Δt The textbook chooses to say that the average velocity is

Transcription

1 1-D Motion Basic I Definitions: One dimensional motion (staight line) is a special case of motion whee all but one vecto component is zeo We will aange ou coodinate axis so that the x-axis lies along the diection of motion Thus, we can simplify ou geneal definitions to only involve the x-components as shown in the sections below It is impotant to note that we ae actually doing vecto math in chapte Howeve, scala math is a special case of vecto math fo one dimension so thee is no diffeence between the types of math in this chapte!! This is not the case in futue chaptes whee we will have to use tigonomety to beak vectos into components Position - x x î + 0 ĵ+ 0kˆ x î Using ou definition of the position vecto, we have The textbook chooses to say that the position vecto is given by x This is ok since the diection î is undestood If x is negative then the paticle is located in the î diection B Displacement - Δ x Using ou definition of the displacement vecto, we have Δ 1 î + 0 ĵ+ 0kˆ î The textbook chooses to say that the displacement vecto is given by Δ x This is ok since the diection î is undestood C veage Velocity - V av Using ou definition of the displacement vecto, we have Vav Δ î + 0 ĵ+ 0kˆ î The textbook chooses to say that the aveage velocity is Vav This is ok since the diection î is undestood

2 D Instantaneous Velocity V Using ou definition of the displacement vecto, we have Δ î + 0 ĵ+ 0kˆ V lim lim 0 0 The textbook chooses to say that the velocity is V lim 0 This is ok since the diection î is undestood E veage cceleation - av lim î 0 Using ou definition of the displacement vecto, we have av x î + 0 ĵ+ 0kˆ Since we only have x-components, the subscipt on the change in velocity is not equied This gives us av î The textbook chooses to say that the aveage velocity is av This is ok since the diection î is undestood F Instantaneous cceleation - Using ou definition of the displacement vecto, we have lim lim 0 0 x î + 0 ĵ+ 0kˆ x î lim 0 Since we only have x-components, the subscipt on the change in velocity is not equied This gives us lim 0 The textbook chooses to say that the aveage velocity is lim 0 This is ok since the diection î is undestood NOTE: You don t have to lean any of these fomulas so long as you know the vecto definitions and can do vecto math!!! You have to know the vecto definitions î x î

3 to be successful in Chapte 3 and the est of the couse Vecto math skills will also help you though out the couse You only have a limited amount of time to study so spend you time wisely studying in a way that pays multiple dividends and not memoizing II Gaphical nalysis of Motion In ode to analyze the motion of a paticle using ou definitions, we must collect data (infomation) about the paticle that we wish to analyze The most common data collected is the position of the paticle as a function of time which we collect using ou eyes Othe ways of collecting this infomation is to collect it using global positions satellites (GPS), ada, o a sonic ange finde (motion senso) Fo special cases like constant acceleation, we may have seen the gaph so often that we ae able to develop a fomulas that allows us to solve fo the slope of chods and tangent lines, and aeas fo the gaph without gaphing Howeve, these ae just special esults Fundamentally, we can find eveything fom the motion gaphs!! Position-Time Gaph x(m) 1 Position To find the paticles position at a paticula time you just ead the value fom the gaph using the y-axis Displacement Read the paticle s final and initial position using the gaph and then use the definition of the displacement to find Example: Find the ball s displacement ove the time inteval fom t s to t 5s Solution: If we looked at the gaph and found that the ball was located at x 4m at t s and x 1 m and t 5s, then the answe would be x x1 1m 4m 3m

4 3 veage Velocity The aveage velocity is found by using the definition of aveage velocity: Vav Gaphically, this is the fomula fo finding the slope of a chod connecting the initial location of the paticle on the gaph to the final location of the paticle as shown below by the blue chod fo the time inteval t 1 to t x(m) x x 1 t 1 t 4 Velocity The velocity at some time t is found by using the definition of velocity: V lim 0 Gaphically, this is the fomula fo finding the slope of a tangent line If the cuve is a staight line at this instant in time then the cuve is its own tangent line and the velocity is just the slope of the cuve Fo instance, the slope of the cuve at point and hence the velocity is zeo as shown below x(m) t If the cuve is not a staight line at the paticula instant in question, then you must daw a tangent line as shown below in blue The tangent line touches the cuve at the instant in time that you want to find the velocity but doesn t coss the cuve You then find the slope of the tangent line that you have dawn

5 x(m) x x 1 t 1 t 5 cceleation In geneal the value fo the acceleation can not be found diectly fom a position-time gaph You must fist constuct a velocity-time gaph The exception is if the acceleation is zeo!! nytime the position-time gaph is a staight line then duing that time inteval, the velocity is constant and the acceleation is zeo If the position-time gaph is not a staight line gaph then the paticle is acceleating!! B Velocity -Time Gaph V(m/s) 1 Velocity To find the paticle s velocity at a paticula time you just ead the value fom the gaph using the y-axis Change in Velocity Read the paticle s final and initial velocity using the gaph and then use the definition of the change in velocity to find

6 Example: Find the ball s change in velocity ove the time inteval fom t s to t 5s Solution: If we looked at the gaph and found that the ball was moving at V 4m/s at t s and V 1 m/w and t 5s, then the answe would be V V1 1m/s 4m/s 3m/s 3 veage cceleation The aveage acceleation is found by using the definition V(m/s) of aveage acceleation: av Gaphically, this is the fomula fo finding the slope of a chod connecting the initial velocity of the paticle on the gaph to the final velocity of the paticle as shown below by the blue chod fo the time inteval t 1 to t V V 1 t 1 t 4 cceleation The acceleation at some time t is found by using the definition of acceleation: lim 0 Gaphically, this is the fomula fo finding the slope of a tangent line If the cuve is a staight line at this instant in time then the cuve is its own tangent line and the acceleation is just the slope of the cuve Fo instance, the slope of the cuve at point and hence the acceleation is zeo as shown below

7 V(m/s) t If the cuve is not a staight line at the paticula instant in question, then you must daw a tangent line as shown below in blue The tangent line touches the cuve at the instant in time that you want to find the acceleation but doesn t coss the cuve You then find the slope of the tangent line that you have dawn V(m/s) V V 1 t 1 t 5 Displacement The method fo finding the displacement comes by solving the definition of velocity fo displacement The pocess whee a limit is imposed upon a atio as in the case of the definition of velocity is called diffeentiation and is one of the main topics of Calculus Since this isn t a Calculus based couse, it is enough fo you to know that this is a pocess just like taking the squae of a numbe To undo a pocess, we must know the invese pocess Fo the squaing pocess, the invese pocess is to take the squae oot Fo diffeentiation, the invese pocess is called integation and its value is the aea between the cuve and the hoizontal axis Thus, displacement is the aea between the cuve of a velocity-time gaph and the time axis If the aea is above the time axis then the aea is positive If the aea is below the time axis, the aea is negative lthough Calculus gives ticks fo finding aeas unde cuves, you can always appoximate the aea by beaking the aea up into a seies of small tiangles and ectangles This is an application of a pocess called the Method of Exhaustion by the ancient Geeks

8 Example: What was the paticle s displacement ove the time inteval fom t 0s to t 5s? V(m/s) 5 Solution: We beak the aea unde the gaph above into a tiangle followed by a ectangle The aea unde the cuve and hence the displacement is found by 1 ( s)( m/s) + ( 3s)( m/s) 8m 6 veage Velocity To find the aveage velocity fom a velocity-time gaph, you must fist find the displacement and then use the definition of aveage velocity Example: Find the paticle s aveage velocity ove the time inteval fom t 0 s to t 5s fo ou last example Solution: Using ou past esults and the definition of aveage velocity, we have NOTE: 80m 50s Vav V av V + V1 16 m s except fo the case whee the velocity-time gaph is a staight line (ie constant acceleation) C cceleation -Time Gaph We will see in Chapte 4 that the acceleation-time gaph takes on special meaning in the field of physics because of the connection between acceleation and the extenal foces acting on the paticle (Newton s nd Law) If we know the foces acting on a paticle then we know the shape of the paticle s acceleationtime cuve even though we may not have measued its position o velocity In this

9 (m/s ) case we ae woking the poblem backwads and tying to find out what the paticle s velocity-time and position-time gaph will look without measuing them n example of this appoach is the fact that we know the foce of gavity acting on a planet so we know its acceleation-time gaph We can then use ou definitions to detemine the location and velocity of the planet in the futue Thus, the equations of physics in a sense allow us to act as a sot of fotune telles to see the futue as well as to be able to look into the past and locate the position of planets and stas 1 cceleation To find the paticle s acceleation at a paticula time you just ead the value fom the gaph using the y-axis Change in cceleation Read the paticle s final and initial acceleation using the gaph and then use the definition of the change in acceleation to find Δ 5 Change in Velocity The change in velocity is the aea between the cuve of a velocity-time gaph and the time axis Example: What was the paticle s change in velocity ove the time inteval fom t 0s to t 5s? (m/s ) 5 Solution: We beak the aea unde the gaph above into a tiangle followed by a ectangle The aea

10 unde the cuve and hence the displacement is found by 1 ( s)( m/s ) + ( 3s)( m/s ) 8m/s 3 veage cceleation To find the aveage acceleation fom an acceleationtime gaph, you must fist find the displacement and then use the definition of aveage velocity Example: Find the paticle s aveage acceleation ove the time inteval fom t 0 s to t 5s fo ou last example Solution: Using ou past esults and the definition of aveage acceleation, we have 80m/s 50s av m 16 s