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1、Chapter 3 Motion in Two and Three Dimensions,Main Points of Chapter 3,Position and displacement Velocity and acceleration Trajectories Motion with constant acceleration Projectile motion Circular motion Relative motion,3-1 Position and Displacement,1. Position Vector,Reference Object, the Cartesian
2、unit vectors,magnitude,direction,Cartesian coordinate system,(the Equations of Motion),In Cartesian coordinate system,Example: A particle moves in the xy plane so that its x and y coordinates vary with time according to x(t)=Rcost and y(t)=Rsint . Find the equation of its path.,x,y,z,O,A t,B t+t,Sup
3、pose the particle is located at position A at time t1, and it moves along its path to position B at time t2(t +t). We define the displacement vector as the change in position that occurs in this interval :,2. Displacement,trajectory,Discussion:,(2) Position vectors depend on origin of coordinate sys
4、tem, but displacement vector does not,(1) Displacement is a vector. It is not the same as the distance.,O,trajectory,3-2 Velocity and Acceleration,1. Velocity,trajectory,Average Velocity,Instantaneous Velocity,O,s,A,B,Instantaneous speed,trajectory,Direction,Magnitude,Magnitude,Direction,Where,Discu
5、ssion:,1) It is a vector, instantaneous and relative quantity.,2) Note the difference between the velocity and speed.,In Cartesian coordinate system,(a) first; (b) third,ACT The figure shows a circular path taken by a particle. If the instantaneous velocity of the particle is , through which quadran
6、t is the particle moving when it is traveling (a) clockwise and (b) counterclockwise around the circle? For both cases, draw on the figure.,(draw tangent to path, tail on path),S=S(t),In the intrinsic coordinate system,Know the trajectory of a particle,Its origin must on the trajectory, Equation of
7、Motion,In the planar polar coordinate system,y,x,z,o,A,B,2. Acceleration,In Cartesian coordinate system,Magnitude,Direction,Discussion:,1) It is a vector, instantaneous and relative quantity.,2) It describes the change in velocity with time (either magnitude or direction).,In the intrinsic coordinat
8、e system,Angle between,curvature,Radius of curvature,The change of magnitude of velocity,The change of direction of velocity,Tangential acceleration,Normal acceleration,Acceleration,Magnitude,Direction,Recap in 3-D Kinematics,The position, velocity, and acceleration of a particle in 3 dimensions can
9、 be expressed as:,Recap in 2-D Kinematics,3-3 Motion with Constant Acceleration,When an object moves with constant acceleration, it can move only in a line or a plane. The plane is formed by the initial velocity vector and the acceleration vector.,For constant acceleration we can integrate to get:,3
10、-4 Projectile Motion,A projectile that moves under the effect of gravity,Trajectory for projectile motion lies in the plane formed by the initial velocity vector and the acceleration vector,A stroboscopic photograph of an orange golf ball bouncing off a hard surface. Between impacts, the ball has pr
11、ojectile motion.,The Trajectory is a parabola,In projectile motion, the horizontal motion and the vertical motion are independent of each other (known from experiment) .We can separate it into horizontal and vertical components.,Example: Throwing a baseball (neglecting air resistance) with an initia
12、l velocity elevation angle Acceleration is constant (gravity) Choose y axis up: ay = -g Choose x axis along the ground in the direction of the throw,Range is distance object travels over level ground,Range depends on launch angle and the initial speed,Flight Time,Maximum Height,ACT Figure shows thre
13、e paths for a football kicked from ground level. Ignoring the effects of air on the flight, rank the paths according to (a) time of flight, (b) initial vertical velocity component, (c) initial horizontal velocity component, and (d) initial speed, greatest first.,(a) All tie,(b) All tie,(c) 3,2,1,(c)
14、 3,2,1,Example: Shooting the Monkey,Where does the zookeeper aim if he wants to hit the monkey? ( He knows the monkey will let go as soon as he shoots ! ),If there were no gravity, simply aim at the monkey,Example: Shooting the Monkey,With gravity, still aim at the monkey!,Example: Shooting the Monk
15、ey,x = x0 y = -1/2 g t2,This may be easier to think about. Its exactly the same idea!,x = v0 t y = -1/2 g t2,Example: Shooting the Monkey,CAI,3, 2, 1; (b) 1, 2, 3; (c) all tie; (d) 6, 5, 4,ACT In Figure below, a cream tangerine is thrown up past windows 1, 2, and 3, which are identical in size and r
16、egularly spaced vertically. Rank those three windows according to (a) the time the cream tangerine takes to pass them and (b) the average speed of the cream tangerine during the passage, greatest first. The cream tangerine then moves down past windows 4, 5, and 6, which are identical in size and irr
17、egularly spaced horizontally. Rank those three windows according to (c) the time the cream tangerine takes to pass them and (d) the average speed of the cream tangerine during the passage, greatest first.,Two-kind Problems in Kinematics,1. Differential Problems,Solution,(a),(b),(c),When t =2 s,h,x,E
18、xample The figure shows a rope which hangs over a pulley and fixes its one end with a boat. The other end is pulling with constant speed u by a man who stands on a platform(height is h above the face of the lake). The length between the boat and the pulley is l 0 at the beginning. What is the veloci
19、ty and acceleration of the boat at time t ?,O,Solution,The coordinate system is given in right figure.,Velocity,Acceleration,2. Integral problems,Initial Conditions,Example,Solution,By using the initial condition, we have,Question: If acceleration , repeat calculation above.,A particle moves in an x
20、y plane with acceleration . If its velocity and position vector are and respectively when t =0. Please give and changing with time.,ACTA particle moves along the positive x axis with acceleration a =4x (SI) (x is position coordinate). If its initial speed and position coordinate are v0 =2 m/s and x0
21、 =1 m respectively. Find the speed when time t =1 s .,Solution,Question: If acceleration , repeat calculation above.,Example: A golf ball undergoing the projectile motion, the initial velocity of magnitude and elevation angle are , find the radius of curvature at the starting point o.,Solution:,o,A,
22、B,Example: Figure below shows a trajectory of a projectile. A, B are starting point and landing point, find the meaning of the three integrals. t1 is the flight time.,trajectory,3-5 Circular Motion,1. Angular Coordinate and Angular Displacement,In the Polar Coordinate System,r,The Equation of motion
23、,For circular motion,Angular Displacement,Angular Coordinate,2. Angular Velocity,Motion along a circular path, or a segment of a circular path,Tangential acceleration,Centripetal acceleration,4. The relations between angular quantities and linear quantities,3. Angular Acceleration,Circular Motion,Un
24、iform Circular Motion,Period,Frequency,Recap: The relations between angular quantity and linear quantity,(a) yes; (b) no; (c) yes,ACT a) Is it possible to be accelerating while traveling at constant speed? Is it possible to round a curve with (b) zero acceleration and (c) a constant magnitude of acc
25、eleration?,Example A wheel with radius R rolls along a road, with the center moving in a straight line at a uniform speed of v. What are the position vector, the velocity vector, and the acceleration vector of a fixed point on the rim of the wheel relative to a fixed point on the straight line follo
26、wed by the wheel on the road?,We will use a coordinate system with the origin at the point A on the ground that was in contact at t = 0 with the point B on the rim, as shown in the diagram,Because the wheel is rolling, the length of arc from the current contact point to point B must equal the horizo
27、ntal distance the center traveled:,The position of B can be treated as the sum of three displacements:,Solution,A particle is moving in a circle of radius 2m counterclockwise on a level frictionless table. It has an angular velocity that changes square of time; that is , =kt2, in which k is an const
28、ant. If its velocity is 32m/s when t=2s , calculate its velocity and acceleration when t=0.5 s respectively.,Question: If its angular acceleration a=kt2, calculate its velocity and acceleration when t=0.5 s respectively.,Example,Solution,(b) Suppose the angle between acceleration and radius is 45o w
29、hen time t,(b) When =? ，the angle between acceleration and radius is 45o.,(a) t =2s an , a, the magnitude of ;,A particle moves in a circle of radius 0.1m assuming that the equation of motion is given by,(a),ACT,Solution,A puck is moving in a circle of radius R with initial velocity v0. The angle be
30、tween the direction of velocity and the direction of acceleration is a constant. Write expressions for the magnitude of its velocity as function of time.,Example,Solution,Observers moving with respect to each other will describe motion differently:,3-6 Relative Motion,Rain appears to fall at angle q
31、 when the observer is moving in the horizontal directions.,The velocity of an object measured by an observer depends not only on the motion of the object, but also on the motion of the observer.,A man on a train tosses a ball straight up in the air. View this from two reference frames:,1. Basic conc
32、eptions,One object,Two reference frame,(translating relatively),Absolute reference frame S Relative reference frame S,Three-kind motion,position of P with respect to S,position of P with respect to S,position of O with respect to O,The position of P as measured by S is equal to the position of P as
33、measured by S plus position of S as measured by S.,?,NOTES: Two reference frames are supposed when t=t=0,2. The Law of Transformation of Velocity,?,Observers disagree on velocities but agree on accelerations if relative velocity is constant,Consider a problem An airplane flying on a windy day.,The p
34、lane is moving north in the IRF attached to the air: Vp, a is the velocity of the plane w.r.t. the air.,But suppose the air is moving east in the IRF attached to the ground. Va,g is the velocity of the air w.r.t. the ground (i.e. wind).,What is the velocity of the plane in an IRF attached to the gro
35、und? Vp,g is the velocity of the plane w.r.t. the ground.,Vp,g = Vp,a + Va,g Is a vector equation relating the airplanes velocity in different reference frames.,Example The compass in an airplane indicates that it is headed due east; its air speed indicator reads 215 km/h. A steady wind of 65 km/h i
36、s blowing due north. (a) What is the velocity of the plane with respect to the ground? (b) If the pilot wishes to fly due east, what must be the heading? That is, what must the compass read?,Solution,Velocity of wind with respect to the ground,Velocity of the plane with respect to wind,The orientati
37、on north of due east,(a),?,Velocity of wind with respect to the ground,Velocity of the plane with respect to wind,Velocity of plane with respect to the ground,The orientation south of due east,(b),ACT You are swimming across a 50m wide river in which the current moves at 1 m/s with respect to the sh
38、ore. Your swimming speed is 2 m/s with respect to the water. You swim across in such a way that your path is a straight perpendicular line across the river. How many seconds does it take you to get across ? (a) (b) (c),2 m/s,1 m/s,50 m,Act Tractor 1,Which direction should I point the tractor to get
39、it across the table fastest? A) 30 degrees left B) Straight across C) 30 degrees right,Act Tractor 2 (moving table),Which direction should I point the tractor to get it across the table fastest? A) 30 degrees left B) Straight across C) 30 degrees right,*Example Suppose that three people are at the a
40、pexes of a right triangle with sides of length l. They begins to chase the front one clockwise with speed v relative the surface of the earth. Find the distance they traveled when they meet.,O,Solution,The velocity of B with respect to C s,In the frame of the surface of the earth s,Summary of Chapter 3,Position vector of particle moving in space:,Velocity:,Instantaneous acceleration:,Summary of Chapter 3, cont.,For constant acceleration:,In absence of air resistance, projectile moves with constant acceleration near Earths surface,Summary of Chapter 3, cont.,Circular motion,Relative motion,