Chapter 4 – 2D and 3D Motion

1. Chapter 4 – 2D and 3D Motion • Definitions • Projectile motion • Uniform circular motion • Relative motion

2. Motion in Two Dimensions • Using + or – signs is not always sufficient to fully describe motion in more than one dimension • Vectors can be used to more fully describe motion • Still interested in displacement, velocity, and acceleration • Will serve as the basis of multiple types of motion in future chapters

3. Position and Displacement • The position of an object is described by its position vector,r • The displacement of the object is defined as the change in its position Δr = rf - ri

4. General Motion Ideas • 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 direction

5. Definitions Position vector:extends from the origin of a coordinate system to the particle. Displacement vector:represents a particle’s position change during a certain time interval.

6. Average 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, Δr

7. Average Velocity • 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

8. Definitions Average velocity:

9. Instantaneous Velocity • The instantaneous velocity is the limit of the average velocity as Δt approaches zero The direction of the instantaneous velocity is along a line that is tangent to the path of the particle’s direction of motion

10. Instantaneous velocity: The direction of the instantaneous velocity of a particle is always tangent to the particle’s path at the particle’s position

11. Average 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.

12. Average Acceleration • As a particle moves,Δvcan be found in different ways • The average acceleration is a vector quantity directed alongΔv

13. Instantaneous Acceleration • The instantaneous acceleration is the limit of the average acceleration as Δtapproaches zero

14. Instantaneous acceleration:

15. Producing 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 simultaneously

16. Kinematic 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

17. Kinematic Equations • Position vector • Velocity • Since acceleration is constant, we can also find an expression for the velocity as a function of time: vf = vi + at

18. Kinematic Equations • The velocity vector can be represented by its components vfis generally not along the direction of eitherviorat

19. Kinematic Equations • The position vector can also be expressed as a function of time: rf = ri + vit + ½ at2 • This indicates that the position vector is the sum of three other vectors: • The initial position vector • Thedisplacement resulting from vit • The displacement resulting from ½ at2

20. Kinematic Equations • The vector representation of the position vector • rfis generally not in the same direction asvior asai • rfandvfare generally not in the same direction

21. Kinematic Equations, Components • The equations for final velocity and final position are vector equations, therefore they may also be written in component form • This shows that two-dimensional motion at constant acceleration is equivalent to two independent motions • One motion in the x-direction and the other in the y-direction

22. Kinematic Equations, Component Equations • vf = vi + at becomes vxf = vxi + axt and vyf = vyi + ayt • rf = ri + vit + ½ at2becomes xf = xi + vxit + ½ axt2 and yf = yi + vyit + ½ ayt2

23. Projectile Motion • An object may move in both thexand ydirections simultaneously • The form of two-dimensional motion we will deal with is called projectile motion

24. Assumptions of Projectile Motion • The free-fall acceleration g is constant over the range of motion And is directed downward • 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 trajectory

25. Verifying the Parabolic Trajectory • Reference frame chosen yis vertical with upward positive • Acceleration components ay = -gandax = 0 • Initial velocity components vxi = vi cos qandvyi = vi sin q

26. II. Projectile motion Motion of a particle launched with initial velocity, v0,and free fall accelerationg. The horizontal and vertical motions are independent from each other. Horizontal motion: ax=0vx=v0x

27. II. Projectile motion Range (R):horizontal distance traveled by a projectile before returning to launch height.

28. II. Projectile motion Vertical motion:ay= -g

29. Trajectory:projectile’s path. We can findyas a function ofxby eliminating time

30. Horizontal range: R = x-x0; Vertical displacement: y-y0=0. (Maximum for a launch angle of 45º )

31. Projectile Motion – Problem Solving Hints • Select a coordinate system • Resolve the initial velocity into x and ycomponents • Analyze the horizontal motion using constant velocity techniques • Analyze the vertical motion using constant acceleration techniques • Remember that both directions share the same time

32. A third baseman wishes to throw to first base, 127 feet distant. His best throwing speed is 85 mi/h. (a) If he throws the ball horizontally 3 ft above the ground, how far from first base will it hit the ground? (b) From the same initial height, at what upward angle must the third baseman throw the ball if the first baseman is to catch it 3 ft above the ground? (c) What will be the time of flight in that case? y v0 h=3ft B1 B3 x xmax 0 xB1=38.7m

33. (b) From the same initial height, at what upward angle must the third baseman throw the ball if the first baseman is to catch it 3 ft above the ground? (c) What will be the time of flight in that case? y v0 θ h=3ft x B3 38.7m B1

34. y v0 x θ=45º x=R=R’? N7: In Galileo’s Two New Sciences, the author states that “for elevations (angles of projection) which exceed or fall short of 45º by equal amounts, the ranges are equal…” Prove this statement.

35. A ball is tossed from an upper-story window of a building. The ball is given an initial velocity of8.00 m/sat an angle of20.0°below the horizontal. It strikes the ground3.00 slater. (a) How far horizontally from the base of the building does the ball strike the ground? (b) Find the height from which the ball was thrown. (c) How long does it take the ball to reach a point10.0 mbelow the level of launching?

36. Uniform Circular Motion Uniform circular motion occurs when an object moves in a circular path with a constant speed • An acceleration exists since the direction of the motion is changing • This change in velocity is related to an acceleration • The velocity vector is always tangent to the path of the object

37. Changing Velocity in Uniform Circular Motion • The change in the velocity vector is due to the change in direction • The vector diagram showsDv = vf- vi

38. Centripetal Acceleration • The acceleration is always perpendicular to the path of the motion • The acceleration always points toward the center of the circle of motion • This acceleration is called thecentripetal acceleration

39. Centripetal Acceleration • The magnitude of the centripetal acceleration vector is given by • The direction of the centripetal acceleration vector is always changing, to stay directed toward the center of the circle of motion

40. Period • The period,T, is the time required for one complete revolution • The speed of the particle would be the circumference of the circle of motion divided by the period • Therefore, the period is

41. Tangential Acceleration • The magnitude of the velocity could also be changing • In this case, there would be atangential acceleration

42. Total Acceleration • The tangential acceleration causes the change in the speed of the particle • The radial acceleration comes from a change in the direction of the velocity vector

43. Total Acceleration, equations • The tangential acceleration: • The radial acceleration: • The total acceleration: • Magnitude

44. Total Acceleration, In Terms of Unit Vectors • Define the following unit vectors • r lies along the radius vector • q is tangent to the circle • The total acceleration is

45. Uniform circular motion Motion around a circle at constant speed. Magnitude of velocity and acceleration constant. Direction varies continuously. Velocity:tangent to circle in the directionof motion. Acceleration:centripetal Period of revolution:

46. 54.A cat rides a merry-go-round while turning with uniform circular motion. At time t1= 2s, the cat’s velocity is: v1= (3m/s)i+(4m/s)j, measured on an horizontal xy coordinate system. At time t=5s its velocity is: v2= (-3m/s)i+(-4m/s)j. What is (a) the magnitude of the cat’s centripetal acceleration? v2 x v1 y

47. Figure represents the total acceleration of a particle moving clockwise in a circle of radius 2.50 m at a certain of time. At this instant, find (a) the radial acceleration, (b) the speed of the particle, and (c) its tangential acceleration.

48. A ball swings in a vertical circle at the end of a rope1.50 mlong. When the ball is36.9past the lowest point on its way up, its total acceleration is . At that instant, (a) sketch a vector diagram showing the components of its acceleration, (b) determine the magnitude of its radial acceleration, and (c) determine the speed and velocity of the ball.

49. Relative Velocity • Two observers moving relative to each other generally do not agree on the outcome of an experiment • For example, observers A and B below see different paths for the ball

50. Relative Velocity • Reference frameSis stationary • Reference frameS’is moving atvo • This also means thatSmoves at–vorelative toS’ • Define timet = 0as that time when the origins coincide