Lecture 3:

1. Lecture 3:

2. Freely Falling Objects Free fall from rest: Free fall is the motion of an object subject only to the influence of gravity. The acceleration due to gravity is a constant, g. g = 9.8 m/s2 For free falling objects, assuming your x axis is pointing up, a = -g = -9.8 m/s2

3. 1-D motion of a vertical projectile

4. v v v 1-D motion of a vertical projectile v b: a: c: d: t t t t Question 1:

5. v v v 1-D motion of a vertical projectile v b: a: c: d: t t t t Question 1:

6. Basic equations

7. A ball is dropped from a height of 5.0 m. How long does it take to reach the floor? How fast will it be going when it hits? At what height will it be going half this speed?

8. A ball is dropped from a height of 5.0 m. How long does it take to reach the floor? {t,y} How fast will it be going when it hits the floor? {v,t} At what height will it be going half this speed? {y,v}

9. A ball is dropped from a height of 7.0 m. How long does it take to reach the floor? How fast will it be going when it hits? At what height will it be going half this speed?

10. Question 2Free Fall I a) its acceleration is constant everywhere b) at the top of its trajectory c) halfway to the top of its trajectory d) just after it leaves your hand e) 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?

11. Question 2Free Fall I a) its acceleration is constant everywhere b) at the top of its trajectory c) halfway to the top of its trajectory d) just after it leaves your hand e) 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.

12. Alice Bill v0 vA vB Question 3Free Fall II 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? a) Alice’s ball b) it depends on how hard the ball was thrown c) neither—they both have the same acceleration d) Bill’s ball

13. Follow-up: which one has the greater velocity when they hit the ground? Alice Bill v0 vA vB Question 3Free Fall II 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? a) Alice’s ball b) it depends on how hard the ball was thrown c) neither—they both have the same acceleration d) Bill’s ball Both balls are infree fallonce they are released, therefore they both feel theacceleration due to gravity(g).This acceleration is independent of the initial velocity of the ball.

14. Question 4Throwing Rocks I You drop a rock off a bridge. When the rock has fallen 4 m, you drop a second rock. As the two rocks continue to fall, what happens to their separation? a) the separation increases as they fall b) the separation stays constant at 4 m c) the separation decreases as they fall d) it is impossible to answer without more information

15. At any given time, the first rock always has a greater velocity than the second rock, therefore it will always be increasing its lead as it falls. Thus, the separation will increase. Question 4Throwing Rocks I You drop a rock off a bridge. When the rock has fallen 4 m, you drop a second rock. As the two rocks continue to fall, what happens to their separation? a) the separation increases as they fall b) the separation stays constant at 4 m c) the separation decreases as they fall d) it is impossible to answer without more information

16. A hot-air balloon has just lifted off and is rising at the constant rate of 2.0 m/s. Suddenly one of the passengers realizes she has left her camera on the ground. A friend picks it up and tosses it straight upward with an initial speed of 13 m/s. If the passenger is 2.5 m above her friend when the camera is tossed, how high is she when the camera reaches her?

17. Solution: we know how to get position as function of time balloon camera Find the time when these are equal A hot-air balloon has just lifted off and is rising at the constant rate of 2.0 m/s. Suddenly one of the passengers realizes she has left her camera on the ground. A friend picks it up and tosses it straight upward with an initial speed of 13 m/s. If the passenger is 2.5 m above her friend when the camera is tossed, how high is she when the camera reaches her?

18. Recall: Scalars Versus Vectors Scalar: number with units Example: Mass, temperature, kinetic energy Vector: quantity with magnitude and direction Example: displacement, velocity, acceleration

19. C = A + B Vector addition C B A

20. B A Adding and Subtracting Vectors C = A + B tail-to-head visualization Parallelogram visualization

21. -B is equal and opposite to B Adding and Subtracting Vectors D = A - B D = A - B C = A + B If then D = A +(- B)

22. Question 5 Vectors I a) same magnitude, but can be in any direction b) same magnitude, but must be in the same direction c) different magnitudes, but must be in the same direction d) same magnitude, but must be in opposite directions e) different magnitudes, but must be in opposite directions If two vectors are given such that A + B = 0, what can you say about the magnitude and direction of vectors A and B?

23. Question 5 Vectors I a) same magnitude, but can be in any direction b) same magnitude, but must be in the same direction c) different magnitudes, but must be in the same direction d) same magnitude, but must be in opposite directions e) different magnitudes, but must be in opposite directions If two vectors are given such that A + B = 0, what can you say about the magnitude and direction of vectors A and B? The magnitudes must be the same, but one vector must be pointing in the opposite direction of the other in order for the sum to come out to zero. You can prove this with the tip-to-tail method.

24. The Components of a Vector Can resolve vector into perpendicular components using a two-dimensional coordinate system: characterize a vector using magnitude |r| and direction θr or by using perpendicular components rx and ry

25. Magnitude (length) of a vector A is |A|, or simply A Ay Ax Calculating vector components Length, angle, and components can be calculated from each other using trigonometry: relationship of magnitudes of a vector and its component A2 = Ax2 + Ay2 Ax = A cos θ Ay = A sin θ tanθ = Ay / Ax

26. Adding and Subtracting Vectors • Find the components of each vector to be added. • Add the x- and y-components separately. • Find the resultant vector.

27. Scalar multiplication of a vector Multiplying unit vectors by scalars: the multiplier changes the length, and the sign indicates the direction.

28. ^ Ax = Ax x ^ Ay = Ay y A Unit Vectors Unit vectors are dimensionless vectors of unit length.

29. Question 6Vector Addition a) 0 b) 18 c) 37 d) 64 e) 100 You are adding vectors of length 20 and 40 units. Of the following choices, only one is a possible result for the magnitude. Which is it?

30. Question 6 Vector Addition a) 0 b) 18 c) 37 d) 64 e) 100 You are adding vectors of length 20 and 40 units. Of the following choices, only one is a possible result for the magnitude. Which is it? The minimum resultant occurs when the vectors are opposite, giving 20 units. The maximum resultant occurs when the vectors are aligned, giving 60 units. Anything in between is also possible for angles between 0° and 180°.

31. Position vector points from the origin to a location. The displacement vector points from the original position to the final position. Displacement and change in position

32. Average velocity vector: So is in the same direction as . t1 t2 Average Velocity

34. Instantaneous velocity vector v is always tangent to the path. t1 t2 Instantaneous

35. Average Acceleration Average acceleration vector is in the direction of the change in velocity:

36. 2-Dimensional Motion (sections 4.1-4.5)

37. Question 7: Vector Components II a) 30° b) 180° c) 90° d) 60° e) 45° A certain vector has x and y components that are equal in magnitude. Which of the following is a possible angle for this vector in a standard x-y coordinate system?

38. Question 7: Vector Components II a) 30° b) 180° c) 90° d) 60° e) 45° A certain vector has x and y components that are equal in magnitude. Which of the following is a possible angle for this vector in a standard x-y coordinate system? The angle of the vector is given by tan Θ = y/x. Thus, tan Θ = 1 in this case if x and y are equal, which means that the angle must be 45°.

39. Question 8: Acceleration and Velocity Vectors a) point 1 b) point 2 c) point 3 d) point 4 e) I cannot tell from that graph. Below is plotted the trajectory of a particle in two dimensions, along with instantaneous velocity and acceleration vectors at 4 points. For which point is the particle speeding up?

40. Question 8: Acceleration and Velocity Vectors a) point 1 b) point 2 c) point 3 d) point 4 e) I cannot tell from that graph. Below is plotted the trajectory of a particle in two dimensions, along with instantaneous velocity and acceleration vectors at 4 points. For which point is the particle speeding up? At point 4, the acceleration and velocity point in the same direction, so the particle is speeding up

41. The Components of Velocity Vector v vy vx Motion along each direction becomes a 1-D problem

42. Projectile Motion: objects moving under gravity y g Assumptions: • ignore air resistance • g = 9.81 m/s2, downward • ignore Earth’s rotation • y-axis points upward, x-axis points horizontally • acceleration in x-direction is zero • Acceleration in y-direction is -9.81 m/s2 vy x vx

43. These, then, are the basic equations of projectile motion:

45. Launch angle: direction of initial velocity with respect to horizontal

46. Zero Launch Angle In this case, the initial velocity in the y-direction is zero. Here are the equations of motion, with x0 = 0 and y0 = h:

47. Zero Launch Angle Eliminating t and solving for y as a function of x: This has the form y = a + bx2, which is the equation of a parabola. The landing point can be found by setting y = 0 and solving for x:

48. Trajectory of a zero launch-angle projectile horizontal points equally spaced vertical points not equally spaced parabolic y = a + bx2

49. ball 1 ball 2 Question 9: Drop and not a) Ahead (to the left) of Ball 2 b) Behind (to the right) of Ball 2 c) On top of Ball 2 d) impossible to say from the given information Where will Ball 1 land on the lower surface?

50. ball 1 ball 2 Question 9: Drop and not a) Ahead (to the left) of Ball 2 b) Behind (to the right) of Ball 2 c) On top of Ball 2 d) impossible to say from the given information Where will Ball 1 land on the lower surface?