1 Chapter 3 Kinematics I: rectilinear motion Kinematics is the theory about the description of motion. Physical theories are creations of the human intellect; they must be invented rather than discovered. Motion implies change, and it is change make life-and physics-visible and interesting. 2 §3.1 Position and displacement of a particle 1. Several concepts Frame of reference —any object that is chosen for reference of a motion. Particle—a single mass point without shape. An object whose part are all move in exactly the same way can be treated as a particle; or a complex object can be treated as a particle if there are no internal motions or the internal motions can be neglected for the problem which you are discussing. Coordinate system—the abstract of reference 2. The position vector and displacement vector of a particle in rectilinear motion x O ixr ii ? = r x O ixr ff ? = r Note: ixixxrrr ifif ?? )( ?? =?=?= rrr Displacement—the change of the position vector §3.1 Position and displacement of a particle 1 The magnitude of the displacement is not necessarily equal to the total distance traveled by the particle during the time interval ?t . .zeronotbemay,0when sr ?? = r s? 3 Note: 3 The position of a moving particle is a function of time, r r ?2 is independent of the the specific coordinate system we choose. itxtr ? )()( = r x t Agraphofx versus t §3.1 Position and displacement of a particle 4the path of the particle is a straight line. 2. Instantaneous velocity and instantaneous speed §3.2 The speed and velocity of a moving particle 1. Average speed and average velocity Define: t s v ? ? = ave t r v ? ? r r = ave aveave vvrs rr Q ≠∴≠ ?? t x 1 t? 2 t? )(tx )( 1 tx )( 2 tx itxtrr i ? )()( == rr ittxttrr f ? )()( ?? +=+= rr 4 itxttxtrttrr ? )]()([)()( ?+=?+= ??? rrr t itxttx t r v ? ? ? ? ? )]()([ ave ?+ == r r i t x t r t trttr vtv tt ? d d d d)()( limlim)( s0 ave s0 == ?+ == →→ rrr rr ? ? ?? Instantaneous velocity: The component of the velocity: t tx tv x d )(d )( = §3.2 The speed and velocity of a moving particle v t s t r t r v srr tt t ====∴ == →→ → d d d d limlim ddlim s0 ave s0 s0 rr r rr Q ? ? ? ?? ? Instantaneous speed: t ts t s vtv tt d )(d limlim)( s0 ave s0 === →→ ? ? ?? The direction of the velocity: 0 d )(d > t tx 0 d )(d < t tx same as , i ? i ? opposite of §3.2 The speed and velocity of a moving particle 5 §3.3 The acceleration in rectilinear motion Define: t vv t v a if ?? ? rr r r ? == ave 1. Average acceleration i t tvtv iaa ixfx x ? )()( ? aveave ? ? == r 2. Instantaneous acceleration if ixvttvvvv ittvvitvv xxif xfxi ? )]()([ ? )(, ? )( ?+=?= +== ?? ? rrr rr then t tv t itvttv t v a xx tt d )(d ? )]()([ limlim s0s0 rr r = ?+ == →→ ? ? ? ? ?? or i t tx t tr i t tv t tv a x ? d )(d d )(d d )(d d )(d 2 2 2 2 ==== r r r r The instantaneous acceleration of a particle is the time rate of change of the velocity vector or the first derivative of the instantaneous velocity vector with respect to time;or the second derivative of the instantaneous position vector with respect to time. §3.3 The acceleration in rectilinear motion 6 2 2 d )(d d )(d )( t tx t tv ta x x == The component of the acceleration: §3.3 The acceleration in rectilinear motion 3. How to detect and measure the instantaneous acceleration Accelerometer—a plum bob Direction of deviation of plum bob from vertical is opposite to the direction of the acceleration of the particle in the horizontal plane. a r θ a r θ §3.4 Rectilinear motion with a constant acceleration 1. Some rules §3.4 Rectilinear motion with a constant acceleration 1ignore the effects of air resistance 2the origin of the coordinate could be chosen discretionarily 3write the constant acceleration as 4choose the initial time instant to be iaa x ? = r 0= i t tttt if =?=? 5let 00 )0()()0()( xxixi vvtvxxtx ==== 7 2. Rectilinear motion with a constant acceleration from x x a t tv = d )(d We have tavtv tatv xxx x tv v x x x += = ∫∫ 0 t 0 )( )( d)(d 0 Likewise, from )( d )(d tv t tx x = We have 2 00 0 0 t 0 )( 2 1 )( d)(d)(d 0 tatvxtx ttavtvtx xx t xxx tx x ++= +== ∫∫∫ §3.4 Rectilinear motion with a constant acceleration One –dimensional motion with constant acceleration: itvtvtavtv xxxx ? )()(where)( 0 =+= r itxtrtatvxtx xx ? )()(where 2 1 )( 2 00 =++= r iaaa xx ? whereconstant == r §3.4 Rectilinear motion with a constant acceleration Eliminate t in equations about v x and x )(2 0 2 0 2 xxavv xxx ?=? 8 §3.5 Geometric interpretations 1. The change in the position vector component ∫∫ ==?= f i f i t t x tx tx x ttvxxttvx d)(dd)(d )( )( 2. The change in the velocity component ∫∫ ==?= f i fx ix t t x tv tv xxxx ttavvtav d)(ddd )( )( tt )(tv x )(tv x i t i tf t f t Constant speed motionAccelerated motion t x v x d d = t v a x x d d = tt )(ta x )(ta x i t i tf t f t Constant acceleration motionAccelerated motion tt )(ta x )(tv x Accelerated motion §3.5 Geometric interpretations 3. What does the negative areas mean? 9 x a x v x §3.5 Geometric interpretations §3.5 Geometric interpretations A rebounding ball bearing 10 Exercise 1 A geologist at the top of a 100-meter-deep crevasse cannot resist the temptation to hurl a gneiss rock down to the bottom. The rock has an initial downward speed of 10.0m/s as it leaves the geologist’s hand when t =0s. Find a. The time interval during which the particle is in flight, called the time of flight; and b. The speed of the particle at the instant just before impact. R. P96 A rocket is launched from rest from an underwater base a distance of 125m below the surface of a body of water. It moves vertically upward with an unknown but assumed constant acceleration , and it reaches the surface in a time of 2.15s. When it breaks the surface its engines automatically shut off and it continues to rise. What maximum height does it reach? Exercise 2 (H. P29)