ch3 - Kinematics in 2D; Vectors Chapter Outline GENERAL...

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Unformatted text preview: Kinematics in 2D; Vectors Chapter Outline GENERAL PHYSICS Vectors and Scalars PHY 302K Addition and Subtraction of Vectors Chapter 3: Kinematics in 2D; Vectors Components of a Vector Displacement, Velocity, and Acceleration in 2D Maxim Tsoi Motion in 2D; Projectile Motion Relative Velocity Physics Department, The University of Texas at Austin http://www.ph.utexas.edu/~tsoi/302K.htm 302K - Ch.3 302K - Ch.3 Vector and Scalar Quantities Some Properties of Vectors Magnitude and Direction Equality of two vectors • Two vectors A and B are equal if they have the same magnitude • A scalar quantity is completely specified and point in the same direction (A=B, A||B) by a single value with an appropriate unit and has no direction (e.g., temperature, volume, mass, speed, time intervals) • A vector quantity is completely specified by a number and appropriate units plus a direction (e.g., velocity, A, A , A, |A| displacement) 302K - Ch.3 302K - Ch.3 Some Properties of Vectors Some Properties of Vectors Laws of vector addition • The rules for adding vectors are conveniently described by graphical methods • To add vector B to vector A: Adding more than two vectors • The rules for adding vectors are conveniently described by graphical methods • To add four (N) vectors A, B, C, and D: • Draw 2nd (3rd, etc.) vector with its tail • Draw B with its tail starting from the starting from the tip of the 1st (2nd, etc) tip of A • R=A+B+C+D is the vector drawn from • The vector R=A+B is the vector the tail of the first vector to the tip of drawn from the tail of A to the tip of B the last vector Total displacement is the vector sum of the individual displacements 302K - Ch.3 Total displacement is the vector sum of the individual displacements 302K - Ch.3 1 Some Properties of Vectors Some Properties of Vectors Commutative law of addition Associative law of addition • The rules for adding vectors are conveniently described by graphical methods • When two vectors are added, the sum is • The rules for adding vectors are conveniently described by graphical methods • When three or more vectors are added, their independent of the order of the addition: sum is independent of the way in which the individual vectors are grouped together: • A+B=B+A • A+(B+C)=(A+B)+C Total displacement is the vector sum of the individual displacements 302K - Ch.3 • All vectors must have the same units and type of quantity Total displacement is the vector sum of the individual displacements 302K - Ch.3 Some Properties of Vectors Some Properties of Vectors Negative of a vector Subtracting vectors • The rules for adding vectors are conveniently described by graphical methods • The negative of the vector A is defined as the • The rules for adding vectors are conveniently described by graphical methods • Make use of the definition of the negative of a vector that when added to A gives zero for the vector vector sum: • We define subtraction A-B as vector –B added • A+(-A)=0 to vector A: • A – B = A + (-B) • The vectors A and -A have the same magnitude but point in opposite directions A -A 302K - Ch.3 302K - Ch.3 Some Properties of Vectors Components of a Vector Multiplying a vector by a scalar Adding vectors using their projections • The rules for adding vectors are conveniently described by graphical methods • The projections of vector A along coordinate axes are called the components Ax and Ay of the vector • m – positive the product mA is a vector that has the • Any vector can be completely described by its same direction as A and magnitude mA components • -m – negative the product -mA is directed opposite A Ax A cos A 5A B Ay A sin -½ B 2 2 A Ax Ay Ay Ax tan 1 A=Ax+Ay the signs of the components depend on the angle a vector A can be specified either with Ax and Ay or with A and ` 302K - Ch.3 302K - Ch.3 2 Coordinate Systems Components of a Vector Using components to add vectors Description of a location in space Rx Ax Bx • Cartesian coordinate system (rectangular coordinates) (x, y) R y Ay B y • Polar coordinate system (r, ) x r cos y r sin tan y x r x2 y2 2 2 R Rx R y Example 3.2 302K - Ch.3 Components of a Vector Ax Bx 2 Ay B y 2 tan Ry Rx Ay B y Ax B x 302K - Ch.3 Position, Velocity, and Acceleration Rotation of a coordinate system Position and displacement vectors • Motion of a particle is completely known if its position is • In many applications/problems it is convenient to express the components known as a function of time of a vector A in a coordinate system having axes that are not horizontal and vertical but are still perpendicular to each other • Position of a particle is described by its position vector r • In different coordinate systems the components of the same vector must be modified accordingly r is drawn from the y’ origin of a coordinate x’ system to the particle Ax A cos Ax ' A Ay A sin Ay ' 0 • Displacement vector is the difference between its final position vector and its initial position vector: r r f ri 302K - Ch.3 302K - Ch.3 Position, Velocity, and Acceleration Position, Velocity, and Acceleration Average and instantaneous velocity vectors • Average velocity of a particle during time interval t is the displacement of the particle divided by the time interval: r v t Average and instantaneous acceleration vectors • Average acceleration of a particle is the change in its instantaneous velocity vector divided by the time interval: • Instantaneous velocity is the limit of the average velocity as t approaches zero: r v lim t 0 t v f v i v a t f ti t • Instantaneous acceleration is the limit of the average acceleration as t approaches zero: v a lim t 0 t • The magnitude of the instantaneous velocity Various changes can occur when a particle accelerates: magnitude and/or direction! vector v=|v| is called speed 302K - Ch.3 302K - Ch.3 3 2D Motion with Constant Acceleration Projectile Motion Kinematic equations in components v f vi a t e. g., a baseball in motion Assumptions v xf v xi a x t v yf v yi a y t r f ri v i t 1 a t 2 2 • The free-fall acceleration g is constant over the range of motion and is directed downward 2 1 x f xi v xi t 2 a x t 2 y f yi v yi t 1 a y t 2 • The effect of air resistance is negligible • 2D motion at constant acceleration is equivalent to two independent motions – in x and y directions – having constant accelerations ax and ay Example 3.5 302K - Ch.3 The path of a projectile, called trajectory, is always a parabola Example 3.6 302K - Ch.3 Projectile Motion Projectile Motion Trajectory of a projectile Trajectory of a projectile • Vector expression for the position vector of the projectile: • Reference frame: y is vertical and positive is upward ay=-g, ax=0 r f ri v i t 1 g t 2 2 • At t=0 the projectile leaves the origin (xi=yi=0) with velocity vi which makes an angle i with x xi 0 v xi v i cos i yi 0 v yi v i sin i 2 1 x f xi v xi t 2 a x t 2 y f yi v yi t 1 a y t 2 x f v xi t v i cos i t y f v yi t 1 a y t 2 v i sin i t 1 gt 2 2 2 • Projectile motion is superposition of two motions: 2 g y tan i x 2 2v cos 2 x i i constant-velocity motion in the horizontal direction y=ax-bx2 parabola free-fall motion in the vertical direction 302K - Ch.3 302K - Ch.3 Projectile Motion Projectile Motion Horizontal range and maximum height of a projectile Horizontal range and maximum height of a projectile • Various trajectories of a projectile launched with the same speed at different angles • Two points of the projectile trajectory are especially interesting to analyze: The peak point (R/2, h) The landing point (R, 0) • R – horizontal range • h – maximum height v yf v yi a y t 0 v i sin i gt A h v yi t 1 a y t 2 v i sin i 2 tA v i sin i 1 v i sin i 2 g g g R v xi t B v i cos i 2t A v i cos i 302K - Ch.3 v i sin i g 2 v 2 sin 2 i i 2g 2v i sin i v i2 sin 2 i g g R Example 3.7 v i2 sin 2 i g RMAX v i2 g 302K - Ch.3 4 Relative Velocity and Relative Acceleration Relative Velocity and Relative Acceleration Observations made in different frames of reference • Observers in different frames of reference may measure different Observations made in different frames of reference • Observers in different frames of reference may measure different positions, velocities, and accelerations for a given particle positions, velocities, and accelerations for a given particle • Two observers moving relative to each other generally • Two observers moving relative to each other generally do not agree on the outcome of a measurement 302K - Ch.3 do not agree on the outcome of a measurement 302K - Ch.3 Relative Velocity and Relative Acceleration SUMMARY Galilean transformation equations • A particle located at point (A) • SCALAR quantities have only a numerical value; no direction • Two reference frames S and S’ • VECTOR quantities have both magnitude (>0) and direction; obey • S’ moves to the right relative the laws of vector addition to S with constant v0 • R=A+B is the vector drawn from the tail of A to the tip of B • at time t: r ' r v0 t r' r v0 t t v' v t t Vectors • The x (y) component Ax (Ay) of the vector A is equal to the projection of A along the x (y) axis of a coordinate system v' v v 0 • Components of a vector A: • The acceleration of a particle measured Ax A cos Ay A sin by an observer in one frame of reference is the same as that measured by any other observer moving with constant a' a velocity relative to the first frame 302K - Ch.3 302K - Ch.3 SUMMARY Kinematics in 2D • If a particle moves with constant acceleration a and velocity vi and position ri at t=0, its velocity and position vectors at some later time t are: v f vi a t r f ri v i t 1 a t 2 2 • For 2D motion in xy plane component expressions of these two vector expressions describe two independent 1D motions in x and y directions • Projectile motion type of 2D motion under constant acceleration where ax=0 and ay=-g • Velocity of a particle in different frames of reference moving with constant velocity relative to each other are related by: v ' v v0 302K - 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