accelerates from rest at one station at a rate of 1. m/s for half the distance to the next station, then decelerates at this same rate for the final half. If the stations are 1100m apart, find a) the time of travel between stations and b) the maximum speed of the train. Assignment: Homework B 5 Galileo and his famous experiment 6 4

5 Falling objects accelerate at the same rate, g = 9.8 m/s The acceleration is uniform This is not obvious; Aristotle thought falling objects acquired some characteristic falling speed. He was thinking of terminal velocity, perhaps? Galileo: a rock falling m drives a stake much more than a rock falling 10 cm. Linear accelerations (or any type of acceleration) can be expressed in g s: fighter pilots, etc. 7 g is altitude and latitude-dependent 8 g is altitude and latitude-dependent g is depth-dependent 9 30 ONLY IN VACUUM Astronaut David Scott, 1971: hammer and feather video Painting by astronaut Alan Bean g on the Moon Apollo

6 Feather Drop Ranking Task 36 Falling Problems Equations a = -g can be written explicitly into kinematics equations This assumes + is up also replace x with y Examples Make a table with columns for t,y,v,a. Fill in values for t = 0,1,,3,4 A wrench is dropped down a 45 m elevator shaft. How fast does it hit? How long does it take to hit? A cliff-diver steps off a cliff and falls 3.1s before hitting water. How high is the cliff? How fast is the diver falling when he hits? 38 Falling Problems A stone is thrown downward from a bridge with a speed of 14.7m/s. If the stone hits the water in.00 s, what is the height of the bridge? What if it was 14.7m/s up? What does this answer mean? A pitcher throws a baseball straight up, with v o = 5 m/ s. a) How long does it take to reach its highest point? b) How high does the ball rise above its release point? c) How long will it take for the ball to reach a point 5m above its release point? A rock is dropped from a 100m cliff. How long does it take to fall the first 50m and the last 50m? Apply to hang time. Reaction time and the dollar trick. Assignment: Homework C 39 6