PHYS 1441 Section 00 Lecture #6 Monday, Feb. 4, 008 Examples for 1-Dim kinematic equations Free Fall Motion in Two Dimensions Maximum ranges and heights Today s homework is homework #3, due 9pm, Monday, Feb. 11!! Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 1
E-mail distribution list Announcements Test message sent out last Thursday 4 of you confirmed the receipt If you did not get the message, you could be missing important information related to class Please make sure that you are on the distribution list Quiz Wednesday, Feb 6, early in the class Coers from CH1 what we learn today Monday, Feb. 4, 008 PHYS 1441-00, Spring 008
Reminder: Special Problems for Extra Credit Derie the quadratic equation for yx -zx+0 5 points Derie the kinematic equation + from first principles and the known kinematic equations 10 points You must show your work in detail to obtain the full credit Due next Wednesday, Feb. 6 a x x ( ) 0 0 Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 3
Kinematic Equations of Motion on a Straight Line Under Constant Acceleration t + at () 0 1 1 x x ( ) 0 t + 0 t 1 x x0 + 0t+ at + a x x Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 ( ) 0 0 Velocity as a function of time Displacement as a function of elocities and time Displacement as a function of time, elocity, and acceleration Velocity as a function of Displacement and acceleration You may use different forms of Kinematic equations, depending on the information gien to you in specific physical problems!! 4
How do we sole a problem using a kinematic formula under constant acceleration? Identify what information is gien in the problem. Initial and final elocity? Acceleration? Distance, initial position or final position? Time? Conert the units of all quantities to SI units to be consistent. Identify what the problem wants Identify which kinematic formula is appropriate and easiest to sole for what the problem wants. Frequently multiple formulae can gie you the answer for the quantity you are looking for. Do not just use any formula but use the one that can be easiest to sole. Sole the equations for the quantity or quantities wanted. Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 5
Example 8 An Accelerating Spacecraft A spacecraft is traeling with a elocity of +350 m/s. Suddenly the retrorockets are fired, and the spacecraft begins to slow down with an acceleration whose magnitude is 10.0 m/s. What is the elocity of the spacecraft when the displacement of the craft is +15 km, relatie to the point where the retrorockets began firing? x a o t +15000 m -10.0 m/s? +350 m/s Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 6
x a o t +15000 m -10.0 m/s? +350 m/s o + ax Sole for o ± + ax ± 350 m s + 10.0m s 15000 m ( ) ( )( ) ± 500m s What do two opposite signs mean? Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 7
Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 8
xf xf Example Suppose you want to design an air-bag system that can protect the drier in a headon collision at a speed 100km/hr (~60miles/hr). Estimate how fast the air-bag must inflate to effectiely protect the drier. Assume the car crumples upon impact oer a distance of about 1m. How does the use of a seat belt help the drier? How long does it take for the car to come to a full stop? The initial speed of the car is We also know that Using the kinematic formula The acceleration is a x 0 m / Thus the time for air-bag to deploy is xi and t Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 s xi xf ( xf xi) x f xi As long as it takes for it to crumple. 100 km / h 100000 m 8 m/ s 3600s 1m xi + ax xf xi ( ) 0 8 m/ s 1m xf a ( ) xi 390 m/ s 0 8 m/ s 390 m/ s 0.07s 9
Free Fall Free fall is a motion under the influence of graitational pull (graity) only; Which direction is a freely falling object moing? A motion under constant acceleration All kinematic formulae we learned can be used to sole for falling motions. Graitational acceleration is inersely proportional to the distance between the object and the center of the earth The magnitude of the graitational acceleration is g 9.80m/s on the surface of the Earth, most of the time. The direction of graitational acceleration is ALWAYS toward the center of the earth, which we normally call (-y); when the ertical directions are indicated with the ariable y Thus the correct denotation of the graitational acceleration on the surface of the earth is g - 9.80m/s g 3.ft s Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 Down to the center of the Earth! Note the negatie sign!! 10
Free Falling Bodies Down to the center of the Earth g 9.80m s Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 11
Example 10 A stone is dropped from the top of a tall building. After 3.00s of free fall, what is the displacement y of the stone? Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 1
y a o t? -9.8 m/s 0 m/s 3.00 s Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 13
y a o t? -9.80 m/s 0 m/s 3.00 s Which formula would you like to use? y t o + 1 at t+ o 1 gt ( 0m s)( 3.00 s) 1 ( 9.80m s )( 3.00 s) + 44.1 m Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 14
Example 1 How high does it go? The referee tosses the coin up with an initial speed of 5.00m/s. In the absence if air resistance, how high does the coin go aboe its point of release? Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 15
y a o t? -9.8 m/s 0 m/s + 5.00 m/s Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 16
y a o t? -9.80 m/s 0 m/s +5.00 m/s o + ay Sole for y y o a y o a 0m s 5.00m s ( ) ( ) ( ) 9.80m s 1.8 m Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 17
Conceptual Example 14 Acceleration Versus Velocity There are three parts to the motion of the coin. On the way up, the coin has a ector elocity that is directed upward and has decreasing magnitude. At the top of its path, the coin momentarily has zero elocity. On the way down, the coin has downward-pointing elocity with an increasing magnitude. In the absence of air resistance, does the acceleration of the coin, like the elocity, change from one part to another? Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 18
Example for Using 1D Kinematic Equations on a Falling object Stone was thrown straight upward at t0 with +0.0m/s initial elocity on the roof of a 50.0m high building, What is the acceleration in this motion? f g-9.80m/s (a) Find the time the stone reaches at the maximum height. What is so special about the maximum height? V0 yi ayt + + 0.0 9.80t 0.00 m/ s Sole for t (b) Find the maximum height. t 0.0 9.80.04s 1 1 y f yi+ yit+ ayt 50.0 + 0.04 + ( 9.80) (.04) 50.0 + 0.4 70.4( m) Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 19
Example of a Falling Object cnt d (c) Find the time the stone reaches back to its original height. yf t.04 4.08s (d) Find the elocity of the stone when it reaches its original height. yi + 0.0( m/ s) + ayt 0.0 ( 9.80) 4.08 (e) Find the elocity and position of the stone at t5.00s. Velocity Position yf y f yi + 1 y i + yit + ayt m s + ayt 0.0 ( 9.80) 5.00 9.0( / ) 1 50.0 + 0.0 5.00 + ( 9.80) (5.00) + 7.5( m ) Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 0
Conceptual Example 15 Taking Adantage of Symmetry Does the pellet in part b strike the ground beneath the cliff with a smaller, greater, or the same speed as the pellet in part a? Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 The same!! 1
Dimensional Kinematic Quantities r o initial position r final position Δ r r r o displacement Monday, Feb. 4, 008 PHYS 1441-00, Spring 008
Aerage elocity is the displacement diided by the elapsed time. D Aerage Velocity r r o Δr t t o Δt Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 3
The instantaneous elocity indicates how fast the car moes and the direction of motion at each instant of time. lim Δ t 0 Δr Δt Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 4
D Instantaneous Velocity lim Δ t 0 Δr Δt Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 5
D Aerage Acceleration Δ o a o Δ t t o Δt Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 6
Displacement, Velocity, and Acceleration in -dim Displacement: Aerage Velocity: Instantaneous Velocity: Aerage Acceleration Instantaneous Acceleration: Δ r a a r f r i Δ r r f Δ t t f Δ r lim Δ t 0 Δ t Δ f Δ t t lim Δ t 0 Δ Δt f r t i t i i i How is each of these quantities defined in 1-D? Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 7
Kinematic Quantities in 1D and D Quantities Displacement Aerage Velocity Inst. Velocity Aerage Acc. Inst. Acc. a 1 Dimension Δ x x x Dimension x Δ r r f ri Δx x f xi Δ r r f r i Δt t t Δ t t t x Δx lim Δt 0 Δt Δ x xf Δt t t a x x lim f Δ t 0 f f Δ x Δt i i i xi a f i Δ r lim Δt 0 Δt Δ f Δ t t f ti Δ a lim Δ t 0 Δt i What is the difference between 1D and D quantities? Monday, Feb. 4, 008 PHYS 1441-00, Spring 008 8