Chapter 3 Motion in two or three dimensions Lecture by Dr. Hebin Li
Announcements As requested by the Disability Resource Center: In this class there is a student who is a client of Disability Resource Center. Would someone please volunteer to share his or her notes with that student for the semester? If willing, please meet with me after this class. Then, visit Disability Resource Center, copy your notes there at no cost, and leave them there for the student to retrieve. Or, Disability Resource Center can provide special carbonless (NCR) paper that provides immediate copies of your notes. Thank you. Lecture slides (for Monday class) and recitation outline (for Wednesday class) can be downloaded at http://faculty.fiu.edu/~hebli/teaching/phy-2048-schedule/ Pop quizzes can be in either lectures (Monday) or recitations (Wednesday). Since they are random, I cannot tell you the exact dates.
Assignment Due at 11:59pm on Sunday, September 14 Homework assignment on Masteringphysics Due before the lecture on Monday, September 15 Read Chapter 4 (p 104~126) Read Chapter 5 (p 134~161)
Goals for Chapter 3 To use vectors to represent the position of a body To determine the velocity vector using the path of a body To investigate the acceleration vector of a body To describe the curved path of projectile To investigate circular motion To describe the velocity of a body as seen from different frames of reference
Describe motion in general What determines the trajectory of a basketball and where the basketball lands? A satellite is flying in a circular orbit at constant speed, is it accelerating? How is the motion of a particle described by different moving observers? We need to extend the description of motion to two and three dimensions.
Going from 1D to 2D/3D Position 1D x 2D/3D r Time t t Velocity v x v Acceleration a x a
Drawing vectors Draw a vector as a line with an arrowhead at its tip. The length of the line shows the vector s magnitude. The direction of the line shows the vector s direction. Figure 1.10 shows equal-magnitude vectors having the same direction and opposite directions.
Adding vectors graphically Two vectors may be added graphically using either the parallelogram method or the head-to-tail method. Subtracting two vectors: reverse the direction of one vector Adding three vectors: two at a time or adding all vectors directly
Ending point Displacement Starting point Order does not matter in vector addition Thanks to Dr. Narayanan for the animation.
Components of a vector Adding vectors graphically provides limited accuracy. Vector components provide a general method for adding vectors. Any vector can be represented by an x-component A x and a y-component A y. Use trigonometry to find the components of a vector: A x = Acos θ and A y = Asin θ, where θ is measured from the +xaxis toward the +y-axis.
Unit vectors A unit vector has a magnitude of 1 with no units. The unit vector i points in the +x-direction, j points in the +ydirection, and k points in the +z-direction. Any vector can be expressed in terms of its components as A = A x i + A y j + A z k This is equivalent to the expression: A = (A x, A y, A z )
Calculations with unit vectors and components Given vectors A and B A = A x i + A y j + A z k B = B x i + B y j + B z k The magnitude of A A = A x 2 + A y 2 + A z 2 Adding two vectors A + B = (A x +B x ) i + (A y +B y ) j + (A z + B z ) k Multiplying a scalar ca = ca x i + ca y j + ca z k
Position vector The position of an object in a 3D space is given by a position vector. The vector has three components. r = x i + y j + z k
Average velocity The average velocity between two points is the displacement divide by the time interval between the two points. It has the same direction as the displacement. r = r 2 r 1 v av = r t In terms of components: r = x i + y j + z k v av x = x t v av y = y t v av z = z t
Instantaneous velocity The instantaneous velocity is the instantaneous rate of change of position vector with respect to time. The instantaneous velocity is always tangent to the path of motion. r v = lim t 0 t = dr dt In terms of components: v x = dx dt v y = dy dt v z = dz dt
Average acceleration The average acceleration during a time interval t is defined as the velocity change divided by t. a av = v t In terms of components: a av x = v x t a av y = v y t a av z = v z t
Instantaneous acceleration The instantaneous acceleration is the instantaneous rate of change of the velocity with respect to time. An object following a curved path is accelerating, even if the speed is constant. v a = lim t 0 t = dv dt In terms of components: a x = dv x dt = d2 x dt 2 a y = dv y dt = d2 y dt 2 a z = dv z dt = d2 z dt 2
Parallel and perpendicular components of acceleration The acceleration can be resolved into a component parallel to the path and a component perpendicular to the path The parallel component tells about changes in the speed The perpendicular component tells about changes in the direction of motion
Summary: 1D -> 2D -> 3D 1D 2D 3D = = = = = =
Projectile motion A projectile is any body given an initial velocity that then follows a path determined by the effects of gravity (for now, we neglect the air resistance and the curvature and rotation of the earth)
PhET simulation http://phet.colorado.edu/
The x and y motion of a projectile Projectile motion can be analyzed by separating x and y motion Horizontal direction (x) Constant velocity, a x = 0 v x = v 0x x = x 0 + v 0x t Vertical direction (y) Initial velocity, a y = g v y = v 0y gt y = y 0 + v 0y t 1 2 gt2
Analyzing projectile motion These equations describe projectile motion Derivation : The strategy is to consider a projectile motion in x and y directions separately. In x direction: Initial x-component velocity: v 0x = v 0 cos α 0 x-component acceleration: a x = 0 Thus, v x = v 0 cos α 0 and x = v 0x t = (v 0 cos α 0 )t x v 0 cos 0 t y v 0 sin 0 t 1 2 gt2 v x v 0 cos 0 v y v 0 sin 0 gt In y direction: Initial y-component velocity: v 0y = v 0 sin α 0 y-component acceleration: a y = g Thus, v y = v 0y + a y t = v 0 sin α 0 gt; y = v 0y t + 1 2 a yt 2 = (v 0 sin α 0 )t 1 2 gt2.
Trajectory of projectile motion An equation of the trajectory in terms of x and y by eliminating t. g y = (tan α 0 )x 2v 2 0 cos 2 x 2 α 0 Derivation: From (1), we got t = x v 0 cos α 0 Plug it into (2), we have y = v 0 sin α 0 x v 0 cos α 0 1 2 g x v 0 cos α 0 2 x v 0 cos 0 t y v 0 sin 0 t 1 2 gt2 v x v 0 cos 0 v y v 0 sin 0 gt = tanα 0 x g 2v 0 2 cos 2 α 0 x 2 (1) (2) The trajectory is parabolic
The effects of air resistance Calculations become more complicated. Acceleration is not constant. Effects can be very large. Maximum height and range decrease. Trajectory is no longer a parabola.
Circular motion & uniform circular motion v v v a a a For uniform circular motion, the speed is constant and the acceleration is perpendicular to the velocity
Centripetal acceleration Comparing the two triangles in (a) and (b), v v 1 = s R => v = v 1 s R v a = lim t 0 t = lim v 1 s t 0 R t = v2 R with v 1 = v 2 = v Centripetal acceleration Magnitude: a rad = v2 R or a rad = 4π2 R T 2 The direction is perpendicular to the velocity and inward along the radius. It is changing.
PhET simulation http://phet.colorado.edu/
Nonuniform circular motion If the speed varies, the motion is nonuniform circular motion. The acceleration has both the radial and tangential components. a rad = v2 R a tan = d v dt
Relative velocity The velocity of a moving body seen by a particular observer is called the velocity relative to that observer, or simply the relative velocity. A frame of reference is a coordinate system plus a time scale.
Relative velocity in one dimension The position of point P relative to frame B is x P/B, the position of the origin of frame B relative to frame A is x B/A, then the position of point P relative to frame A is given by x P/A = x P/B + x B/A If P is moving relative to frame B and frame B is moving relative to frame A, then the x-velocity of P relative to frame A is v P/A x = v P/B x +v B/A x
Relative velocity in 2/3 dimensions The concept of relative velocity can be extended into 2/3 dimensions by using vectors r P/A = r P/B + r B/A v P/A = v P/B + v B/A