1. PHYS 172: Modern Mechanics
Fall 2011
PHYS 172: Modern Mechanics
Lecture 2 – Vectors Momentum & Relativity Read:

2. Example of Vectors r Definitions: A Dr B
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”
Points from “old value” to “new value” Dr B A

3. 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

4. Definitions: Vector Operations 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

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

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

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

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

9. Indicators of interaction
Change of velocity
Change of identity H2 + O2  H2O
Change of shape bending a wire
Change of temperature heating pot of water on a hot stove
Lack of change when change is expected balloon floating in sky
Uniform motion: velocity is constant
Buoyancy – will address later

10. Today
Velocity
Momentum
Principle of Relativity

11. 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:

12. Velocity has Magnitude and Direction
Velocity is a Vector z x y
100 m in 10 s
The law should not depend on the choice of the coordinate system – can move origin to arbitrary location…
Definition:
average velocity

13. Example x y 1 2 3 4 5 6 7 8m 9 7m -1 -2

14. 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 and t
Instantaneous velocity:
derivative

15. Acceleration = Change in Velocity
Express as
Now use the chain rule to take the derivative:
Rate of change of direction
Will address later…
Rate of change of magnitude of velocity (speed)
is parallel to the velocity.

16. Predicting new position
The position update formula
Note: position update fromula is not postulated – it is derived from the definition of average velocity. The definition was postulated.

17. 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
Need quantitative description
(Legal Disclaimer: there's more to momentum
for objects near the speed of light!)

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

19. 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:

20. The principle of relativity
Physical laws work in the same way for observers in uniform motion as for observer at rest
Stop 2010
(when showing time the appearance of the guy (when cart is on the left), and the trajectory (also when cart is on left side)
Mention Einstein extensions: Special relativity, General relativity
Cats tail is at rest in cats frame – but moving in our reference frame
At pointes where velocity changes - violation
Another example: map on the dash in car – when turns, tends to go straight

21. 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: This lside (and the next) employ sound effect. Turn on audio and set the audio to PC. In room 114 you probably need to rise volume to ~80-90% of max.
A guy on a train throws a rock at a duck. If he knows the average velocity at which he throws the rock he can predict its motion using the position update formula in his own reference frame. It does not matter if train is at rest or moves at constant velocity – it is still an inertial reference frame.

22. 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
Now train moves at constant velocity. For the guy in the train everything looks the same – he throws the rock at the same speed, and of course after the same time the rock hits the goose.
However, in the reference frame of the guy standing on ground the rock must pass much longer distance in the same time. Still the same equation will predict the new position exactly – because in his reference frame velocity will be much larger!
So the same equation works in any inertial reference frame, but the parameters – coordinates, velocities etc. must be all measured in the respective frame to predict motion correctly.
Note: all parameters must be measured in respect to the selected reference frame to predict motion in respect to that reference frame 22

23. 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?

24. 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

25. Momentum – The Whole Story
Definition of momentum:
(Lorentz factor)
For v << c, 1, approximation: p=
v, m/s  1
300
30,000
3×107
1.005
0.9999c
70.7
YOU ARE HERE
30 km/s is about the fastest speed any man made device can reach.
The fastest manmade spaceship is robotic space probe Voyager 1 (722 kg) launched on Sept , it is still operational. Its mission was to visit Jupiter and Saturn. It is fastest because it used gravity boosts. Current speed – 17 km/s, currently about three times farther from Sun than Pluto (at distance ~110 au, Pluto is 40 au). The fastest speed was (due to gravity buoosts) 62,000 km/h = (
The record for the fastest spacecraft of all is held by the twin Helios probes that were placed in orbit around the Sun. Both these vehicles reached top speeds of around 150,000 mph (250,000 km/h=70 km/s) at closest approach to the Sun in their highly elliptical orbits, Helios 2 being marginally the swifter. ( p=
No mass can reach speed of light!