General Physics (PHY 2130)
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1 General Physics (PHY 130) Lecture 4 Motion in one dimension Position and displacement Velocity average instantaneous Acceleration motion with constant acceleration
2 Department of Physics and Astronomy announces the Winter 013 opening of The Physics Resource Center on Monday, January 14 in Room 17 of Physics Research Building. Undergraduate students taking PHY130 will be able to get assistance in this Center with their homework, labwork and other issues related to their physics course. The Center will be open from Monday, January 14until the end of the semester.
3 Lightning Review Last lecture: 1. Motion in one dimension: displacement: depends only on x f x i average velocity: displacement over time interval instantaneous velocity: same as above for a very small time interval average acceleration: velocity change over time interval instantaneous acceleration: same as above for a very small time interval
4 Review: Displacement vs path
5 Average Acceleration Changing velocity (non-uniform) means an acceleration is present Average acceleration is the rate of change of the velocity a average Δv Δt v f v Δt i Average acceleration is a vector quantity (i.e. described by both magnitude and direction)
6 Average Acceleration When the sign of the velocity and the acceleration are the same (either positive or negative), then the speed is increasing When the sign of the velocity and the acceleration are opposite, the speed is decreasing Units SI Meters per second squared (m/s ) CGS Centimeters per second squared (cm/s ) US Customary Feet per second squared (ft/s )
7 Instantaneous and Uniform Acceleration Instantaneous acceleration is the limit of the average acceleration as the time interval goes to zero a inst lim Δt 0 Δ v Δt lim Δt 0 v f v i Δt When the instantaneous accelerations are always the same, the acceleration will be uniform The instantaneous accelerations will all be equal to the average acceleration
8 Graphical Interpretation of Acceleration Average acceleration is the slope of the line connecting the initial and final velocities on a velocity-time graph Instantaneous acceleration is the slope of the tangent to the curve of the velocitytime graph
9 Example 1: Motion Diagrams Uniform velocity (shown by red arrows maintaining the same size) Acceleration equals zero
10 Example : Velocity and acceleration are in the same direction Acceleration is uniform (blue arrows maintain the same length) Velocity is increasing (red arrows are getting longer)
11 Example 3: Acceleration and velocity are in opposite directions Acceleration is uniform (blue arrows maintain the same length) Velocity is decreasing (red arrows are getting shorter)
12 NEAT OBSERVATION: the area under a velocity versus time graph (between the curve and the time axis) gives the displacement in a given interval of time. Why is that? For the constant velocity: v x (m/s) v x Δx Δt Δx v x Δt Divide the graph into small rectangular shapes Area S i (v x ) i Δt i Δx i Δx total i i i t (sec) This whole area determines displacement! 1
13 Example: Speedometer readings are obtained and graphed as a car comes to a stop along a straight-line path. How far does the car move between t 0 and t 16 seconds? Solution: since there is not a reversal of direction, the area between the curve and the time axis will represent the distance traveled. The rectangular portion has an area of Lw (0 m/s)(4 s) 80 m. The triangular portion has an area of ½bh ½(8 s) (0 m/s) 80 m. Thus, the total area is 160 m. This is the distance traveled by the car. 13
14 One-dimensional Motion With Constant Acceleration a a If acceleration is uniform (i.e. ): a v t f f v t 0 o v f t v o thus: v v + f o at Shows velocity as a function of acceleration and time
15 One-dimensional Motion With Constant Acceleration Used in situations with uniform acceleration Δx v average t vo + v 1 Velocity changes Δ x vot + at v v + aδx f o f t v f vo + uniformly!!! at
16 Notes on the equations vo + vf Δ x vaverage t t Gives displacement as a function of velocity and time 1 Δ x vot + at Gives displacement as a function of time, velocity and acceleration v v + aδx f o Gives velocity as a function of acceleration and displacement
17 Example: The graph shows speedometer readings as a car comes to a stop. What is the magnitude of the acceleration at t 7.0 s? The slope of the graph at t 7.0 sec is a av Δv Δt x v t v t 1 1 ( 0 0) ( 1 4) m/s s.5 m/s 17
18 Summary of kinematic equations
19 Free Fall All objects moving under the influence of only gravity are said to be in free fall All objects falling near the earth s surface fall with a constant acceleration This acceleration is called the acceleration due to gravity, and indicated by g
20 Acceleration due to Gravity Symbolized by g g 9.8 m/s² (can use g 10 m/s² for estimates) g is always directed downward toward the center of the earth
21 Free Fall -- an Object Dropped Initial velocity is zero Frame: let up be positive Use the kinematic equations Generally use y instead of x since vertical y v o 0 a g x 1 Δy at a 9.8m s
22 Free Fall -- an Object Thrown Downward a g With upward being positive, acceleration will be negative, g -9.8 m/s² Initial velocity 0 With upward being positive, initial velocity will be negative
23 Free Fall -- object thrown upward Initial velocity is upward, so positive The instantaneous velocity at the maximum height is zero a g everywhere in the motion g is always downward, negative v 0
24 Example: A penny is dropped from the observation deck of the Empire State Building 369 m above the ground. With what velocity does it strike the ground? Ignore air resistance. Given: v iy 0 m/s, Solution: y a y -9.8 m/s, Δy -369 m Find: V yf? Use: v v fy yf v iy a + a y a Δy y y Δy Δy 369 m a y x v yf a Δy y ( 9.8 m/s )( 369) m 85.0 m/s 4
25 5 How long does it take for the penny to strike the ground? 8.7 sec 1 1 Δ Δ Δ Δ + Δ Δ y y y iy a y t t a t a t v y Example continued: Given: v iy 0 m/s, a y -9.8 m/s, Δy -369 m Unknown: Δt
26 Thrown upward The motion may be symmetrical then t up t down then v f -v o The motion may not be symmetrical Break the motion into various parts generally up and down
27 Fun QuickLab: Reaction time d 1 g t, g d t g 9.8m s
28
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