Lecture 3: Free Fall & Vectors in Physics

1. Lecture 3: Free Fall & Vectors in Physics (sections 2.6-2.7, 3.1-3.6)

2. Basic equations

3. 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

4. Free-fall must exclude air resistance An object falling in air is subject to air resistance (and therefore is not freely falling).

5. 1-D motion of a vertical projectile S

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

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

8. Basic equations

9. Free Fall

10. Freely falling Object - more

11. Freely falling Object – even more

12. 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?

13. 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.

14. 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

15. 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.

16. 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

17. 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

18. 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?

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

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

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

22. -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)

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?

24. 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.

25. 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

26. 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

27. The Components of a Vector Signs of vector components:

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

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

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

31. 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?

32. 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°.

33. Assignment 2 on MasteringPhysics. Due Tuesday, September 9. • Reading, for next class (4.1-4.5) • When you exit, please use the REAR doors!