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

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

Department of Physics and Applied Physics , 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

Department of Physics and Applied Physics , 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

Department of Physics and Applied Physics , 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)

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

Department of Physics and Applied Physics , 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)

Department of Physics and Applied Physics , 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

Department of Physics and Applied Physics , 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

Department of Physics and Applied Physics , 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

Department of Physics and Applied Physics , 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

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

Department of Physics and Applied Physics , 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

Department of Physics and Applied Physics , 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

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

Department of Physics and Applied Physics , 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

Department of Physics and Applied Physics , 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

Department of Physics and Applied Physics , 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

Department of Physics and Applied Physics , 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

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

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

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

Department of Physics and Applied Physics , 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

Department of Physics and Applied Physics , 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

Department of Physics and Applied Physics , 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

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

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

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

Department of Physics and Applied Physics , 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:

Department of Physics and Applied Physics , 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

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

Department of Physics and Applied Physics , 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?

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