Physics 207, Lecture 3

1. Physics 207, Lecture 3 • 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 Assignment: • For Thursday: Read Chapter 3 (carefully) through 4.4 • Homework Set 2 available

2. Position, Displacement, Velocity, Acceleration y time (sec) 1 2 3 4 5 6 position 1 2 3 4 5 6 (meters) x position vectors origin displacement vectors velocity vectors 1 2 3 4 5 6 and, this case, there is no change in velocity

3. v0 v1 v1 v1 v3 v5 v2 v4 v0 Acceleration • 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 • Here, a “particle” with smooth decreasing speed

4. v1 v0 v3 v5 v2 v4 a a a a a a a Dt t t 0 Constant Acceleration • Changes in a particle’s motion often involve acceleration • The magnitude of the velocity vector may change • A particle with smoothly decreasing speed: v(Dt)=v0 + a Dt v a Dt = area under curve = Dv (an integral)

5. Instantaneous Acceleration • The instantaneous acceleration is the limit of the average acceleration as ∆v/∆t approaches zero • 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)

6. And given a constant acceleration we can integrate to get explicit vx and ax x x0 t vx 0 t ax t

7. x t vx t ax t Position, velocity & acceleration for motion along a line • If the positionx is known as a function of time, then we can find both the instantaneous velocityvxand instantaneous accelerationax as a function of time!

8. Position, displacement, velocity & acceleration • All are vectors and so vector algebra is a must ! • These cannot be used interchangeably (different units!) (e.g., position vectors cannot be added directly to velocity vectors) • 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 • Given x(t)  vx(t)  ax (t) • Given ax (t)  vx (t)  x(t)

9. ax t Rearranging terms gives two other relationships • For constant acceleration: • From which we can show (caveat: a constant acceleration) Slope of x(t) curve

10. Free Fall • When any object is let go it falls toward the ground !! The force that causes the objects to fall is called gravity. • This acceleration on the Earth’s surface, caused by gravity, is typically written as “little” g • 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.

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

12. Gravity facts: • g does not depend on the nature of the material ! • Galileo (1564-1642) figured this out without fancy clocks & rulers! • Feather & penny behave just the same in vacuum • Nominally, g= 9.81 m/s2 • At the equator g = 9.78 m/s2 • At the North pole g = 9.83 m/s2

13. y 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? Exercise 1Motion in One Dimension • Bothv = 0anda = 0 • v  0, but a = 0 • v = 0, but a  0 • None of the above

14. t a v v v t 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 myacceleration vs timeis given by, Exercise 2 More complex Position vs. Time Graphs •  •   •   • Question: My velocity vs time graph looks like which of the following ?

15. Exercise 3 1D Freefall • Alice and Bill are standing at the top of a cliff of heightH. Both throw a ball with initial speedv0, Alice straightdownand Bill straightup. The speed of the balls when they hit the ground arevAandvBrespectively. • vA < vB • vA= vB • vA > vB Alice v0 Bill v0 H vA vB

16. Exercise 3 1D Freefall : Graphical solution • Alice and Bill are standing at the top of a cliff of heightH. Both throw a ball with initial speedv0, Alice straightdownand Bill straightup. back at cliff cliff Dv= -g Dt turnaround point v0 vx t identical displacements (one + and one -) -v0 vground ground ground

17. Problem Solution Method: Five Steps: • Focus the Problem - draw a picture – what are we asking for? • Describe the physics • what physics ideas are applicable • what are the relevant variables known and unknown • Plan the solution • what are the relevant physics equations • Execute the plan • solve in terms of variables • solve in terms of numbers • Evaluate the answer • are the dimensions and units correct? • do the numbers make sense?

18. Example of a 1D motion problem • 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. • What is the speed of the cart during the 2nd half of the trip? • Dynamical relationships: And

19. The picture v1 ( > v0 ) a1=0 m/s2 v0 a0=0 m/s2 • Plus the average velocity • Knowns: • x0 = 0 m • t0 = 0 s • v0 = 20 m/s • t2 = 10 s • vavg = 25 m/s • relationship between x1 and x2 • Four unknowns x1 v1 t1 & x2 and must find v1 in terms of knowns x0 t0 x1 t1 x2 t2

20. v1 ( > v0 ) a1=0 m/s2 v0 a0=0 m/s2 x0 t0 x1 t1 x2 Using • Four unknowns • Four relationships t2

21. v1 ( > v0 ) a1=0 m/s2 v0 a0=0 m/s2 x0 t0 x1 t1 x2 Fini • Plus the average velocity • Given: • v0 = 20 m/s • t2 = 10 s • vavg = 25 m/s t2

22. Using v1 ( > v0 ) a1=0 m/s2 v0 a0=0 m/s2 • Minimizing algebra x0 t0 x1 t1 x2 t2 1 2 3 4 4 3

23. Minimizing algebra v1 ( > v0 ) a1=0 m/s2 v0 a0=0 m/s2 x0 t0 x1 t1 125 m 125 m x2 t2 2 1 1 2

24. Minimizing algebra v1 ( > v0 ) a1=0 m/s2 v0 a0=0 m/s2 x0 t0 x1 t1 125 m 3.75 s 125 m 6.25 s x2 t2 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!

25. Tips: • 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. • Watch your units (dimensional analysis) ! • Always check the units of your answer, and carry the units along with your numbers during the calculation. • Ask questions !

26. Coordinate Systems and vectors • In 1 dimension, only 1 kind of system, • Linear Coordinates (x) +/- • In 2 dimensions there are two commonly used systems, • Cartesian Coordinates (x,y) • Circular Coordinates (r,q) • In 3 dimensions there are three commonly used systems, • Cartesian Coordinates (x,y,z) • Cylindrical Coordinates (r,q,z) • Spherical Coordinates (r,q,f)

27. Vectors • In 1 dimension, we can specify direction with a + or - sign. • In 2 or 3 dimensions, we need more than a sign to specify the direction of something: • To illustrate this, consider the position vectorr in 2 dimensions. Example: Where is Boston? • Choose origin at New York • Choose coordinate system Boston is 212 milesnortheastof New York [ in (r,q) ]OR Boston is 150 milesnorth and 150 mileseast of New York [ in (x,y) ] Boston r New York

28. A A Vectors... • There are two common ways of indicating that something is a vector quantity: • Boldface notation: A • “Arrow” notation: A =

29. Recap • Assignment: • For Thursday: Read Chapter 3 (carefully) through 4.4 • Homework Set 2 available