Chapter 4 Motion in Two Dimensions - PowerPoint Presentation

1. Chapter 4Motion in Two Dimensions

2. Motion in Two DimensionsUsing + or – signs is not always sufficient to fully describe motion in more than one dimensionVectors can be used to more fully describe motionStill interested in displacement, velocity, and accelerationWill serve as the basis of multiple types of motion in future chapters

3. Position and DisplacementThe position of an object is described by its position vector, rThe displacement of the object is defined as the change in its positionΔr = rf - ri

4. General Motion IdeasIn two- or three-dimensional kinematics, everything is the same as as in one-dimensional motion except that we must now use full vector notationPositive and negative signs are no longer sufficient to determine the direction

5. Average VelocityThe average velocity is the ratio of the displacement to the time interval for the displacementThe direction of the average velocity is the direction of the displacement vector, Δr

6. Average Velocity, contThe average velocity between points is independent of the path takenThis is because it is dependent on the displacement, also independent of the path

7. Instantaneous VelocityThe instantaneous velocity is the limit of the average velocity as Δt approaches zeroThe direction of the instantaneous velocity is along a line that is tangent to the path of the particle’s direction of motion

8. Instantaneous Velocity, contThe 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 motionThe magnitude of the instantaneous velocity vector is the speedThe speed is a scalar quantity

9. Average AccelerationThe 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.

10. Average Acceleration, contAs a particle moves, Δv can be found in different waysThe average acceleration is a vector quantity directed along Δv

11. Instantaneous AccelerationThe instantaneous acceleration is the limit of the average acceleration as Δv/Δt approaches zero

12. Producing An AccelerationVarious changes in a particle’s motion may produce an accelerationThe magnitude of the velocity vector may changeThe direction of the velocity vector may changeEven if the magnitude remains constantBoth may change simultaneously

13. Kinematic Equations for Two-Dimensional MotionWhen the two-dimensional motion has a constant acceleration, a series of equations can be developed that describe the motionThese equations will be similar to those of one-dimensional kinematics

14. Kinematic Equations, 2Position vectorVelocitySince acceleration is constant, we can also find an expression for the velocity as a function of time: vf = vi + at

15. Kinematic Equations, 3The velocity vector can be represented by its componentsvf is generally not along the direction of either vi or at

16. Kinematic Equations, 4The position vector can also be expressed as a function of time:rf = ri + vit + ½ at2This indicates that the position vector is the sum of three other vectors:The initial position vectorThe displacement resulting from vi tThe displacement resulting from ½ at2

17. Kinematic Equations, 5The vector representation of the position vectorrf is generally not in the same direction as vi or as airf and vf are generally not in the same direction

18. Kinematic Equations, ComponentsThe equations for final velocity and final position are vector equations, therefore they may also be written in component formThis shows that two-dimensional motion at constant acceleration is equivalent to two independent motionsOne motion in the x-direction and the other in the y-direction

19. Kinematic Equations, Component Equationsvf = vi + at becomesvxf = vxi + axt and vyf = vyi + ayt rf = ri + vi t + ½ at2 becomesxf = xi + vxi t + ½ axt2 and yf = yi + vyi t + ½ ayt2

20. Projectile MotionAn object may move in both the x and y directions simultaneouslyThe form of two-dimensional motion we will deal with is called projectile motion

21. Assumptions of Projectile MotionThe free-fall acceleration g is constant over the range of motionAnd is directed downwardThe effect of air friction is negligibleWith these assumptions, an object in projectile motion will follow a parabolic pathThis path is called the trajectory

22. Verifying the Parabolic TrajectoryReference frame choseny is vertical with upward positiveAcceleration componentsay = -g and ax = 0Initial velocity componentsvxi = vi cos q and vyi = vi sin q

23. Verifying the Parabolic Trajectory, contDisplacementsxf = vxi t = (vi cos q) tyf = vyi t + ½ay t2 = (vi sin q)t - ½ gt2Combining the equations gives:This is in the form of y = ax – bx2 which is the standard form of a parabola

24. Analyzing Projectile MotionConsider the motion as the superposition of the motions in the x- and y-directionsThe x-direction has constant velocityax = 0The y-direction is free fallay = -gThe actual position at any time is given by: rf = ri + vit + ½gt2

25. Projectile Motion Vectorsrf = ri + vi t + ½ g t2The final position is the vector sum of the initial position, the position resulting from the initial velocity and the position resulting from the acceleration

26. Projectile Motion Diagram

27. Projectile Motion – Implications The y-component of the velocity is zero at the maximum height of the trajectoryThe accleration stays the same throughout the trajectory

28. Range and Maximum Height of a ProjectileWhen analyzing projectile motion, two characteristics are of special interestThe range, R, is the horizontal distance of the projectileThe maximum height the projectile reaches is h

29. Height of a Projectile, equationThe maximum height of the projectile can be found in terms of the initial velocity vector:This equation is valid only for symmetric motion

30. Range of a Projectile, equationThe range of a projectile can be expressed in terms of the initial velocity vector:This is valid only for symmetric trajectory

31. More About the Range of a Projectile

32. Range of a Projectile, finalThe maximum range occurs at qi = 45oComplementary angles will produce the same rangeThe maximum height will be different for the two anglesThe times of the flight will be different for the two angles

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34. Projectile Motion – Problem Solving HintsSelect a coordinate systemResolve the initial velocity into x and y componentsAnalyze the horizontal motion using constant velocity techniquesAnalyze the vertical motion using constant acceleration techniquesRemember that both directions share the same time

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37. Non-Symmetric Projectile MotionFollow the general rules for projectile motionBreak the y-direction into partsup and down or symmetrical back to initial height and then the rest of the heightMay be non-symmetric in other ways

38. Uniform Circular MotionUniform circular motion occurs when an object moves in a circular path with a constant speedAn acceleration exists since the direction of the motion is changing This change in velocity is related to an accelerationThe velocity vector is always tangent to the path of the object

39. Changing Velocity in Uniform Circular MotionThe change in the velocity vector is due to the change in directionThe vector diagram shows Dv = vf - vi

40. Centripetal AccelerationThe acceleration is always perpendicular to the path of the motionThe acceleration always points toward the center of the circle of motionThis acceleration is called the centripetal acceleration

41. Centripetal Acceleration, contThe magnitude of the centripetal acceleration vector is given byThe direction of the centripetal acceleration vector is always changing, to stay directed toward the center of the circle of motion

42. PeriodThe period, T, is the time required for one complete revolutionThe speed of the particle would be the circumference of the circle of motion divided by the periodTherefore, the period is

43. Quiz

44. Example

45. Tangential AccelerationThe magnitude of the velocity could also be changingIn this case, there would be a tangential acceleration

46. Total AccelerationThe tangential acceleration causes the change in the speed of the particleThe radial acceleration comes from a change in the direction of the velocity vector

47. Total Acceleration, equationsThe tangential acceleration:The radial acceleration:The total acceleration:Magnitude

48. Total Acceleration, In Terms of Unit VectorsDefine the following unit vectorsr lies along the radius vectorq is tangent to the circleThe total acceleration is

49. Example

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51. End of the chapterFinished