Do Now

1. Do Now Starting from rest, a car undergoes a constant acceleration of 10. m/s/s.How far will the car travel in 3.0 s? How fast will the car move in 3.0 s?

2. Do Now Starting from rest, a car undergoes a constant acceleration of 10 m/s/s.How far will the car travel in 3 s? How fast will the car move in 3 s? Given: Solution: Find:

3. Do Now A basketball is dropped from rest from height of 1 m toward motion detector located on the floor. Draw x vs. t, v vs. t, and a vs. t graphs of the motion of the ball.

4. Unit 4: Kinematics in Two Dimensions

5. Unit Plan • Free Fall • Projectile Motion • Solving Problems Involving Projectile Motion • Projectile Motion Is Parabolic

6. Aristotle (382BC-322BC)- Greek natural philosopher. • A student of Plato and teacher of Alexander the Great • Believed that more massive objects fall faster. • Did not account air resistance. A detail of The School of Athens, a fresco by Raphael.

7. Galileo Galilei (1564-1642) – Italian physicist. • Reexamined motion of falling objects • Has been called the “Father of Modern Physics”(used models and experimentation) • Postulated that all objects would fall with the same constant acceleration in the absence if air resistance.

8. Free Fall • Freely falling objects are affected only by gravity. • At a given location on the Earth and in absence of air resistance, all objects fall with the same constant acceleration. • Acceleration due to gravity,or acceleration of free fall

9. Air Resistance • A feather an a coin accelerate equally when there is no air around them (in a vacuum). • For compact objects the effect of air resistance is small enough to be neglected.

10. Accelerated Motion Due to Gravity • We can choose y to be positive in the upward direction or in the downward direction. • Consider motion up to be positive. • acceleration due to gravity • For problem solving, we will approximate • On Earth, acceleration due to gravity always has downward direction(towards center of Earth).

11. Vertical Motion with Gravity • Start with the key equations for 1-dimensional motion. Assume that the motion is only up and down. Since motion is vertical -> , add the subscript y to the velocity, and substitute –g for a.

12. Object Thrown Up A rock is thrown upward with initial velocity 30 m/s.

13. Object Thrown Up A rock is thrown upward with initial velocity 30 m/s.

14. Object Thrown Up A rock is thrown upward with initial velocity 30 m/s.

15. Velocity vs. Time Graph

16. Object Thrown Up. Graphs

17. Position vs. Time Graph

20. Object Thrown Up • What is the instantaneous speed at the highest point? • 0 • How does velocity change during the upward part of its motion? • Decreasing from to 0. • How much does its speed decrease each second? • The speed decreases 10 m/s each second.

21. Object Thrown Up • What is the instantaneous speed of the object at points of equal elevation? • The same. • Are velocities same or different at points of equal elevation? • Same magnitude, opposite directions. • Is acceleration different when the object moving upward or downward? • The same 10 m/s/sdownwards.

22. Dropped Object A rock is dropped from the top of the cliff. How far did it travel in 1s, 2s, and 3s?

23. Dropped Object A rock is dropped from the top of the cliff. How far did it travel in 1s, 2s, and 3s?

24. Dropping with . Find time if you know Δy.

25. Time Up = Time Down • Since for the object thrown upward the motion up and down is symmetrical, you can use the same formula to find the time to go up a certain distance. • If you throw a ball upwards with just enough velocity to go up a distance of 35 m, how long will it take to reach the top?

26. Exercise 1 A ball is thrown upward with an initial velocity of 20 m/s. How long will it take for the ball to reach its maximum height? Given: Solution: Find:

27. Dropping With Initial VelocityExercise 2 • A ball is thrown downward from the top of a roof with a speed of 25 m/s. Find the instantaneous velocity of the ball in 2 seconds.

28. ConcepTest 2.8b Acceleration II 1) both v = 0 and a = 0 2) v ¹ 0, but a = 0 3) v = 0, but a ¹ 0 4) both v ¹ 0 and a ¹ 0 5) not really sure When throwing a ball straight up, which of the following is true about its velocity v and its acceleration a at the highest point in its path?

29. ConcepTest 2.8b Acceleration II y 1) both v = 0 and a = 0 2) v ¹ 0, but a = 0 3) v = 0, but a ¹ 0 4) both v ¹ 0 and a ¹ 0 5) not really sure When throwing a ball straight up, which of the following is true about its velocity v and its acceleration a at the highest point in its path? At the top, clearly v = 0 because the ball has momentarily stopped. But the velocity of the ball is changing, so its acceleration is definitely not zero! Otherwise it would remain at rest!! Follow-up: …and the value of a is…?

30. ConcepTest 2.9a Free Fall I 1) its acceleration is constant everywhere 2) at the top of its trajectory 3) halfway to the top of its trajectory 4) just after it leaves your hand 5) just before it returns to your hand on the way down You throw a ball straight up into the air. After it leaves your hand, at what point in its flight does it have the maximum value of acceleration?

31. ConcepTest 2.9a Free Fall I 1) its acceleration is constant everywhere 2) at the top of its trajectory 3) halfway to the top of its trajectory 4) just after it leaves your hand 5) just before it returns to your hand on the way down You throw a ball straight up into the air. After it leaves your hand, at what point in its flight does it have the maximum value of acceleration? The ball is in free fall once it is released. Therefore, it is entirely under the influence of gravity, and the only acceleration it experiences is g, which is constant at all points.

32. ConcepTest 2.9b Free Fall II Alice Bill v0 vA vB Alice and Bill are at the top of a building. Alice throws her ball downward. Bill simply drops his ball. Which ball has the greater acceleration just after release? 1) Alice’s ball 2) it depends on how hard the ball was thrown 3) neither -- they both have the same acceleration 4) Bill’s ball

33. ConcepTest 2.9b Free Fall II Alice Bill v0 vA vB Alice and Bill are at the top of a building. Alice throws her ball downward. Bill simply drops his ball. Which ball has the greater acceleration just after release? 1) Alice’s ball 2) it depends on how hard the ball was thrown 3) neither -- they both have the same acceleration 4) Bill’s ball Both balls are in free fall once they are released, therefore they both feel the acceleration due to gravity (g). This acceleration is independent of the initial velocity of the ball. Follow-up: Which one has the greater velocity when they hit the ground?

34. ConcepTest 2.10a Up in the Air I 1) more than 10 m/s 2) 10 m/s 3) less than 10 m/s 4) zero 5) need more information You throw a ball upward with an initial speed of 10 m/s. Assuming that there is no air resistance, what is its speed when it returns to you?

35. ConcepTest 2.10a Up in the Air I 1) more than 10 m/s 2) 10 m/s 3) less than 10 m/s 4) zero 5) need more information You throw a ball upward with an initial speed of 10 m/s. Assuming that there is no air resistance, what is its speed when it returns to you? The ball is slowing down on the way up due to gravity. Eventually it stops. Then it accelerates downward due to gravity (again). Since a = g on the way up and on the way down, the ball reaches the same speed when it gets back to you as it had when it left.

36. ConcepTest 2.10b Up in the Air II Alice Bill v0 Alice Bill v0 v0 v0 H H vA vB vA vB 1) vA < vB 2) vA = vB 3) vA > vB 4) impossible to tell Alice and Bill are at the top of a cliff of heightH. Both throw a ball with initial speedv0, Alice straightdownand Bill straightup. The speeds of the balls when they hit the ground arevAandvB.If there is no air resistance,which is true?

37. ConcepTest 2.10b Up in the Air II Alice Bill v0 v0 H vA vB 1) vA < vB 2) vA = vB 3) vA > vB 4) impossible to tell Alice and Bill are at the top of a cliff of heightH. Both throw a ball with initial speedv0, Alice straightdownand Bill straightup. The speeds of the balls when they hit the ground arevAandvB.If there is no air resistance,which is true? Bill’s ball goes up and comes back down to Bill’s level. At that point, it is moving downward with v0, the same as Alice’s ball. Thus, it will hit the ground with the same speed as Alice’s ball. Follow-up: What happens if there is air resistance?

38. Projectile Motion 2-D Kinematics

39. Projectile Motion • A projectile is any object that is given an initial velocity or dropped and then follows a path determined entirely by the effects of gravity. • Projectiles - batted baseball, a thrown football, a package dropped from an airplane, a bullet shot from a rifle. • The path followed by a projectile is called its trajectory. • The trajectory of a projectile is a parabola.

41. Horizontally Launched Projectile

42. Horizontal and Vertical Motion • We can analyze projectile motion as a combination of horizontal motion with constant velocity and vertical motion with constant acceleration.

43. 3-5 Projectile Motion It can be understood by analyzing the horizontal and vertical motions separately.

44. Independence of Horizontal and Vertical Components The vertical force acts perpendicular to the horizontal motion and will not affect it since perpendicular components of motion are independent of each other. Thus, the projectile travels with a constant horizontal velocity and a downward vertical acceleration.

45. Independence of Horizontal and Vertical Motion Demo • Two balls released simultaneously. One ball dropped freely, another projected horizontally • Both balls fall the same vertical distance in equal times.

46. 3-5 Projectile Motion The speed in the x-direction is constant; in the y-direction the object moves with constant accelerationg. This photograph shows two balls that start to fall at the same time. The one on the right has an initial speed in the x-direction. It can be seen that vertical positions of the two balls are identical at identical times, while the horizontal position of the yellow ball increases linearly.

47. Projectile Motion Vertical motion: Vertical downward acceleration: Vertical velocity is constantly changing Horizontal motion: Horizontal velocity is never changing

48. Practice Problem The boy on a tower (h = 5m) throws a ball a distance of 20m. At what speed is the ball thrown? Given: Solution: Vertical: Horizontal: Find: t= 1.00 s

49. Do Now A stone is thrown horizontally at a speed of +5.0 m/s from the top of a cliff 80.0 m high. • How long does it take the stone to reach the bottom of a cliff? • How far from the base of the cliff does the stone strike the ground?

50. Do Now A stone is thrown horizontally at a speed of +5.0 m/s from the top of a cliff 80.0 m high. • How long does it take the stone to reach the bottom of a cliff? • How far from the base of the cliff does the stone strike the ground?