Lecture 3: Free Fall & Vectors in Physics (sections , )

Basic equations

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/s 2 For free falling objects, assuming your x axis is pointing up, a = -g = -9.8 m/s 2

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

1-D motion of a vertical projectile S

v t a: v t b: v t c: v t d: Question 1:

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

Basic equations

Free Fall

Freely falling Object - more

Freely falling Object – even more

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. 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?

Question 3 Free 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 v0v0 BillAlice vAvA vBvB

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. 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 v0v0 BillAlice vAvA vBvB Follow-up: which one has the greater velocity when they hit the ground? Question 3 Free Fall II

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 Question 4 Throwing Rocks I

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. 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 Question 4 Throwing Rocks I

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? Solution: we know how to get position as function of time balloon camera Find the time when these are equal

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

Vector addition A B C C = A + B

tail-to-head visualization Parallelogram visualization Adding and Subtracting Vectors B A

D = A - B If then D = A +(- B) C = A + B D = A - B -B is equal and opposite to B

If two vectors are given such that A + B = 0, what can you say about the magnitude and direction of vectors A and B ? 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 Question 5Vectors I

If two vectors are given such that A + B = 0, what can you say about the magnitude and direction of vectors A and B ? 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 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. Question 5Vectors I

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 θ r or by using perpendicular components r x and r y

Calculating vector components Length, angle, and components can be calculated from each other using trigonometry: A 2 = A x 2 + A y 2 A x = A cos θ A y = A sin θ tanθ = A y / A x AxAx AyAy Magnitude (length) of a vector A is |A|, or simply A relationship of magnitudes of a vector and its component

The Components of a Vector Signs of vector components:

Adding and Subtracting Vectors 1. Find the components of each vector to be added. 2. Add the x- and y-components separately. 3. Find the resultant vector.

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

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

Question 6Vector Addition Question 6 Vector Addition 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? a) 0 b) 18 c) 37 d) 64 e) 100

Question 6Vector Addition Question 6 Vector Addition a) 0 b) 18 c) 37 d) 64 e) 100 minimum opposite20 unitsmaximum aligned60 units 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°. 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?

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