1. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Physics I 95.141 LECTURE 3 9/14/09

2. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Clicker Test Today’s lecture could possibly go worse than Wednesday’s? –True/False

3. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Outline Freely Falling Body Problems Vectors and Scalars Addition of vectors (Graphical) Adding Vectors by Components Unit Vectors What Do We Know? –Units/Measurement/Estimation –Displacement/Distance –Velocity (avg. & inst.), speed –Acceleration

4. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Review of Lecture 2 Last week we discussed how to describe the position and motion of an object Reference Frames Position Velocity Acceleration Constant Acceleration

5. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 LP-HW 2.65 A stone is dropped from a cliff and the splash is heard 3.4s later. If the speed of sound is 340m/s, how high is the cliff? Two part problem! Draw a diagram! Coordinate system Knowns/Unknowns A) B)

6. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 LP-HW 2.65 Choose equations A) B)

7. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Example Problem II Batman launches his grappling bat-hook upwards, if the beam it attaches to is 50m above Batman’s Batbelt, at what bat-velocity must the hook be launched at in order to make it to the beam? (Ignore the mass of the cord and air resisitance) 1) Choose coordinate system 2) Knowns and unknowns 3) Choose equation(s)

8. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Example Problem II Batman launches his grappling bat-hook upwards, if the beam it attaches to is 50m above Batman’s Batbelt, at what bat-velocity must the hook be launched at in order to make it to the beam? (Ignore the mass of the cord and air resisitance) 3) Choose equation(s) 4) Solve

9. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Vectors and Scalars A quantity that has both direction and magnitude, is known as a vector. –Velocity, acceleration, displacement, Force, momentum –In text, we represent vector quantities as Quantities with no direction associated with them are known as scalars –Speed, temperature, mass, time

10. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Vectors and Scalars In the previous chapter we dealt with motion in a straight line –For horizontal motion (+/- x) –For vertical motion (+/- y) Velocity, displacement, acceleration were still vectors, but direction was indicated by the sign (+/-). We will first understand how to work with vectors graphically

11. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Addition of vectors In one dimension –If the vectors are in the same direction –But if the vectors are in the opposite direction

12. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Addition of Vectors (2D) In two dimensions, things are more complicated

13. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Addition of Vectors “Tip to tail” method –Draw first vector –Draw second vector, placing tail at tip of first vector –Arrow from tail of 1 st vector to tip of 2 nd vector is

14. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Commutative property of vectors “Tip to tail” method works in either order –Draw first vector –Draw second vector, placing tail at tip of first vector –Arrow from tail of 1 st vector to tip of 2 nd vector is

15. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Three or more vectors Can use “tip to tail” for more than 2 vectors + + =

16. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Subtraction of vectors For a given vector the negative of the vector is a vector with the same magnitude in the opposite direction. - =+ Difference between two vectors is equal to the sum of the first vector and the negative of the second vector

17. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Multiplying a vector by a scalar You can also multiply a vector by a scalar When you do this, you don’t change the direction of the vector, only its magnitude c=2 c=4 c=-2

18. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Adding vectors by components Adding vectors graphically is useful to understand the concept of vectors, but it is inherently slow (not to mention next to impossible in 3D!!) Any 2D vector can be decomposed into components

19. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Determining vector components So in 2D, we can always write any vector as the sum of a vector in the x-direction, and one in the y-direction. Given V(V,θ), we can find V x and V y

20. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Example A vector is given by its magnitude and direction: Write the vector in terms of its components

21. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Determining vector components Or, given V x and V y, we can find V(V,θ).

22. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Example A vector is given by its vector components: Write the vector in terms of magnitude and direction

23. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Adding vectors by components Given V 1 and V 2, how can we find V= V 1 + V 2 ? V1V1 V2V2 V

24. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Example A trucker drives 10miles North, then 40 miles 30º East from South. What is his final displacement? Set up coordinate system

25. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Example A trucker drives 10miles North, then 40 miles 30º East from South. What is his final displacement? Resolve each vector into x,y components

26. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Example A trucker drives 10miles North, then 40 miles 30º East from South. What is his final displacement? Add vector components

27. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 3D Vectors Adding vectors vectors by components is especially helpful for 3D vectors. Also, much easier for subtraction

28. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 3D Vectors Which is traveling fastest? Three cars are traveling with velocities given by:

29. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Unit Vectors Up to this point, we have written vectors in terms of their components as follows: There is an easier way to do this, and this is how we will write vectors for the remainder of the course:

30. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Unit Vectors What are unit vectors? –Unit vectors have a magnitude of 1 and point along major axes of our coordinate system Writing a vector with unit vectors is equivalent to multiplying each unit vector by a scalar

31. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Unit Vectors For a vector with components: Write this is unit vector notation

32. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Example: Vector Addition/Subtraction Displacement –A hiker traces her movement along a trail. The first leg of her hike brings her to the foot of the mountain: –On the second leg, she ascends the mountain, which she figures to be a displacement of: –On the third, she walks along a plateau. –Then she falls of a cliff –What is the hiker’s final displacement?

33. Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Example: Vector Addition/Subtraction