VISUAL PHYSICS ONLINE KINEMATICS DESCRIBING MOTION

Transcription

1 VISUAL PHYSICS ONLINE KINEMATICS DESCRIBING MOTION The language used to describe motion is called kinematics. Surprisingl, ver few words are needed to full the describe the motion of a Sstem. Warning: words used in a scientific sense often have a different interpretation to the use of those words in everda speech. The language needed to full describe motion is outlined in Table 1. In analsing the motion of an object or collection of objects, the first step ou must take is to define our frame of reference. 1

2 Frame of reference Observer Origin O(0,0, 0) reference point Cartesian coordinate aes (X, Y, Z) Unit vectors iˆˆj k ˆ Specif the units 2

3 Phsical Smbol Scalar S.I. Other quantit vector unit units time minute t scalar: 0 second s hour da ear time interval minute t dt scalar: 0 second s hour da ear distance travelled d d scalar: 0 metre m mm km displacement s r (position vector) s, s, vector m mm km average speed speed (instantaneous) average velocit velocit (instantaneous) average acceleration acceleration (instantaneous) v avg v scalar: 0 scalar: 0 m.s -1 km.h -1 m.s -1 km.h -1 v avg vector m.s -1 km.h -1 v v, v vector m.s -1 km.h -1 a avg vector m.s -2 a a, a vector m.s -2 Table1. Kinematics: terminolog for the complete description of the motion of a Sstem in a plane. 3

4 POSITION DISTANCE DISPLACEMENT Consider two tractors moving about a paddock. To stud their motion, the frame of reference is taken as a XY Cartesian Coordinate Sstem with the Origin located at the centre of the paddock. The stationar observer is located at the centre of the paddock and the metre is the unit for a distance measurement. The positions of the tractors are given b their X and Y coordinates. Each tractor is represented b a dot and the tractors are identif using the letters A and B. Fig. 1. Frame of reference used to analse the motion of the two tractors in a plane. 4

5 Both tractors move from their initial position at the Origin O(0, 0) to their final position at (60, 80) as shown in figure (2). Tractor A follows the red path and tractor B follows the blue path. Event 1 corresponds to the initial instance of the tractor motion and Events 2 and 3 are the instances when the tractors each their final position. Fig. 2. RED path of tractor A and BLUE path of tractor B. Both tractors start at the Origin O(0, 0) and finish at the point (60 m, 80m). 5

6 Event 1 t1 0 s tractors A and B at their initial positions Position of tractors Sstem A A 1 0 m A 1 0 m Sstem B B1 0 m B1 0 m N.B. The first subscript is used to identif the Sstem and the second the time of the Event. Remember we are using a model in our model it is possible for both tractors to occup the same position at the same time. Event 2 t2 100 s tractor A arrives at its final position Sstem A A m A m Event 3 t3 180 s tractor B arrives at its final position Sstem B B m B m Distance travelled Using figure (2), it is simple matter to calculate the distance d travelled b each tractor Sstem A d m 320 m A Sstem B d A d B d m 300 m B 6

7 Displacement Position Vector The change in position of the tractors is called the displacement. The displacement onl depends upon the initial position (Event 1) and final position (Events 2 and 3) of the Sstem and not with an details of what paths were taken during the time interval between the two Events. The displacement is represented b the position vector and is drawn as a straight arrow pointing from the initial to the final position as shown in figure (3). The tractors start at the same position and finish at the same position, therefore, the must have the same displacement, even though the have travelled different distance in different time intervals. Fig. 3. The displacement of the tractors shown as a position vector. 7

8 From figure (3), it is obvious the values for the component of the position vector are s s s s 60 m s 80 m A B The magnitude of the displacement is s s s s m 100 m The direction of the displacement is given b the angle o o that the position vector makes with the X ais s 80 atan atan 53.1 s 60 o N.B. The distance travelled (scalar) and the displacement (vector) are ver different phsical quantities. 8

9 The displacement gives the change in position as a vector, hence we can write the displacements for Sstem A and Sstem B as 2 _ 1_ 2 _ 1_ 3_ 1_ 2 _ 1_ s s s iˆ s s ˆj 60 0 iˆ 80 0 ˆj 60 iˆ 80 ˆj A A A A A s s s iˆ s s ˆj 60 0 iˆ 80 0 ˆj 60 iˆ 80 ˆj B B B B B Multiple subscripts look confusing, but, convince ourself that ou can interpret the meaning of all the smbols. Once ou get our head around it, using multiple subscript means that ou can conve a lot of information ver precisel. s X component of the displacement of Sstem A at A2_ the time of Event 2. 9

10 AVERAGE SPEED AVERAGE VELOCITY Time is a measure of movement Aristotle ( BC) The time interval between Event 1 and Event 2 be given b t t2 t1 t is one smbol delta t t In this time interval, the change in position is given b distance travelled d scalar displacement s vector The definition of the average speed is (1) vavg d t scalar: zero or a positive constant The definition of average velocit is (2) vavg s t vector Warning: the magnitude of the average velocit vavg necessaril equal to the average speed vavg is not during the same time interval. The same smbol is used for average velocit and average speed hence ou need to careful in distinguishing the two concepts. The average speed and average velocit are different phsical quantities. 10

11 From the information for the motion of the two trajectories of tractors A and B shown in figure (3), we can calculate the average speed and average velocities of each Sstem. Sstem A (tractor A) red path Time interval between Event 1 and Event 2 t t2 t1 (100 0) s 100 s Distance travelled d d A 300 m Displacement s s 60iˆ 80 ˆj A s s 100 m 53.1 A A o Using equations (1) and (2) Average speed v avg d 300 m.s 3.00 m.s t scalar Average velocit vector / same direction as the displacement s 60iˆ80 ˆj v m.s ˆ ˆ avg i j m.s t v avg s 100 m.s 1.00 m.s t o

12 Using the components of the average velocit magnitude v m.s 1.00 m.s avg direction 0.8 atan o The average speed and average velocit are different phsical quantities. Sstem B (tractor B) blue path Time interval between Event 1 and Event 3 t t3 t1 (180 0) s 180 s Distance travelled d d B 300 m Displacement s s 60iˆ 80 ˆj B s s 100 m 53.1 B A o s B s A Using equations (1) and (2) Average speed v avg d 300 m.s 1.67 m.s t scalar Average velocit vb va 12

13 INSTANTANEOUS SPEED INSTANTANEOUS VELOCITY On most occasions, we want to know more than just averages, we want details about the dnamic motion of a particle on an instant-b-instant basis. The definition of average velocit is (2) vavg s t vector If we make the time interval t smaller and smaller, the average velocit approaches the instantaneous value at that instant. Mathematicall it is written as v limit t 0 s t This limit is one wa of defining the derivative of a function. The instantaneous velocit is the time rate of change of the displacement (3) ds v definition of instantaneous velocit dt 13

14 In terms of vector components for the displacement and velocit s s iˆ s ˆj s s s v limit limit iˆ ˆ j t0 t t0 t t (4) v v iˆ v ˆj As the time interval approach zero t 0, the distance travelled approaches the value for the magnitude of the displacement d s. Therefore, the magnitude of the instantaneous velocit is equal to the value of the instantaneous speed. This is not the case when referring to average values for the speed and velocit. When ou refer to the speed or velocit it means ou are talking about the instantaneous values. Therefore, on most occasions ou can omit the word instantaneous, but ou can t omit the term average when talking about average speed or average velocit. 14

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16 ACCELERATION An acceleration occurs when there is a change in velocit with time. Object speeds up Object slows down Object change s its direction of motion The average acceleration of an object is defined in terms of the change in velocit and the interval for the change a (6) avg v definition of average velocit t The instantaneous acceleration (acceleration) is the time rate of change of the velocit, i.e., the derivative of the velocit gives the acceleration (equation 7). (7) dv a definition of instantaneous acceleration dt In terms of vector components for the velocit and acceleration v v iˆ v ˆj v v v limit limit iˆ a ˆ j t0 t t0 t t (4) a a iˆ a ˆj 16

17 VISUAL PHYSICS ONLINE If ou have an feedback, comments, suggestions or corrections please Ian Cooper School of Phsics Universit of Sdne 17