Physics 123 Lecture 2 1 Dimensional Motion

Transcription

1 Reiew: Physics 13 Lecture 1 Dimensinal Mtin Displacement: Dx = x - x 1 (If Dx < 0, the displacement ectr pints t the left.) Aerage elcity: (Nt the same as aerage speed) a x t x t 1 1 Dx Dt slpe = a x 1 x x-t diagram: x = x(t) (t ) (t ) > (t 1 ) (t) Instantaneus elcity: lim Dt0 Dx Dt dx dt (t) is slpe f tangent t x-t plt at time t.

2 Crrectin: Calculus is a c-requisite fr this curse, nt a prerequisite. If yu hae nt taken calculus befre, yu will learn abut deriaties this semester in Analytical Physics and in Calculus. Quiz in recitatin next week: Based n last week s hmewrk. Understand hw t sle HW prblems. Be able t sle prblems fairly quickly (there are nly 10 minutes fr the quiz).

3 i-clicker Questin Hw many beans are in the 900 ml beaker? A. Fewer than 1000 B C D E. Mre than 500 Shw estimatin game!

4 is nt cnstant in time = (t) acceleratin acceleratin: time rate f change f elcity aerage acceleratin: aa instantaneus acceleratin: (slpe f line tangent t (t) at time t ) a t t 1 1 lim Dt0 D Dt D Dt d dt a d dt d x dt a is slpe f (t) graph. a is curature f x(t) graph. In -t plt abe, (t) is a straight line cnstant acceleratin i.e.: (t) = [cnst]t d/dt = [cnst] Nt always true! Suppse: (t) = Ct 3 Then: a = a(t) = d/dt = 3Ct cnstant in time!

5 Finding acceleratin n a -t graph The (t) graph may be used t find the instantaneus acceleratin and the aerage acceleratin. Cpyright 01 Pearsn Educatin Inc.

6 Acceleratin frm x-t plt: > 0 > 0 (> A ) > 0 (> B ) > 0 (< C ) = 0 < 0 < 0 > 0 > 0 = 0 < 0 < 0 = 0 > 0 Slpe f -t plt gies instantaneus acceleratin

7 i-clicker Questin Shw mtin sensr!

8 Cnstant acceleratin is an imprtant special case! Deseres special attentin!! differentiate a * (t-t a ) (t) = (t a ) + a * (t-t a ) x (t a ) Let t a = 0 (t) = + at ½at t differentiate x x(t) = x + t + ½at

9 KINEMATIC EQUATIONS in 1D 1 (t) = + at x(t) = x + t + ½ at cnstant acceleratin Other helpful relatinships: 3 4 a xx ; a t cnst. acc. nly ALGEBRA: 3 x x = a t ax subst. frm 4 ( x x ) t 5 rewrite 1 t = ( - ) / a 6 plug 6 int 5 ( x x ) a ( x x ) a

10 Lets put these equatins t wrk! Drag race: Cnstant acceleratin alng 400 m track. = 150 m/s at end. What is the acceleratin? Knwn: (x x ) = 400 m; = 150 m/s; = 0 Need: a =? ( x x ) a (150 m/s) a (400 m) 8 m/s Hw lng des the race take? Knwn: (x x ),,, and a Need: t =? x(t) = x + t + ½at a ( x x ) 0 0 x - x = t + ½at 400 m = ½ (8 m/s ) t t = 5.3 s

11 Yellw Light Driing at 30 m/s Light turns yellw when yu are 30 m frm int. Decelerate at 10 m/s. Will yu stp befre intersectin? Knwn: = 30 m/s; a = -10 m/s ; f = 0 m/s; Need: (x f - x ) =? N! f 0 (30 m/s) ( x f x ) a ( 10 m/s ) 45 m What shuld a be? Knwn: (x f - x ) = 30 m; = 30 m/s; f = 0 m/s Need: a =? a f ( x f x ) 0 (30 m/s) (30 m) 15 m/s If a = -30 m/s, where will I stp? (x f - x ) ~ 1/a s (x f - x ) = 15 m

12 iclicker A mtrcycle traeling alng the x-axis is accelerating at a rate f a = -4m/s. a. The mtrcycle is speeding up. b. The mtrcycle is slwing dwn. c. The mtrcycle is neither speeding up nr slwing dwn. d. The mtrcycle is bth speeding up and slwing dwn. e. The mtrcycle may be slwing dwn r speeding up. a a Slwing dwn Speeding up

13 Freely falling bdies Free fall is the mtin f an bject under the influence f nly graity. In the figure, a strbe light flashes with equal time interals between flashes. The elcity change is the same in each time interal, s the acceleratin is cnstant. Cpyright 01 Pearsn Educatin Inc.

14 FREE FALL Mtin in 1-D under the influence f graity. acceleratin due t graity is cnstant (at Earth s surface) a = -g where g = 9.80 m/s graity acts ertically dwnward (chse y-axis as ertical) Same equatins f mtin BUT: a is replaced with g! (t) = - gt y(t) = y + t - ½gt ( y y ) g

15 EXAMPLE: REACTION TIME (red rulers) Knwn: y = 0 m; = 0 m/s ; a = -g ; y f = m Need: t =?? 0 0 y = y + t - ½gt y f = - ½gt t y g ( 0.10 m) 9.8 m/s t 0.0 s 0.14 s

16 EXAMPLE: Drp a penny frm tp f the Empire State Building! (DO NOT TRY THIS!) Obsere: The penny takes 8.1 s t hit grund Hw tall is building? Knwn: = 0 m/s; a = -g; t = 8.1 s; y = 0 Need: y - y 0 0 y = y + t - ½gt y = - ½gt = -(½)(9.8 m/s )(8.1 s) y = - 30 m What s the elcity f the penny just befre it hits the grund? Knwn: = 0 m/s; a = -g; t = 8.1 s; and (y - y )= -30 m = -gt = - (9.8 m/s )(8.1 s) = -79 m/s

17 What if I first thrw cin upward with speed f 67 mi/hr (=30 m/s)? When will cin reach max height? Knwn: = +30 m/s; a = -g Need: t when = 0 = - gt 0 m = 30 m/s (9.8 m/s )t t g 30 m/s 9.8 m/s y = y + t - ½gt but y = y = 0 0 = t - ½gt = t ( - ½gt) t = 0 r t = 6 s (abe starting pint) When will it pass me n the way dwn? 3s What is elcity just befre hitting grund? ( y y ) g = - 85 m/s

18 Things yu always wanted t knw but were afraid t ask 1. Can a penny drpped frm the Empire State Building kill a persn r embed itself in the sidewalk?. Is it OK t neglect air resistance? Ask the Mythbusters! 18

19 EXAMPLE: Drp a penny frm tp f the Empire State Building! (DO NOT TRY THIS!) Obsere: The penny takes 8.1 s t hit grund Hw tall is building? Knwn: y = 0 m/s; a = -g; t = 8.1 s; y = 0 Need: y - y 0 0 y = y + y t - ½gt y = - ½gt = -(½)(9.8 m/s )(8.1 s) y = - 30 m What s the elcity f the penny just befre it hits the grund? Knwn: y = 0 m/s; a = -g; t = 8.1 s; and (y - y )= -30 m = -gt = - (9.8 m/s )(8.1 s) = -79 m/s BUT: Terminal elcity = -9 m/s!!!