Lesson

1. Lesson Vectors Review

2.  R head tail Scalars vs Vectors • Scalars have magnitude only • Distance, speed, time, mass • Vectors have both magnitude and direction • displacement, velocity, acceleration

3.  x A Direction of Vectors • The direction of a vector is represented by the direction in which the ray points. • This is typically given by an angle.

4. If vector A represents a displacement of three miles to the north… A B Then vector B, which is twice as long, would represent a displacement of six miles to the north! Magnitude of Vectors • The magnitude of a vector is the size of whatever the vector represents. • The magnitude is represented by the length of the vector. • Symbolically, the magnitude is often represented as │A │

5. Equal Vectors • Equal vectors have the same length and direction, and represent the same quantity (such as force or velocity).

6. A -A Inverse Vectors • Inverse vectors have the same length, but opposite direction.

7. B A R Graphical Addition of Vectors • Vectors are added graphically together head-to-tail. • The sum is called the resultant. • The inverse of the sum is called the equilibrant A + B = R

8. Component Addition of Vectors • Resolve each vector into its x- and y-components. Ax = Acos Ay = Asin Bx = Bcos By = Bsin etc. • Add the x-components together to get Rx and the y-components to get Ry. • Use the Pythagorean Theorem to get the magnitude of the resultant. • Use the inverse tangent function to get the angle.

9. Sample problem: Add together the following graphically and by component, giving the magnitude and direction of the resultant and the equilibrant. • Vector A: 300 m @ 60o • Vector B: 450 m @ 100o • Vector C: 120 m @ -120o

11. Lesson Unit Vectors

12. a Consider Three Dimensions Polar Angle z Azimuthal Angle az q ay y f ax xy Projection x

13. Unit Vectors • Unit vectors are quantities that specify direction only. They have a magnitude of exactly one, and typically point in the x, y, or z directions.

14. Unit Vectors z k j i y x

15. Unit Vectors • Instead of using magnitudes and directions, vectors can be represented by their components combined with their unit vectors. • Example: displacement of 30 meters in the +x direction added to a displacement of 60 meters in the –y direction added to a displacement of 40 meters in the +z direction yields a displacement of:

16. Adding Vectors Using Unit Vectors • Simply add all the i components together, all the j components together, and all the k components together.

17. Sample problem: Consider two vectors, A = 3.00 i + 7.50 j and B = -5.20 i + 2.40 j. Calculate C where C = A + B.

18. Sample problem: You move 10 meters north and 6 meters east. You then climb a 3 meter platform, and move 1 meter west on the platform. What is your displacement vector? (Assume East is in the +x direction).

19. Suppose I need to convert unit vectors to a magnitude and direction? • Given the vector

20. Sample problem: You move 10 meters north and 6 meters east. You then climb a 3 meter platform, and move 1 meter west on the platform. How far are you from your starting point?

21. Lesson Position, Velocity, and Acceleration Vectors in Multiple Dimensions

22. x: position x: displacement v: velocity a: acceleration r: position r: displacement v: velocity a: acceleration In Unit Vector Notation • r = x i + y j + z k • r = x i + y j + z k • v = vxi + vyj + vzk • a = axi + ayj + azk 1 Dimension 2 or 3 Dimensions

23. Sample problem: The position of a particle is given byr = (80 + 2t)i – 40j - 5t2k. Derive the velocity and acceleration vectors for this particle. What does motion “look like”?

24. Sample problem: A position function has the form r = x i + y j with x = t3 – 6 and y = 5t - 3. a) Determine the velocity and acceleration functions. b) Determine the velocity and speed at 2 seconds.

25. Miscellaneous • Let’s look at some video analysis. • Let’s look at a documentary. • Homework questions?

26. Lesson Multi-Dimensional Motion with Constant (or Uniform) Acceleration

27. Sample Problem: A baseball outfielder throws a long ball. The components of the position are x = (30 t) m and y = (10 t – 4.9t2) m a) Write vector expressions for the ball’s position, velocity, and acceleration as functions of time. Use unit vector notation! b) Write vector expressions for the ball’s position, velocity, and acceleration at 2.0 seconds.

28. Sample problem: A particle undergoing constant acceleration changes from a velocity of 4i – 3j to a velocity of 5i + j in 4.0 seconds. What is the acceleration of the particle during this time period? What is its displacement during this time period?

29. g g g g g Trajectory of Projectile • This shows the parabolic trajectory of a projectile fired over level ground. • Acceleration points down at 9.8 m/s2 for the entire trajectory.

30. Trajectory of Projectile • The velocity can be resolved into components all along its path. Horizontal velocity remains constant; vertical velocity is accelerated. vx vx vy vy vx vy vx vy vx

31. y y x x t t Position graphs for 2-D projectiles. Assume projectile fired over level ground.

32. Velocity graphs for 2-D projectiles. Assume projectile fired over level ground. Vy Vx t t

33. Acceleration graphs for 2-D projectiles. Assume projectile fired over level ground. ay ax t t

34. Vo,y = Vo sin  Vo,x = Vo cos  Remember…To work projectile problems… • …resolve the initial velocity into components. Vo 

35. Sample problem: A soccer player kicks a ball at 15 m/s at an angle of 35o above the horizontal over level ground. How far horizontally will the ball travel until it strikes the ground?

36. Sample problem: A cannon is fired at a 15o angle above the horizontal from the top of a 120 m high cliff. How long will it take the cannonball to strike the plane below the cliff? How far from the base of the cliff will it strike?

37. Lesson Monkey Gun Experiment – shooting on an angle

38. Lesson A day of derivations

39. Sample problem: derive the trajectory equation.

40. Sample problem: Derive the range equation for a projectile fired over level ground.

41. Sample problem: Show that maximum range is obtained for a firing angle of 45o.

42. Will the projectile always hit the target presuming it has enough range? The target will begin to fall as soon as the projectile leaves the gun.

43. Punt-Pass-Kick Pre-lab • Purpose: Using only a stopwatch, a football field, and a meter stick, determine the launch velocity of sports projectiles that you punt, pass, or kick. • Theory: Use horizontal (unaccelerated) motion to determine Vx, and vertical (accelerated) motion to determine Vy. Ignore air resistance. • Data: Prepare your lab book to collect xi, xf, yo, and Dt measurements for each sports projectile. Analyze the data fully for at least three trials. • Make sure you dress comfortably tomorrow!

44. Lesson Punt-pass-kick lab

45. Lesson Review of Uniform Circular Motion Radial and Tangential Acceleration

46. Uniform Circular Motion • Occurs when an object moves in a circle without changing speed. • Despite the constant speed, the object’s velocity vector is continually changing; therefore, the object must be accelerating. • The acceleration vector is pointed toward the center of the circle in which the object is moving, and is referred to as centripetal acceleration.

47. v v a a a a v v Vectors inUniform Circular Motion a = v2 / r

48. Sample Problem The Moon revolves around the Earth every 27.3 days. The radius of the orbit is 382,000,000 m. What is the magnitude and direction of the acceleration of the Moon relative to Earth?

49. Sample problem: Space Shuttle astronauts typically experience accelerations of 1.4 g during takeoff. What is the rotation rate, in rps, required to give an astronaut a centripetal acceleration equal to this in a simulator moving in a 10.0 m circle?

50. Tangential acceleration • Sometimes the speed of an object in circular motion is not constant (in other words, it’s not uniform circular motion). • An acceleration component may be tangent to the path, aligned with the velocity. This is called tangential acceleration. It causes speeding up or slowing down. • The centripetal acceleration component causes the object to continue to turn as the tangential component causes the speed to change. The centripetal component is sometimes called the radial acceleration, since it lies along the radius.