PHYS 172 Modern Mechanics Lecture 2 Vectors Momentum
- Slides: 26
PHYS 172: Modern Mechanics Lecture 2 – Vectors, Momentum, & Relativity Summer 2012 Read: 1. 6 -1. 11
Example of Vectors r • Definitions: Position Vector: A vector that gives the position of an object relative to an origin. common symbol units: meters (m) A 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 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”. 4 m 400 cm
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.
Vectors Unit vectors in the direction of the axes: General unit vector:
Indicators of interaction Ø Ø Ø Change of velocity Change of identity Change of shape Change of temperature Lack of change when change is expected H 2 + O 2 H 2 O bending a wire heating pot of water on a hot stove balloon floating in sky Uniform motion: velocity is constant
Q 1. 5. d What is the magnitude of the vector < 3, 5, − 2 >? A) 5. 48 B) 6. 16 C) 6. 00 D) 30. 00 E) 38. 00
Q 1. 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. 1. D. The object has no net interaction with the rest of the 2. 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:
Velocity has Magnitude and Direction Velocity is a Vector z y 100 m in 10 s x Definition: average velocity
Example y 9 8 m 7 6 5 4 3 2 -2 -1 x 1 2 3 4 5 6 7 m
Instantaneous vs. average velocity The trajectory of a ball through air: Instantaneous velocity at point B It is tangent to trajectory at point B It's the SLOPE! The average velocity will depend on the choice of Instantaneous velocity: derivative and t
Predicting new position The position update formula Units?
Interactions: changing velocity Newton’s first law of motion is qualitative: An object moves in a straight line and at constant speed except to the extent that it interacts with other objects 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 Units: Kg*m/s (Legal Disclaimer: there's more to momentum for objects near the speed of light!)
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: Units: Instantaneous rate of change of momentum:
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. ” (=in all inertial reference frames) The position update formula
RELATIVITY “Physical laws work in the same way for observers in uniform motion as for observers at rest. ” (=in all inertial reference frames) The position update formula 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. Physical laws work in the same way in any inertial frame Are you in an inertial reference frame right now?
Special theory of relativity Inertial frame moves at constant velocity. Speed of light = constant in all inertial reference frames! SPACE AND TIME WARP TO ENSURE THIS STAYS TRUE Time dilation: time runs slower in moving reference frames Length contraction: object length becomes shorter in moving reference frame
Momentum – The Whole Story Definition of momentum: (Lorentz factor) For v << c, 1, approximation: v, m/s 0 300 p= 1 1. 0000005 30, 000 1. 00005 3× 107 1. 005 0. 9999 c 70. 7 No mass can reach speed of light! p=
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