L02_Kinematics1 - Lecture 2 Describing Motion: Kinematics...

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Unformatted text preview: Lecture 2 Describing Motion: Kinematics in One Dimension Kinematics : describing how things move Reference Frame: We always need a reference point. That point is the origin of our coordinate system y O x 1. Position: [m] Distance: The length of the total path traveled (m) Displacement: Difference between final and initial position (m) Example: Distance vs. Displacement You live 20 mi away from the Al. campus. Today you have only one class. You come to school, take it, and then go home. What is the distance you traveled? What is your displacement? Distance: mi you are at home after school you are at home before school; Displacement: mi 2. Speed and Velocity m/s] (measurement of how quickly the distance changes) Average Speed: Average Velocity: Example: Consider the previous example. If it took you 2 hours overall to come to school and go back home, how much is your average speed? How much the average velocity? 1 mi hr mi hr mi/hr mi/hr Instantaneous Velocity: lim Note: When the time interval is very small, the distance and displacement are the same and the instantaneous velocity and speed are the same. Question: (Q3) When an object moves with constant velocity, does its average velocity differ from its instantaneous velocity at any instance? Ans.: No. Plots: Velocity as a function of time Constant Velocity 50 40 Velocity (km/h) 30 20 10 0 Velocity (km/h) 0 0.1 0.2 0.3 0.4 0.5 Time (h) Varying velocity 60 50 Velocity (km/h) 40 30 20 10 0 0 0.1 0.2 Time (h) 0.4 0.5 Velocity (km/h) Aver. velocity Question: Q1 Does a car speedometer measure speed, velocity, or both? Ans.: speed only. 2 3. Acceleration [m/s ] (measurement of how quickly the velocity changes) average acceleration m/s Instantaneous acceleration: lim Question: (Q3) Can an object have a northward velocity and a southward acceleration? Ans: yes. If the car is moving northward and it is slowing down, the acceleration is pointing southward (opposite to the direction of motion). ------------------------------------------------------------------------------Example: P.16 A sports car accelerates from rest to 95 km/h in 6.2 s. What is its average acceleration in m/s ? Solution: 1. Convert the velocity in regular units (m/s) 95 km ! " 1000 m ! " %# m/s #$ m/s % h 3600 s 2. Calculate the average acceleration m/s 26.39 m/s #$ s & #$ % $ m/s $ !# m/s 6.2 s Question: (a) If the acceleration is zero, is the velocity zero, as well? Ans.:No (b) If the velocity is zero, is the acceleration zero, as well? Ans.: No ------------------------------------------------------------------------------Example: P.19 A sports car moving at constant speed travels 110 m in 5.0 s. If it then brakes and comes to a stop in 4.0s, what is its acceleration in m/s ? Express the answer in terms of "g's", where 1.00g $ m/s ' Solution: m/s !$ s m/s $ $s m/s m/s 3 9.8 m/s 5.5 m/s 2 -9.8 m/ s 1g ?g 1g 2 -5.5 m/ s xg $ ' !$ ! & !$ ! $ ' $ "g's" !# Deceleration: When an object is slowing down. It is the same as "negative acceleration". 4. Motion at Constant Acceleration Acceleration: ( $ $( % ) s -" Velocity: Average velocity: & & * +, , . / ( $ !$ ) *+ -" & & " 4 Position: -" & & & -" " Velocity-acceleration-distance relation: ------------------------------------------------------------------------------Example: p.22 A car slows down from 23 m/s to rest in a distance of 85 m. What was its acceleration, assumed constant? Solution: Given: % m/s m/s '! Find: ? Solution: Which formula relates directly: ) ) ? m/s % m/s " '! m % ! % '! & m/s '! 0 ------------------------------------------------------------------------------Example: p.24 A world-class sprinter can burst out of the blocks to essentially top speed (of about 11.5 m/s) in the first 15.0 m of the race. What is the average acceleration of this sprinter, and how long does it take her to reach that speed? Solution: Given: m/s $ m/s ! !$ m Find: ? ? Solution: 5 acceleration: $ m/s ! $ m/s $ ! %$ & " $ ! %$ !$ m m/s time interval: we know initial velocity, final velocity, and acceleration. Which formula relates these? $ m/s ! $ ! & $ m/s $ s 6 Formulae from the previous lecture: lim lim ---------------------------------------------* Formulae in red must be memorized! Example: Q15: You travel from point A to point B in a car moving at a constant speed of 70 km/h. Then you travel the same distance from point B to another point C, moving at a constant speed of 90 km/h. Is your average speed for the entire trip from A to C 80 km/h? Explain why or why not? Answer: 1 A B: av. speed 02 3& h 0 B A: av. speed 23& ( 4 ( average speed " 0 0 "0 " 0 $ % $ km/h 5. Falling Objects Galileo experiments: Bodies fall in the same time no matter their mass (no resistance) Conclusion: at a given location on the Earth and in the absence of air resistance, all objects fall with the same constant acceleration. Gravitational acceleration: $ m/s ' ------------------------------------------------------------------------------Q17: Which one of these motions is not at constant acceleration: a rock falling from a cliff, and elevator moving from the second floor to the fifth floor making stops along the way, a dish resting on a table? ------------------------------------------------------------------------------Example P.34 7 If a car rolls gently ( km/h? m/s) off a vertical cliff, how long does it take it to reach 85 Given: Direction "up" considered positive! $ m/s ' m/s '! km/h '! " %# m s Want: ? Assuming %$ # $ ' & & %$ # $ ' %$ m/s # s ------------------------------------------------------------------------------Example: A person throws a ball upward the air with an initial velocity of 15.0 m/s. Calculate (a) how high it goes (b) how long the ball is in the air before it comes back (c) how much time it takes for the ball to reach the maximum height (d) the velocity of the ball when it returns to the thrower's hand (e) at what time t the ball passes a point 8.00 m above the person's hand Given: $ m/s ' const. !$ m/s m/s m (a) at the top of the trajectory: !$ m/s !$ m/s $ m/s ' $ m/s ' $!m 5 56 6 Note: check for units: 5 6 56 (b) in the beginning and in the end, the position is the same: ? for the entire flight? !$ $ ' $ ' !$ & $ s, s %$ # s (c) at the top, 8 ? for the flight to the top? $ !$ $ ' & $s !% Note: The time to the top is half the total time! (d) velocity when it returns? !$ $ %$ # ' !$ m/s Conclusion: If there is no resistance: 1. The velocity with which the balls is thrown has the same magnitude as the velocity when it returns, only the direction is opposite 2. The time of the total flight is twice as long as the time to climb to highest point ------------------------------------------------------------------------------Q18: An object that is thrown vertically upward will return to its original position with the same speed as it had initially if air resistance is negligible. If air resistance is appreciable, will this result be altered, and if so, how? ------------------------------------------------------------------------------Note: Misconceptions: 1. The acceleration and the velocity are always in the same direction!!! 2. When the velocity is zero, the acceleration is also zero!!! WRONG! WRONG! WRONG! Correct: 1. The acceleration and the velocity can be in different directions 2. The acceleration can be non-zero even if the velocity is zero. If an object is not moving it means its velocity is zero. In order to determine its acceleration - we must wait some time. If the object is still not moving - the acceleration is zero. BUT if the object is moving after some time interval, it means there was some acceleration Remember: the acceleration is a measurement of how the velocity changes! ------------------------------------------------------------------------------Q19: Can an object have zero velocity and nonzero acceleration at the same time? Give examples. ------------------------------------------------------------------------------Example: P.39 9 A helicopter is ascending vertically with a speed of 5.20 m/s. At a height of 125 m above the Earth, a package is dropped from a window. How much time does it take for the package to reach the ground? Given: Assuming "up" is positive. s !$ m/s !m $ m/s ' m Find: t ? or in other words: Solution: y y $ ) ? !m !$ !$ 7 !$ !$ m/s ! $ ! $ m/s ' !$ 7 $ ' $ 00 $ $s ! !$ s # Answer: s ------------------------------------------------------------------------------Graphical Analysis of Linear Motion 1. Velocity is the slope of vs. graph 2. Acceleration is the slope of vs. graph 3. Displacement is the area under vs. graph. ------------------------------------------------------------------------------Example: Displacement using vs. graph. A space probe accelerates uniformly from 50 m/s at t=0 s to 150 m/s at t=10 s. How far did it move between t=2.0 s and t=6.0 s? Given: s ! m/s s 10 $s Find: Solution: ! m/s #$ s $s ? Velocity vs. Time 200 150 Velocity (m/s) 100 50 0 Data A 0 2 4 Time (h) 6 8 10 The acceleration is the slope of the graph. The slope does not change & ! ! m/s $ #$ 0 m/s m/s & $ #$ ! ! Displacement is the area under the graph. TRAPEZOID AREA average of height width 0 & " #$ $ %# m ------------------------------------------------------------------------------- 11 ...
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