Summer 2012 PHYS 172: Modern Mechanics Lecture 2 – Vectors, Momentum, & Relativity Read:

Example of Vectors Definitions: Position Vector: A vector that gives the position of an object relative to an origin. common symbol units: meters (m) Displacement Vector: Gives position of one point relative to another. common symbol “Delta r” units: meters (m) Points from “old value” to “new value” r A rr B

Vector Operations Definitions: Equality: Two vectors are equal if their magnitudes are equal and their directions are the same. remember- magnitude includes the units Negative of a Vector: The vector denotes the vector having the same magnitude as, but the opposite direction. But no such thing as a “negative vector”. 4m 400cm

Vector Operations Definitions: Multiplication of a Vector by a number: The vector denotes a vector having magnitude |m||A| and: 1) same direction as A if m is positive 2) opposite direction of A if m is negative

Vector Operations Definitions: Sum of Vectors: (Graphical representation of sum) 1.Redraw arrows “head to tail” (keep same direction and length) 2.Draw new arrow from tail of first arrow to tip of second arrow. 3.This arrow represents the vector sum.

Vector Operations Properties: Vector addition is commutative: Vector addition is associative:

Vector Operations Properties: Order of addition and multiplication: Definitions: Difference between Vectors: (Graphical representation of subtraction) 1.Redraw arrows “tail to tail” (keep same direction and length) 2.Draw new arrow from tail of second arrow to tip of first arrow. 3.This arrow represents the vector difference.

Unit vectors in the direction of the axes: General unit vector: Vectors

Indicators of interaction  Change of velocity  Change of identity H 2 + O 2  H 2 O  Change of shape bending a wire  Change of temperatureheating pot of water on a hot stove  Lack of change when change is expectedballoon floating in sky Uniform motion: velocity is constant

Q1.5.d What is the magnitude of the vector ? A) 5.48 B) 6.16 C) 6.00 D) E) 38.00

Q1.2.b Which of the following can NOT be true for an object moving in a straight line at a constant speed? A. Nothing is interacting with the object (it is in interstellar space, far from all other objects). B. The object is experiencing a net interaction. C. The object is experiencing multiple interactions, and these interactions add up to zero. D. The object has no net interaction with the rest of the world.

Today Velocity Momentum Principle of Relativity

Velocity has Magnitude and Direction Magnitude of Velocity = Speed (a scalar) 100 m in 10 s Average speed: If we know speed we can predict future: If we know speed we can reconstruct past:

100 m in 10 s z x y Definition: average velocity Velocity has Magnitude and Direction Velocity is a Vector

Example x y m 9 7m -2

Instantaneous vs. average velocity The average velocity will depend on the choice of and  t The trajectory of a ball through air: Instantaneous velocity at point B Instantaneous velocity: derivative It is tangent to trajectory at point B It's the SLOPE!

Acceleration = Change in Velocity Express as Now use the chain rule to take the derivative: Rate of change of magnitude of velocity Rate of change of direction

Predicting new position The position update formula Units?

Interactions: changing velocity An object moves in a straight line and at constant speed except to the extent that it interacts with other objects Newton’s first law of motion is qualitative: Interactions can change velocity! ? What factors make it difficult to change an object velocity? Mass! Introduce new parameter that involves product of mass and velocity: momentum (Legal Disclaimer: there's more to momentum for objects near the speed of light!) Units: Kg*m/s

Momentum p ≈ mv Momentum is in the same direction as velocity! Momentum can change in Magnitude, direction, or both! Δp ≈ mΔv

Average rate of change of momentum The stronger the interaction, the faster is the change in the momentum Average rate of change of momentum: Instantaneous rate of change of momentum: Units:

The principle of relativity Physical laws work in the same way for observers in uniform motion as for observer at rest

RELATIVITY “Physical laws work in the same way for observers in uniform motion as for observers at rest.” The position update formula (=in all inertial reference frames)

RELATIVITY “Physical laws work in the same way for observers in uniform motion as for observers at rest.” The position update formula (=in all inertial reference frames) Note: all parameters must be measured in respect to the selected reference frame to predict motion in respect to that reference frame

Inertial reference frame Inertial frame moves at constant velocity. Are you in an inertial reference frame right now? Physical laws work in the same way in any inertial frame

Special theory of relativity Speed of light = constant in all inertial reference frames! Time dilation: time runs slower in moving reference frames Length contraction: object length becomes shorter in moving reference frame Inertial frame moves at constant velocity. SPACE AND TIME WARP TO ENSURE THIS STAYS TRUE

Momentum – The Whole Story Definition of momentum: (Lorentz factor) For v << c,  1, approximation: v, m/s  , × c70.7 No mass can reach speed of light! p=