Physics 1408-002 Principles of Physics Sung-Won Lee [email protected] Lecture 4 – Chapter 3 – January 20, 2009 Announcement I Lecture note is on the web Handout (4 slides/page) http://highenergy.phys.ttu.edu/~slee/1408/ HW Assignment #1 is placed on MateringPHYSICS, and is due by 11:59pm on Tuesday, 1/20 (“TODAY”) *** Class attendance is strongly encouraged and will be taken randomly. Also it will be used for extra credits. Announcement II SI session by Reginald Tuvilla Monday 4:30 - 6:00pm - Holden Hall 106 Thursday 4:00 - 5:30pm - Holden Hall 106 SI sessions will be at the following times and location. On-lime Homework !! To access MateringPHYSICS, you must register at http://www.masteringphysics.com/ !! Instructions are in the Student Access Kit. !! Your course ID is LEE2009 !! Once you are registered, you will be able to download the HW assignment. !! 158 out of 198 registered so far… !! If you do not have the Student Access Kit which comes with a new textbook, you can purchase one on the MasteringPHYSICS site. Please do it ASAP. Chapter 3 1.! Vectors and Scalars 2.! Addition of Vectors – Graphical Methods 3.! Subtraction of Vectors, and Multiplication of a Vector by a Scalar 4.! Adding Vectors by Components 5.! Projectile Motion Kinematics in 2, 3-Dimensions ; Vectors 3.1 Vectors and Scalars A vector has magnitude as well as direction. Some vector quantities: displacement, velocity, force, momentum A scalar has only a magnitude. Some scalar quantities: mass, time, temperature Car traveling on a road, slowing down to round the curve. The green arrows represent the velocity vector at each position.Vector"a vector is a quantity that has both magnitude (size) and direction it is represented by an arrow –! length of the arrow is the magnitude –! arrow indicates direction symbol for a vector is a letter with an arrow over it A 3.2 Addition of Vectors – Graphical Methods Vector addition by two different methods, (a) and (b). Part (c) is incorrect. 3.3 Subtraction of Vectors Then we add the negative vector: In order to subtract vectors, we define the negative of a vector, which has the same magnitude but points in the opposite direction. Subtracting two vectors: v2 – v1 Vector Subtraction"equivalent to adding negative vector "C = A – B" = A + ( – B)" Cx = Ax – Bx"Cy = Ay – By"A"B"– B"A – B"Cz = Az – Bz"3.3 Multiplication of a Vector by a Scalar A vector can be multiplied by a scalar c; the result is a vector c that has the same direction but a magnitude cV. If c is negative, the resultant vector points in the opposite direction. (see below) !V!VScalar Multiplication, again Multiplication of a vector A by a scalar ! Result is a vector B B = ! A changes magnitude not direction3.4 Adding Vectors by Components •! It is useful to use rectangular components –! These are the projections of the vector along the x- and y-axes Terminology •! Ax and Ay are the component vectors of A Any vector can be expressed as the sum of two other vectors, which are called its components. The x-component of a velocity vector is the projection along the x-axis: Vx=Vcos#$The y-component of a velocity vector is the projection along the y-axis: Vy=Vsin# 3.4 Adding Vectors by Components The components of v = v1 + v2 are vx = v1x + v2x vy = v1y + v2y 3.4 Adding Vectors by Components 3.5 Unit Vectors"•! A unit vector is a dimensionless vector with a magnitude of exactly 1. •! Unit vectors are used to specify a direction and have no other physical significance Using unit vectors, any vector can be written in terms of its components: Unit Vectors in Vector Notation •! Ax is the same as Ax and Ay is the same as Ay •! The complete vector can be expressed as Vector Addition"Analytically Cx = Ax + Bx"Cy = Ay + By"= ( Ax + Bx ) i + ( Ay + By ) j "= ( Ax i + Ay j ) + ( Bx i + By j ) "C = A + B"""""""""""""= Cx i + Cy j"""""Cz = Az + Bz"The position vector r points to the particle’s position at a particular time. We can now write r mathematically: The x-component of r is simply the x-component of the position: rx=x Similarly, the y-component of r is the y-coordinate: ry = y 3-6 Vector Kinematics The displacement vector %r connects two points on the trajectory: Recall. The average velocity of the particle moving through a displacement through a displacement %r in time %t is then 3-6 Vector Kinematics 3-6 Vector Kinematics As !t and !r become smaller and smaller, the average velocity approaches the instantaneous velocity. (a) As we take !t and !r smaller and smaller, we see that the direction of !r and of the instantaneous velocity (!r/ !t, where !t "0) is (b) tangent to the curve at P1 . We can also write the velocity in terms of its components: The magnitude of velocity is the particle’s speed: The direction is determined by the components: May be + or – But speed is always positive 3-6 Vector Kinematics 3-6 Vector Kinematics !v!vThe instantaneous acceleration is in the direction of ! = 2 – 1, and is given by: !v(a) Velocity vectors v1 and v2 at instants t1 and t2 for a particle at points P1 and P2, as in Fig. (b) The direction of the average acceleration is in the direction of !v = v2 – v1. 3-6 Vector Kinematics Using unit vectors,Generalizing the one-dimensional equations for constant acceleration: 3-6 Vector Kinematics A projectile is an object that moves in 2-dim under the influence of gravity. Free fall is one example of projectile motion, but there are others too… 3-7 Projectile Motion Examples of projectile motion. Notice the effects of air resistance. 3-7 Projectile Motion It can be understood by analyzing the horizontal and vertical motions separately. 3-7 Projectile Motion The speed in the x-direction is constant; in the y-direction the object moves with constant acceleration g. Photograph shows two balls that start to fall at the same time. The one on the right has an initial speed in the x-direction. It can be seen that vertical positions of the two balls are identical at identical times, while the horizontal position of the yellow ball increases linearly. 3-7 Projectile Motion Projectile move along parabolic trajectories. The launch angle ! is the angle of the initial velocity vi above the horizontal (i.e. x-axis). The components of vi are then 3-7 Projectile MotionProjectile Motion Diagram If an object is launched at an initial angle of !0 with the