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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 positionrfirrrAverage 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 pathavgtrvInstantaneous 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 curve0limtdtdtrrv2Instantaneous 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 vvInstantaneous 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 time0limtdtdtvvatvProducing 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:ˆˆxyrijˆˆxydvvdtrvijfitvvaKinematic 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 fvivaKinematic 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 directionfrivafvfrGraphical 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= vicosand 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

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