Physics 207: Lecture 3, Pg 1 Physics 207, Lecture 3 Assignment: 1.For Thursday: Read Chapter 3 (carefully) through Homework Set 2 available l Goals (finish Chap. 2 & 3)   Understand the relationships between position, velocity & acceleration in systems with 1-dimensional motion and non-zero acceleration (usually constant)  Solve problems with zero and constant acceleration (including free-fall and motion on an incline)  Use Cartesian and polar coordinate systems  Perform vector algebra

Physics 207: Lecture 3, Pg 2 Position, Displacement, Velocity, Acceleration time (sec) position (meters) origin position vectors displacement vectors velocity vectors and, this case, there is no change in velocity x y

Physics 207: Lecture 3, Pg 3 Acceleration l Changes in a particle’s motion often involve acceleration  The magnitude of the velocity vector may change  The direction of the velocity vector may change (true even if the magnitude remains constant)  Both may change simultaneously l Here, a “particle” with smooth decreasing speed v0v0 v1v1 v1v1 v1v1 v3v3 v5v5 v2v2 v4v4 v0v0

Physics 207: Lecture 3, Pg 4 Constant Acceleration l Changes in a particle’s motion often involve acceleration  The magnitude of the velocity vector may change l A particle with smoothly decreasing speed: v1v1v1v1 v0v0v0v0 v3v3v3v3 v5v5v5v5 v2v2v2v2 v4v4v4v4 a a a a a a v t 0 v(  t)=v 0 + a  t a  t = area under curve =  v (an integral) a t tt

Physics 207: Lecture 3, Pg 5 Instantaneous Acceleration l The instantaneous acceleration is the limit of the average acceleration as ∆v/∆t approaches zero l Quick Comment: Instantaneous acceleration is a vector with components parallel (tangential) and/or perpendicular (radial) to the tangent of the path (more in Chapter 6)

Physics 207: Lecture 3, Pg 6 And given a constant acceleration we can integrate to get explicit v x and a x x axax vxvx t t t 0 x0x0

Physics 207: Lecture 3, Pg 7 Position, velocity & acceleration for motion along a line If the position x is known as a function of time, then we can find both the instantaneous velocity v x and instantaneous acceleration a x as a function of time! vxvx t x t axax t

Physics 207: Lecture 3, Pg 8 Position, displacement, velocity & acceleration l All are vectors and so vector algebra is a must ! l These cannot be used interchangeably (different units!) (e.g., position vectors cannot be added directly to velocity vectors) l But we can determined directions  “Change in the position” vector gives the direction of the velocity vector  “Change in the velocity” vector gives the direction of the acceleration vector l Given x(t)  v x (t)  a x (t) l Given a x (t)  v x (t)  x(t)

Physics 207: Lecture 3, Pg 9 Rearranging terms gives two other relationships l For constant acceleration: l From which we can show (caveat: a constant acceleration) Slope of x(t) curve axax t

Physics 207: Lecture 3, Pg 10 Free Fall l When any object is let go it falls toward the ground !! The force that causes the objects to fall is called gravity. l This acceleration on the Earth’s surface, caused by gravity, is typically written as “little” g l Any object, be it a baseball or an elephant, experiences the same acceleration (g) when it is dropped, thrown, spit, or hurled, i.e. g is a constant.

Physics 207: Lecture 3, Pg 11 l A “biology” experiment l Hypothesis: Older people have slower reaction times l Distance accentuates the impact of time differences l Equipment: Ruler and four volunteers  Older student  Younger student  Record keeper  Statistician  Expt. require multiple trials to reduce statistical errors.

Physics 207: Lecture 3, Pg 12 Gravity facts: l g does not depend on the nature of the material !  Galileo ( ) figured this out without fancy clocks & rulers! l Feather & penny behave just the same in vacuum l Nominally, g = 9.81 m/s 2  At the equatorg = 9.78 m/s 2  At the North poleg = 9.83 m/s 2

Physics 207: Lecture 3, Pg 13 When throwing a ball straight up, which of the following is true about its velocity v and its acceleration a at the highest point in its path? A. Both v = 0 and a = 0 B. v  0, but a = 0 C. v = 0, but a  0 D. None of the above y Exercise 1 Motion in One Dimension

Physics 207: Lecture 3, Pg 14 In driving from Madison to Chicago, initially my speed is at a constant 65 mph. After some time, I see an accident ahead of me on I-90 and must stop quickly so I decelerate increasingly “fast” until I stop. The magnitude of my acceleration vs time is given by, A.  B.  C.  a t Exercise 2 More complex Position vs. Time Graphs Question: My velocity vs time graph looks like which of the following ? v t v v

Physics 207: Lecture 3, Pg 15 Exercise 3 1D Freefall A. v A < v B B. v A = v B C. v A > v B. l Alice and Bill are standing at the top of a cliff of height H. Both throw a ball with initial speed v 0, Alice straight down and Bill straight up. The speed of the balls when they hit the ground are v A and v B respectively. v0v0v0v0 v0v0v0v0 BillAlice H vAvAvAvA vBvBvBvB

Physics 207: Lecture 3, Pg 16 Exercise 3 1D Freefall : Graphical solution l Alice and Bill are standing at the top of a cliff of height H. Both throw a ball with initial speed v 0, Alice straight down and Bill straight up. vxvx t cliff back at cliff turnaround point ground v0v0 -v 0 v ground  v= -g  t identical displacements (one + and one -)

Physics 207: Lecture 3, Pg 17 Problem Solution Method: Five Steps: 1) Focus the Problem - draw a picture – what are we asking for? 2) Describe the physics -what physics ideas are applicable -what are the relevant variables known and unknown 3) Plan the solution -what are the relevant physics equations 4) Execute the plan -solve in terms of variables -solve in terms of numbers 5) Evaluate the answer -are the dimensions and units correct? -do the numbers make sense?

Physics 207: Lecture 3, Pg 18 Example of a 1D motion problem l A cart is initially traveling East at a constant speed of 20 m/s. When it is halfway (in distance) to its destination its speed suddenly increases and thereafter remains constant. All told the cart spends a total of 10 s in transit with an average speed of 25 m/s. l What is the speed of the cart during the 2 nd half of the trip? l Dynamical relationships: And

Physics 207: Lecture 3, Pg 19 The picture l Plus the average velocity l Knowns:  x 0 = 0 m  t 0 = 0 s  v 0 = 20 m/s  t 2 = 10 s  v avg = 25 m/s  relationship between x 1 and x 2 l Four unknowns x 1 v 1 t 1 & x 2 and must find v 1 in terms of knowns t2t2 t0t0 t1t1 v 1 ( > v 0 ) a 1 =0 m/s 2 v 0 a 0 =0 m/s 2 x0x0 x1x1 x2x2

Physics 207: Lecture 3, Pg 20 Using l Four unknowns l Four relationships t2t2 t0t0 t1t1 v 1 ( > v 0 ) a 1 =0 m/s 2 v 0 a 0 =0 m/s 2 x0x0 x1x1 x2x2

Physics 207: Lecture 3, Pg 21 Fini l Plus the average velocity l Given:  v 0 = 20 m/s  t 2 = 10 s  v avg = 25 m/s t2t2 t0t0 t1t1 v 1 ( > v 0 ) a 1 =0 m/s 2 v 0 a 0 =0 m/s 2 x0x0 x1x1 x2x2

Physics 207: Lecture 3, Pg 22 Using l Minimizing algebra t2t2 t0t0 t1t1 v 1 ( > v 0 ) a 1 =0 m/s 2 v 0 a 0 =0 m/s 2 x0x0 x1x1 x2x

Physics 207: Lecture 3, Pg 23 Minimizing algebra t2t2 t0t0 t1t1 v 1 ( > v 0 ) a 1 =0 m/s 2 v 0 a 0 =0 m/s 2 x0x0 x1x1 x2x m 1 2

Physics 207: Lecture 3, Pg 24 Minimizing algebra t2t2 t0t0 t1t1 v 1 ( > v 0 ) a 1 =0 m/s 2 v 0 a 0 =0 m/s 2 x0x0 x1x1 x2x2 125 m 6.25 s 125 m 3.75 s Using the numerical information is a bit easier to follow but it doesn’t provide the “general solution” If something this difficult were even to appear on an exam (an unlikely event) then “setting it up” would be all that was necessary. This is physics….not math!

Physics 207: Lecture 3, Pg 25 Tips: l Read !  Before you start work on a problem, read the problem statement thoroughly. Make sure you understand what information is given, what is asked for, and the meaning of all the terms used in stating the problem. l Watch your units (dimensional analysis) !  Always check the units of your answer, and carry the units along with your numbers during the calculation. l Ask questions !

Physics 207: Lecture 3, Pg 26 Coordinate Systems and vectors l In 1 dimension, only 1 kind of system,  Linear Coordinates (x) +/- l In 2 dimensions there are two commonly used systems,  Cartesian Coordinates(x,y)  Circular Coordinates(r,  ) l In 3 dimensions there are three commonly used systems,  Cartesian Coordinates(x,y,z)  Cylindrical Coordinates (r, ,z)  Spherical Coordinates(r,  )

Physics 207: Lecture 3, Pg 27 Vectors l In 1 dimension, we can specify direction with a + or - sign. l In 2 or 3 dimensions, we need more than a sign to specify the direction of something: r l To illustrate this, consider the position vector r in 2 dimensions. Example Example: Where is Boston? New York  Choose origin at New York  Choose coordinate system Boston is 212 miles northeast of New York [ in (r,  ) ] OR Boston is 150 miles north and 150 miles east of New York [ in (x,y) ] Boston New York r

Physics 207: Lecture 3, Pg 28 Vectors... l There are two common ways of indicating that something is a vector quantity: A  Boldface notation: A  “Arrow” notation: A A =A A

Physics 207: Lecture 3, Pg 29 Recap l Assignment:  For Thursday: Read Chapter 3 (carefully) through 4.4  Homework Set 2 available