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Kinematics in Two Dimensions 01/09/2008 03:00 PM Kinematics in Two Dimensions Kinematics in 2-dimensions. By the end of this you will 1. Remember your Trigonometry 2. Know how to handle vectors 3. be able to handle problems in 2-dimensions 4. understand projectile motion Swish! by alanfreed Two (or Three) Dimensional Motion For example: Projectiles, Planets, Pendulum, .... Need to have some new mathematical techniques to do this: however you may need to revise your basic trigonometry Basic Trigonometry Basic Definitions of trig relationships A useful mnemomic SohCahToa θ = a R Sin = opposite = d hypotenuse R Cos = adjacent = R-δ hypotenuse R Tan = opposite = d adjacent R-δ Basic Trig. Useful relationships sin2θ + cos2θ = 1 (all values of θ) tan θ = sin θ / cos θ file:///Users/watson/Documents/Physics/1000_level/BIT_Kinematics/BIT1002_2-Dkinematics.html?id=0 Page 1 of 10 Kinematics in Two Dimensions 01/09/2008 03:00 PM These are not so useful cot θ = 1/tan θ sec θ = 1/cos θ cosec θ = 1/sin θ Special Values sin(450) = sin(π/4) = 1/√2 cos(450) = cos(π/4) = 1/√2 tan(45 0) = tan(π/4) = 1 sin(300) = sin(π/6) = 1/2 cos(600) = cos(π/3) = 1/2 sin(600) = cos(300) = √3/2 tan(30 0) = tan(π/3) = 1/√3 tan(60 0) = tan(π/6) = √3 sin(900) = sin(π/2) = 1 cos(900) = cos(π/2) = 0 tan(90 0) = tan(π/2) = ∞ These are worth writing down, but they are very easy to work out. We often have a convenient simplification when the angles are small θ = a/R ~ sin(θ) = d/R ~ tan(θ) = d (R-δ) cos(θ) ~ 1 Note that this only works if we measure angles in radians: 2π radians = 360 0 Vectors Need new mathematics to describe three dimensional objects: Scalars: quantities with only magnitude Vectors have direction as well Scalars Vectors Temperature Force file:///Users/watson/Documents/Physics/1000_level/BIT_Kinematics/BIT1002_2-Dkinematics.html?id=0 Page 2 of 10 Kinematics in Two Dimensions 01/09/2008 03:00 PM Speed Velocity Density Acceleration Time? Time? Need a new symbol: can use ~ which is obvious a a which gets confused with other symbols a, which is hard to read We will use both ~ and a, a Need to be able to describe vectors in terms of scalar quantities: can do this in terms of components: the projection of the vector along each axis These are the components of the vector ~ a Note that the components of a vector are scalars Also can do this in terms of length of the vector and angle(s). These two descriptions are related a x = a cos(θ) a y = a sin(θ) file:///Users/watson/Documents/Physics/1000_level/BIT_Kinematics/BIT1002_2-Dkinematics.html?id=0 Page 3 of 10 Kinematics in Two Dimensions 01/09/2008 03:00 PM Adding vectors: put them nose to tail. Easy diagramatically If we want to add them algebraically, we just add the components: ~ = ~ +~ c a b means cx = a x + b x cy = a y + b y To subtract vectors, Flip the vector round: the negative of a vector must add to the vector to give zero ~ - ~ = ~ + ( -a) = 0 a a a ~ Note that this means that the negative of a vector just has all its components reversed The length of a vector is given by Pythagoras: 3-D analog of Pythagoras: file:///Users/watson/Documents/Physics/1000_level/BIT_Kinematics/BIT1002_2-Dkinematics.html?id=0 Page 4 of 10 Kinematics in Two Dimensions 01/09/2008 03:00 PM Write this as q j~j = r x 2 + y2 + z 2 Note: The length of a vector is a scalar. Displacement is a vector. The length of the displacement vector is the distance Examples e.g. a boat sails 10 km North -East and 5 km South: how must it sail to get to its start? How do we write this as a vector sum? file:///Users/watson/Documents/Physics/1000_level/BIT_Kinematics/BIT1002_2-Dkinematics.html?id=0 Page 5 of 10 Kinematics in Two Dimensions 01/09/2008 03:00 PM N.E means "equal components along the x and y directions" so first step is e.g. Motion by a car. A car travels 5 km N, 10 km E, and then 15 km S. The components of vector ~ that describes this a are 1 . a x = -10 a y = 20 2 . a x = 10 a y = -10 3 . a x = 10 a y = 10 4 . a x = -10 a y = -10 We write this as Projectile Motion (can usually be treated as 2-D) file:///Users/watson/Documents/Physics/1000_level/BIT_Kinematics/BIT1002_2-Dkinematics.html?id=0 Page 6 of 10 Kinematics in Two Dimensions 01/09/2008 03:00 PM Treat position, r, velocity v and acceleration as 2-D vectors. In general,motion in one direction can be treated independently of motion in a second. file:///Users/watson/Documents/Physics/1000_level/BIT_Kinematics/BIT1002_2-Dkinematics.html?id=0 Page 7 of 10 Kinematics in Two Dimensions 01/09/2008 03:00 PM We can also see this quantitatively In general there are three independent vector quantities: The position r, the velocity v and the acceleration a. However we have to treat the components separately. It is easiest to treat the two motions as independent 1-D motions vy = v0y + ay t vx = v0x + ax t 1 1 y = v0y t + 2 ay t2 x = v0x t + 2 ax t2 a = À g(usually) ax = 0(usually) y = À 9:8ms À2 file:///Users/watson/Documents/Physics/1000_level/BIT_Kinematics/BIT1002_2-Dkinematics.html?id=0 Page 8 of 10 Kinematics in Two Dimensions 01/09/2008 03:00 PM = À 9:8ms Can combine these to give (e.g) an equation for the range R: What is value of t when the height y = 0? 2v0 sin ( Ò ) t= g (or t = 0: what does this mean?) Hence Range 2 v0 sin ( 2Ò ) R= g e.g. a ball is thrown at 15 ms-1, at 300 to the horizontal. How far does it go How long is it in the air? Relative Velocity Example: a woman who can swim at 2 m/s is swimming How fast in a river which flows at .7 m/s. does she swim upstream? How fast does she swim downstream? How about a round trip? A woman swims 100 m upstream at 2 m/s in a river with a current of .7 m/s, and then 100 m downstream to return to her starting point. Compared to swimming 200 m in still water, does her journey 1 . Take longer? 2 . Take a shorter time? 3 . Take exactly the same time? As a somewhat more sophisticated example: a woman who can file:///Users/watson/Documents/Physics/1000_level/BIT_Kinematics/BIT1002_2-Dkinematics.html?id=0 Page 9 of 10 Kinematics in Two Dimensions 01/09/2008 03:00 PM woman who can swim at 2 m/s is swimming in a river which flows at .7 m/s. At what angle should she swim to reach a point on the opposite bank immediately opposite the point from which she starts? Now we want to describe why things move , not just how. file:///Users/watson/Documents/Physics/1000_level/BIT_Kinematics/BIT1002_2-Dkinematics.html?id=0 Page 10 of 10

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posted: | 11/4/2009 |

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