Linear Motion, Speed & Velocity

Transcription

1 Add Important Linear Motion, Speed & Velocity Page: 136 Linear Motion, Speed & Velocity NGSS Standard: N/A MA Curriculum Framework (006): 1.1, 1. AP Phyic 1 Learning Objective: 3.A.1.1, 3.A.1.3 Knowledge/Undertanding Goal: undertand term relating to poition, peed & velocity undertand the difference between peed and velocity Language Objective: Undertand and correctly ue the term poition, ditance, diplacement, peed, and velocity. Accurately decribe and apply the concept decribed in thi ection uing appropriate academic language. Lab, Activitie & Demontration: Note: Walk in the poitive and negative direction (with poitive or negative velocity). Walk and change direction to how ditance v. diplacement. coördinate ytem: a framework for decribing an object poition (location), baed on it ditance (in one or more direction) from a pecifically-defined point (the origin). (You hould remember thee term from math.) direction: which way an object i oriented or moving within it coördinate ytem. Note that direction can be poitive or negative. poition (x): the location of an object relative to the origin (zero point) of it coördinate ytem. We will conider poition to be a zero-dimenional vector, which mean it can be poitive or negative with repect to the choen coördinate ytem. ditance (d ): [calar] how far an object ha moved.

2 Add Important Linear Motion, Speed & Velocity Page: 137 diplacement ( d or x ): [vector] how far an object current poition i from it tarting poition ( initial poition ). Diplacement can be poitive or negative (or zero), depending on the choen coördinate ytem. rate: the change in a quantity over a pecific period of time. motion: when an object poition i changing over time. peed: [calar] the rate at which an object i moving at an intant in time. Speed doe not depend on direction, and i alway nonnegative. velocity: (v ) [vector] an object diplacement over a given period of time. Becaue velocity i a vector, it ha a direction a well a a magnitude. Velocity can be poitive, negative, or zero. uniform motion: motion at a contant velocity (i.e., with contant peed and direction) An object that i moving ha a poitive peed, but it velocity may be poitive, negative, or zero, depending on it poition.

3 Add Important Linear Motion, Speed & Velocity Page: 138 Variable Ued to Decribe Linear Motion Variable Quantity MKS Unit x poition m d, Δx ditance m d, x diplacement m h height m v m velocity v m average velocity The average velocity of an object i it diplacement divided by the time, or it change in poition divided by the (change in) time: d x xo Δx Δx v t t t t (Note that elaped time i alway a difference ( t), though we uually ue t rather than t a the variable.) We can ue calculu to turn v into v by taking the limit a Δt approache zero: x v Lim t 0 t i.e., velocity i the firt derivative of diplacement with repect to time. We can rearrange thi formula to how that diplacement i average velocity time time: d vt dx dt Poition i the object tarting poition plu it diplacement: x x d x vt o where x * 0 mean poition at time = 0. Thi formula i often expreed a: x xo d v t o * x o i pronounced x -zero or x -naught.

4 Add Important Linear Motion, Speed & Velocity Page: 139 x Note that i the lope of a graph of poition (x ) v. time (t ). Becaue t x v, thi mean that the lope of a graph of poition v. time i equal to the t velocity. In fact, on any graph, the quantity you get when you divide the quantity on the x- axi by the quantity on the y-axi i, by definition, the lope. I.e., the lope i y y - axi, which mean the quantity defined by will alway be the lope. x x - axi Recall that velocity i a vector, which mean it can be poitive, negative, or zero. On the graph below, the velocity i + 4 from 0 to, zero from to 4, and m from 4 to 8. m

5 Add Important Linear Motion, Speed & Velocity Page: 140 Sample problem: Q: A car travel 100 m in 60 econd. What i it average velocity? A: v v d t 100 m 0 60 m Q: A peron walk 30 m at an average velocity of 1.5 m. How long did it take? A: How long mean what length of time. d v t t t 56 It took 56 econd for the peron to walk 30 m.

6 Add Important Linear Acceleration Page: 141 NGSS Standard: N/A Linear Acceleration MA Curriculum Framework (006): 1.1, 1. AP Phyic 1 Learning Objective: 3.A.1.1, 3.A.1.3 Knowledge/Undertanding Goal: Skill: what linear acceleration mean what poitive v. negative acceleration mean calculate poition, velocity and acceleration for problem that involve movement in one direction Language Objective: Undertand and correctly ue the term acceleration. Accurately decribe and apply the concept decribed in thi ection uing appropriate academic language. Lab, Activitie & Demontration: Note: Walk with different combination of poitive/negative velocity and poitive/negative acceleration. Drop a dollar bill or meter tick and have omeone try to catch it. Drop two tring of bead, one paced at equal ditance and the other paced at equal time. Drop a bottle of water with a hole near the bottom or bucket of ping-pong ball. acceleration: a change in velocity over a period of time. uniform acceleration: when an object rate of acceleration (i.e., the rate at which it velocity change) i contant.

7 Add Important Linear Acceleration Page: 14 If an object velocity i increaing, we ay it ha poitive acceleration. If an object velocity i decreaing, we ay it ha negative acceleration. velocity velocity velocity time time time poitive acceleration acceleration = zero negative acceleration Note that if the object velocity i negative, then increaing velocity (poitive acceleration) would mean that the velocity i getting le negative, i.e., the object would be lowing down in the negative direction. Variable Ued to Decribe Acceleration Variable Quantity MKS Unit a m acceleration g acceleration m due to gravity By convention, phyicit ue the variable g to mean acceleration due to gravity, and a to mean acceleration caued by omething other than gravity.

8 Add Important Linear Acceleration Page: 143 Becaue acceleration i a change in velocity over a period of time, the formula for acceleration i: v vo v v v dv a and, from calculu: a Lim t t t t 0 t dt The unit mut match the formula, which mean the unit for acceleration mut be velocity (ditance/time) divided by time, which equal ditance divided by time quared. dx Becaue v, thi mean that acceleration i the econd derivative of poition dt dv d d with repect to time: dx d x a ( v) dt dt dt dt dt However, in an algebra-baed phyic coure, we will limit ourelve to problem in which acceleration i contant. We can rearrange thi formula to how that the change in velocity i acceleration time time: v v v Note that when an object velocity i changing, the final velocity, v, i not the ame a the average velocity, v. (Thi i a common mitake that firt-year phyic tudent make.) o at

9 Add Important Linear Acceleration Page: 144 v v i the lope of a graph of velocity (v ) v. time (t ). Becaue a, thi t t mean that acceleration i the lope of a graph of velocity v. time: Note the relationhip between velocity-time graph and poition-time graph. poitive acceleration acceleration = zero negative acceleration velocity velocity velocity time time time concave up linear concave down

10 Add Important Linear Acceleration Page: 145 Note alo that v t i the area under a graph (i.e., the area between the curve and the x-axi) of velocity (v ) v. time (t ). Becaue vt d, thi mean the area under a graph of velocity v. time i the diplacement (Δx). Note that thi work both for contant velocity (the graph on the left) and changing velocity (a hown in the graph on the right). In fact, on any graph, the quantity you get when you multiply the quantitie on the x- and y-axe i, by definition, the area under the graph. In calculu, the area under a curve i the integral of the equation for the curve. Thi mean: where v can be any function of t. d t 0 v dt

11 Add Important Linear Acceleration Page: 146 In the graph below, between 0 and 4 the object i accelerating at a rate of.5 m. Between 4 and 6 the object i moving at a contant velocity (of the acceleration i zero. 10 m ), o a =.5 m d 1 (.5)( ) 5m A 1 ()(5) 5m a =.5 m d 1 (.5)(4 ) 0 m A 1 (4)(10) 0 m a = 0 d vt ( 10)() 0 m A ( )(10) 0 m In each cae, the area under the velocity-time graph equal the total ditance traveled.

12 Add Important Linear Acceleration Page: 147 To how the relationhip between v and v, we can combine the formula for average velocity with the formula for acceleration in order to get a formula for the poition of an object that i accelerating. d vt v at However, the problem i that v in the formula v at i the velocity at the end, which i not the ame a the average velocity v. If the velocity of an object i changing (i.e., the object i accelerating), the average velocity, v (the line over the v mean average ), i given by the formula: vo v v If the object tart at ret (not moving, which mean v 0 ) and it accelerate at a contant rate, the average velocity i therefore the average of the initial velocity and the final velocity: vo v 0 v v v Combining all of thee give, for an object tarting from ret: d vt 1 1 vt at) 1 1 v ( t at If an object wa moving before it tarted to accelerate, it had an initial velocity, or a velocity at time = 0. We will repreent thi initial velocity a v o *. Now, the formula become: x x o d v t o 1 at o ditance the object would travel at it initial velocity additional ditance the object will travel becaue it i accelerating * pronounced v-zero or v-naught

13 Add Important Linear Acceleration Page: 148 Thi equation can be combined with the equation for velocity to give the following equation, which relate initial and final velocity and ditance: v vo ad Finally, when an object i accelerating becaue of gravity, we ay that the object i in free fall. On earth, the average acceleration due to gravity i approximately m at ea level (which we will uually round to 10 m ). Any time gravity i involved (and the problem take place on Earth), aume that a g 10. Extenion Jut a a change in velocity i called acceleration, a change in acceleration with repect to time i called jerk : j a. t While quetion about jerk have not been een on the AP exam, ome AP problem do require you to undertand that the area under a graph of acceleration v. time would be the change in velocity (Δv), jut a the area under a graph of velocity v. time i the change in poition. m

14 Add Important Linear Acceleration Page: 149 Homework Problem: Motion Graph 1. An object motion i decribed by the following graph of poition v. time: a. What i the object doing between and 4? What i it velocity during that interval? b. What i the object doing between 6 and 7? What i it velocity during that interval? c. What i the object doing between 8 and 10? What i it velocity during that interval?

15 Add Important Linear Acceleration Page: 150. An object motion i decribed by the following graph of velocity v. time: a. What i the object doing between 0 and? What are it velocity and acceleration during that interval? b. What i the object doing between and 4? What i it acceleration during that interval? c. What i the object doing between 6 and 9? What i it acceleration during that interval?

16 Add Important Linear Acceleration Page: The graph on the left below how the poition of an object v. time. Sketch a graph of velocity v. time for the ame object on a graph imilar to the one on the right. 4. In 1991, Carl Lewi became the firt printer to break the 10-econd barrier for the 100 m dah, completing the event in The chart below how hi time for each 10 m interval. ditance (m) time () Plot Lewi diplacement v. time and velocity v. time on graph imilar to the one below. Ue thi pace for ummary and/or additional note: