Lecture 3: Free Fall & Vectors in Physics

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

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

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

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.

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

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.

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

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

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?

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

C = A + B Vector addition C B A

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

-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)

Question 5Vectors 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?

Question 5Vectors 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.

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

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

The Components of a Vector Signs of vector components:

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

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

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?

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? 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°.

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 displacement

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

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

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

Instantaneous acceleration Velocity vector is always in the direction of motion; acceleration vector can points in the direction velocity is changing:

Velocity and Acceleration Question 6: Only one vector shown here can represent acceleration if the speed is constant. Which is it? a) 1 b) 2 c) 3 d) 4

Relative Motion Velocity vectors can add, just like displacement vectors The speed of the passenger with respect to the ground depends on the relative directions of the passenger’s and train’s speeds:

Relative Motion This also works in two dimensions:

You are riding on a Jet Ski at an angle of 35° upstream on a river flowing with a speed of 2.8 m/s. If your velocity relative to the ground is 9.5 m/s at an angle of 20.0° upstream, what is the speed of the Jet Ski relative to the water? (Note: Angles are measured relative to the x axis shown.)

Now suppose the Jet Ski is moving at a speed of 12 m/s relative to the water. (a) At what angle must you point the Jet Ski if your velocity relative to the ground is to be perpendicular to the shore of the river? (b) If you increase the speed of the Jet Ski relative to the water, does the angle in part (a) increase, decrease, or stay the same? Explain. (Note: Angles are measured relative to the x axis shown.)

Assignment 2 on MasteringPhysics. Due Monday, September 6. • Reading, for next class (4.1-4.5) • When you exit, please use the REAR doors!

Lecture 3: Free Fall & Vectors in Physics