1. Two Dimensional Kinematics

2. Position and Velocity Vectors If an object starts out at the origin and moves to point A, its displacement can be represented by a position vector. x y z x y z A x y z + +

3. Position and Velocity Vectors As an object moves from one point in space to another, the average velocity of its motion can be described as the displacement of the object over the time it takes to move. (average velocity vector) To find the instantaneous velocity (the velocity at a specific point in time) it requires the time interval to be so small that it can effectively be reduced to 0 which can be represented as a limit expression. (instantaneous velocity vector)

4. Components of Instantaneous Velocity The instantaneous velocity can have three different components: x, y, and z. Each component is shown below, Vector representation:

5. Acceleration Vector Acceleration is the rate at which the velocity is changing, and the average acceleration can be found by taking the difference of the final and initial velocity and dividing it by the time it takes for that event to occur. Just as we can find the velocity at a specific point in time, we can also find the instantaneous acceleration using a limit expression.

6. Components of Instantaneous Acceleration Vector representation:

7. Given. Find and.

8. Given. Findand.

9. Projectile Motion Have you ever thrown an object in the air and watched the trajectory it follows? The path the object travels is a parabolic path. vx vy vx v vy

10. Velocity of a Projectile vx vy vx v vy To explain projectile motion in the vertical direction we can use our knowledge of throwing a ball straight up into the air. We know that eventually the acceleration due to gravity will eventually stop the ball and make it move back towards the Earth. At its apex the ball stops moving in the vertical direction, so for a projectile this would be the same.

11. Velocity of a Projectile vx vy vx v vy To account for the projectile's motion in the horizontal direction we imagine a case where a block moves to the right with a velocity, v. In the absence of a resistant force (e.g. air resistance or friction) we can state that the block moves with a constant velocity (note that gravity does not affect the projectile's horizontal motion).

12. Position of an object in projectile motion The position of the projectile with respect to its starting position can be represented with minor changes to the kinematics equation: Horizontal position: Vertical Position: From these equations we can determine the maximum horizontal distance, the maximum height reached by the projectile, the time to reach its highest point, and the time it hits the floor again.

13. 1 Which of the following statements are true regarding projectile motion? is constant A B Acceleration is +g when the object is rising and -g when falling. C In the absence of friction the trajectory will depend on the object's mass as well as its initial and launch angle. D The velocity of the object is zero at the point of maximum elevation. E The horizontal motion is independent of the vertical motion.

14. 2 A marble is shot and follows a parabolic path shown below. Air resistance is negligible. Point Y is the highest point on the path. A B C D v Which of these indicates the direction of the speed, if any, of the marble at point Y? E None

15. 3 A marble is show and follows a parabolic path shown above. Air resistance is negligible. Point Y is the highest point on the path. A B C D E v Which of the following indicates the direction of the net force on the marble at Point X?

16. H v Time to fall from apex When a projectile is thrown in the horizontal direction

17. 4 Two cannon balls are launched simultaneously off a cliff. The two cannon balls have different masses and different initial velocities. Which will strike the ground first? AThe heaviest one BThe lightest one C The slowest one D The fastest one EThey will both strike the ground at the same time

18. To Find Maximum Height Because at the highest point the vertical component of velocity is zero. (time to attain maximum height)

19. To Find Maximum Displacement vx vy vx v vy

20. To find angle between the velocities vy vx vy vx vy

21. 5 At what angle will a projectile have the greatest vertical displacement? A 0 B30 C 45 D 60 E 90

22. 6 At what angle will a projectile have the greatest horizontal displacement? A0 B30 C 45 D 60 E90

23. 7 Which angles will have the same horizontal displacement? A0 and 90 B 30 and 60 C0 and 45 D 35 and 60 E 30 and 90 F None of the two angles above will have the same displacement.

24. Moving in a Circular Path constant speed decreasing speed increasing speed When an object moves in a circle with constant speed and its acceleration is perpendicular to the velocity this is called Uniform Circular Motion.

25. 8 A car is driving with decreasing velocity on a curved path. Which diagram shows the correct direction for the velocity and acceleration? A B CD v a v a v a v a v a E

26. 9 A car is driving with constant velocity on a curved path. Which diagram shows the correct direction for the velocity and acceleration? A B v a v a v a v a v a C D E

27. 10 A car is driving with increasing velocity on a curved path. Which diagram shows the correct direction for the velocity and acceleration? A B C D v a v a v a v a v a E

28. Uniform Circular Motion (centripetal acceleration) If we plug the equation for the velocity into the acceleration equation we get:

29. Uniform Circular Motion C entripetal Acceleration P1 P2 Knowing that the triangles are similar, we can use ratios of corresponding sides, therefore: To find the instantaneous velocity, we first have to come up with a representation for the average acceleration as before

30. Uniform Circular Motion C entripetal Acceleration To find the instantaneous acceleration we have to take a limit expression of the average acceleration. The limit expression will give us the velocity at a certain point in time, this velocity is the same as v1 (centripetal acceleration)

31. 11 If a ball is swung in a circle of a radius of 1 m with a velocity of 5 m/s what would be the centripetal acceleration? A 5 m/s2 B 0.2 m/s2 C25 m/s2 D 0.04 m/s2 E10 m/s2

32. 12 If a ball is swung in a circle of radius 9 m and its centripetal acceleration was 1 m/s2. What would be its velocity? A3 m/s B9 m/s C81 m/s D18 m/s E√3 m/s

33. 13 If an object is moving in a circle with a velocity of 15 m/s and has a centripetal acceleration of 45 m/s2. What would be its radius? A 5 m B1/3 m C 3 m D 10 m E 15 m

34. arad atan arad atan Non-Uniform Circular Motion When you are on a roller coaster and you come to a circular loop, your velocity is not constant. As you approach the top of the loop your velocity decreases and as you come back down your velocity increases. You still have a radial acceleration but now there is a tangential acceleration which is perpendicular to your radial acceleration.

35. Relative Velocity When you are riding in a car and you look out a window what do you see? If there is another car moving along side you with the same velocity relative to you, the other car appears to stand still, but with respect to the ground both of you are moving. If another car is moving with velocity 2v with respect to the ground, then with respect to your car its moving with a velocity of v.

36. Relative Velocity If a plane is flying through the air and enters a crosswind it will have a velocity straight and one perpendicular to it. Vplane Vcrosswind

37. 14 A plane is moving with a constant speed of 1200 km/h and during part of its flight there is a cross wind blowing at 500 km/h. What is the net velocity during this portion of its flight? A1600 km/h B 1300 km/h C 700 km/h D1700 km/h E 2500 km/h 1200 km/h 500 km/h

38. 15 Two kids are on a boat capable of a maximum speed of 10 kilometers per hour in water, and wish to cross a river 2 kilometers wide to a point directly across from their starting point. If the speed of the water in the river is 9 kilometers per hour how much time is required for the crossing? A0.05 hrs B 0.45 hrs C 1 hr D 10 hrs ENot possible