1Week3, Chapter 4Motion in Two DimensionsLecture Quiz A particle confined to motion along the x axis moves with constant acceleration from x = 2.0 m to x = 8.0 m during a 1-s time interval. The velocity of the particle at x =2.0 m is 2.0 m/s. What is the acceleration during this time interval?A. 4.0 m/s2B. 3.2 m/s2C. 6.4 m/s2D. 8.0 m/s2E. 5.7 m/s2Motion in Two Dimensions In two- or three-dimensional kinematics, everything is the same as in one-dimensional motion except that we must now use full vector notation Positive and negative signs are no longer sufficient to determine the directionPosition and Displacement The position of an object is described by its position vector,  The displacement of the object is defined as the change in its positionrfirrrAverage Velocity The average velocity is the ratio of the displacement to the time interval for the displacement The direction of the average velocity is the direction of the displacement vector The average velocity between points is independent of the path taken This is because it is dependent on the displacement, also independent of the pathavgtrvInstantaneous Velocity The instantaneous velocity is the limit of the average velocity as Δtapproaches zero As the time interval becomes smaller, the direction of the displacement approaches that of the line tangent to the curve0limtdtdtrrv2Instantaneous Velocity, cont The direction of the instantaneous velocity vector at any point in a particle’s path is along a line tangent to the path at that point and in the direction of motion The magnitude of the instantaneous velocity vector is the speed The speed is a scalar quantityAverage Acceleration The average acceleration of a particle as it moves is defined as the change in the instantaneous velocity vector divided by the time interval during which that change occurs.f iavgfitt tvvvaAverage Acceleration, cont As a particle moves, the direction of the change in velocity is found by vector subtraction The average acceleration is a vector quantity directed along   fivv vvInstantaneous Acceleration The instantaneous acceleration is the limiting value of the ratio as Δt approaches zero The instantaneous equals the derivative of the velocity vector with respect to time0limtdtdtvvatvProducing An Acceleration Various changes in a particle’s motion may produce an acceleration The magnitude of the velocity vector may change The direction of the velocity vector may change Even if the magnitude remains constant Both may change simultaneouslyKinematic Equations for Two-Dimensional Motion When the two-dimensional motion has a constant acceleration, a series of equations can be developed that describe the motion These equations will be similar to those of one-dimensional kinematics Motion in two dimensions can be modeled as two independent motions in each of the two perpendicular directions associated with the x and y axes Any influence in the y direction does not affect the motion in the x direction3Kinematic Equations, 2 Position vector for a particle moving in the xyplane The velocity vector can be found from the position vector Since acceleration is constant, we can also find an expression for the velocity as a function of time:ˆˆxyrijˆˆxydvvdtrvijfitvvaKinematic Equations, 3 The position vector can also be expressed as a function of time:This indicates that the position vector is the sum of three other vectors: The initial position vector The displacement resulting from the initial velocity  The displacement resulting from the acceleration 212fi ittrrv aKinematic Equations, Graphical Representation of Final Velocity The velocity vector can be represented by its components is generally not along the direction of either or fvivaKinematic Equations, Graphical Representation of Final Position The vector representation of the position vector is generally not along the same direction as or as  and are generally not in the same directionfrivafvfrGraphical Representation Summary Various starting positions and initial velocities can be chosen Note the relationships between changes made in either the position or velocity and the resulting effect on the otherLecture Quiz A boy on a skate board skates off a horizontal bench at a velocity of 10 m/s. One tenth of a second after he leaves the bench, to two significant figures, the magnitudes of his velocity and acceleration are:A. 10 m/s; 9.8 m/s2.B. 9.0 m/s; 9.8 m/s2.C. 9.0 m/s; 9.0 m/s2.D. 1.0 m/s; 9.0 m/s2.E. 1.0 m/s; 9.8 m/s2.4Projectile Motion An object may move in both the x and ydirections simultaneously The form of two-dimensional motion we will deal with is called projectile motionAssumptions of Projectile Motion The free-fall acceleration is constant over the range of motion It is directed downward This is the same as assuming a flat Earth over the range of the motion It is reasonable as long as the range is small compared to the radius of the Earth The effect of air friction is negligible With these assumptions, an object in projectile motion will follow a parabolic path This path is called the trajectoryProjectile Motion DiagramClicker QuestionIf a baseball player throws a ball with a fixed initial speed, but with variable angles, the ball will move furthest if the angle from horizontal direction is:A: 0 degreesB: 30 degreesC: 45 degreesD: 60 degreesE: 90 degreesAnalyzing Projectile Motion Consider the motion as the superposition of the motions in the x- and y-directions The actual position at any time is given by:  The initial velocity can be expressed in terms of its components vxi= vicosand vyi= visin  The x-direction has constant velocity ax= 0 The y-direction is free fall ay= -g212fi itt rrv g Effects of Changing Initial Conditions The velocity vector components depend on the value of the initial velocity Change the angle and note the effect Change the magnitude and note the effect5Analysis Model The analysis model is the superposition of two motions Motion of a particle under constant velocity in the horizontal