Lecture 3. Motion in more than one dimension

Transcription

1 4/9/19 Phsics 2 Olga Dudko UCSD Phsics Lecture 3 Toda: The vector description of motion. Relative Motion. The principle of Galilean relativit. Motion in more than one dimension 1D: position is specified b a single number (= distance from the origin), + or - sign indicate either of the 2 directions 2D: man possible directions 1

2 Vectors Vectors are quantities whose full specification requires both magnitude and direction. Eamples of vector quantities: position, displacement, velocit, force, momentum Scalars are quantities that onl have magnitude. Eamples of scalar quantities:?time, mass, energ? Vector notations When handwriting: an arrow over it Printed: boldface tpe The magnitude of the vector: in italics or with the vector in the middle of absolute value lines Length of arrow = magnitude of the vector quantit Orientation of arrow = direction of vector quantit 2

3 dding vector quantities When adding two vector quantities: ü make sure ou take their directions into account; ü the units must be the same. Perform the addition geometricall or algebraicall. Geometricall vectors are placed head-to-tail: " " + (resultant) Properties of vector addition Vector addition is commutative: + = + When adding multiple vectors, just keep repeating the process until ou have included all of the vectors. Vector addition is associative: ( + ) + C = + ( + C ) R " " + D = + + C + " " + D C 3

4 Multipling vectors b scalars Multipling a vector b a positive scalar changes the magnitude of the vector: If negative scalar => direction of the vector reverses: Vector subtraction - means adding to the vector -, which " has the same magnitude as but the opposite direction: " = + " " " ( ) 1/2 " 1/2 " " + " Resolving vectors into components We described a vector b its magnitude and direction. lternativel, we can give its components. Vectors are usuall resolved into perpendicular and components. Magnitude & direction -> components: = cos = sin Components -> magnitude & direction: = tan = where is the magnitude of 4

5 Vector addition using components Consider two vectors: and. First, break them up into components (, ) (, ) then add like components ( with, with ): + ( +, + ) Unit vector notation Unit vectors allow vectors to be epressed in compact mathematical form. The unit vectors: have magnitude 1, no units, lie along the coordinate aes. z ˆ j k ˆ ˆ i ˆ i ˆ j k ˆ Vector can be rewritten as ˆ = i ˆ+ j ˆ j ˆ i 5

6 Unit vector notation Eample: epress a vector of r = 5.0 m at +45 o to the -ais in unit vector notations: = r cos = (5.0m)(cos45 ) = 3.5m = r sin = (5.0m)(sin45 ) = 3.5m = (3.5m) i ˆ+ (3.5m) j ˆ To add vectors, just add like unit vector terms: + = i ˆ+ j ˆ = i ˆ+ j ˆ = ( ˆ + ) i ˆ+( + ) j The vector description of motion To describe 2D motion, we need to redefine our variables. Position: r Displacement: Δr = r f r i Position depends on the choice of origin, displacement does not. 0 r i 0 r i (t i ) Δr r f r r f (t f ) 6

7 Velocit vector verage velocit is the change in position divided b the time interval over which it occurred: v avg = Δ r Δ t = r f - r i t f - t i? (now is a vector) = displacement Since Δt > 0 is a scalar, average velocit will alwas point in the direction of displacement. Instantaneous velocit is again the velocit of an object at an instant of time: Δ r v = lim Δt 0 Δ t = d r dt cceleration vector verage acceleration is again the change in velocit divided b the time interval: a avg = Δ v Δ t = v f - v i t f - t i Since Δ t > 0 is a scalar, average acceleration will alwas point in the direction of the change of velocit. Instantaneous acceleration is again the acceleration of an object at an instant of time: Δ v a = lim Δt 0 Δ t = d v dt 7

8 Clicker Question n object is shot verticall upward with a velocit of 12 m/s. While it is rising in the air: ) its velocit and acceleration are both upward. ) its velocit and acceleration are both downward. C) its velocit is downward and its acceleration is upward. D) its velocit is upward and its acceleration is downward. E) none of the choices above are correct. Relative Motion Relative motion: relating measurements of two different observers. Frame of reference = the object or sstem with respect to which velocit is measured. Comet Hakutake, Usuall we make measurements with respect to a stationar frame (e.g., the ground). ut sometimes motion is described most simpl in a moving frame. The ke to relative motion: keeping track of subscripts (although the book doesn t). 8

9 Relative Motion Consider two cars ( and ) moving with different speeds in different directions. Position of car as measured b E: Position of car as measured b E: Position of car as measured b : The positions are related: r E + r = r E r E r E r r = r E r E r E We can then relate velocities b: v = v E v E Observer (E) stationar with respect to Earth r E Observer () sitting in car Eample Relative Motion boat s speed in still water is 20.0 km/hr. If the boat is to travel directl across a river whose current has a speed of 12.0 km/hr, at what upstream angle must the boat head? Solution First, define a coordinate sstem. Choose the direction of the current as + and the direction across the river (where the boat wants to travel) as +. current oat wants to travel directl across the river River current 9

10 Solution We know that Relative Motion v bw = v bs - v ws Object: boat () Stationar frame: shore (S) Moving frame: water (W) quick diagram of the situation: v bs v ws v bw = sin = v ws sin v bw 12.0 km/hr 20.0 km/hr = = sin -1 (0.600) = 36.9º The boat must point 36.9 o as measured from the direction directl across the river. Relative Motion: the book notations In the book s notation: v = velocit of an object with respect to some reference frame, S. V = velocit of the reference frame S moving with respect to reference frame S. v = velocit of the object with respect to reference frame S : v = v - V 10

11 Principle of Galilean relativit What about the acceleration of an object in two different reference frames? We generall consider onl reference frames that are not themselves accelerating - inertial reference frames. Dialogue Concerning the Two Chief World Sstems (1632) v = v - V => a = a - 0 The principle of Galilean relativit: The laws of motion are the same in all inertial reference frames. For Net Time: Register our i>clicker b 4/15 at Keep studing for Quiz 1 (Ch1, 2) Quiz 1: Frida 4/12, 5:00-5:50 PM, Galbraith Hall 242 ( Location for section onl ) Check our UCSD for our 3-digit quiz code, know this code b heart. rrive at least 5 min before start (i.e., b 4:55 pm). Closed book eam. You ma bring a half-a-standard-page of notes (OK to write on both sides). ring a scantron No. F-289-PR-L (red) & #2 pencil. ring standard calculator. No laptop, no cell phone etc. ring our picture ID proctors will check identit. cademic integrit rules will be rigorousl enforced. 11