(2)+Kinematics

PAGE EDITORS: Julian Helguero-Kelley, Alex Lopez, Zack Berman, & Nora Fahim


 * Due Date: 11/02/09**

What is Gravity? Some background on Newton's Theory of Gravity **
 * media type="youtube" key="shhjax7jGZU" height="248" width="438"

**Kinematics:**    **n.**     **(used with a sing. verb)**     **The branch of mechanics that studies the motion of a body or a system of bodies without consideration given to its mass or the forces acting on it**.

__**1-D Motion**__
 * everything is a __particle__: all mass is focused at a single point (no size)

__in words__: __equation__: D x = x - x x 0 means xinitial, where the symbol is referred to as "nought" __units__: meters
 * __Displacement__** __(__ D x)-vector
 * the shortest distance between the starting point and ending point of an object's motion, including direction
 * change in position



__in words__: the displacement that is covered by a given time interval __equation__: Vavg= (displacement)/(time)= D x/ D t __units__:m/s
 * Displacement vs. Total Distance Travelled( **TDT)
 * __Average Velocity__** (Vavg) - vector
 * velocity gets its direction (and sign) from the displacement vector

__equation__: Savg=TDT/ D t __units__: m/s
 * __Average Speed__** (Savg) -scalar


 * __Average Value Equation__**




 * DAY 2 (Derivations of Average Velocity and Speed)**

__Average Velocity__   · speed and velocity are usually the same thing except in two scenarios · cases where the differ: o 1. when displacement does not equal TDT o 2. since velocity is a vector, as is displacement, it will be negative when displacement is negative, but speed will always be positive, being a scalar quantity

__**Average Speed**__ S(t)= __|dx|__ dt



à __V on a graph__

Instantaneous velocity: slope of the tangent line to the graph at a point in time (compare the steepness of the slopes to compare the magnitudes instantaneous velocities)

__in words__: __equation__:
 * __Average Acceleration__**(a)-vector
 * the time rate of change of velocity
 * how fast the velocity changes

aavg= D v/ D t

__units__: m/s² __derivation__:


 * __Instataneous Acceleration__**:

a= lim D v/ D t= dv/dt= d(dx/dt)/dt= d²x/dt² D t→0 v=dx/dt

acceleration is the second derivative of position

Sample Problem in text: page19

a=dv/dt a(dt)=dv integrate both sides at=v+c c=integration constant; to find, set t=0 v=c


 * in physics, always use definite integrals, constant "c" represents an initialvalue for the fuction

__Two cases where acceleration is negative__:
 * 1) Object is speeding up in a specified negative direction
 * 2) Object is slowing down in the positive direction
 * When something is slowing down, acceleration must be in the opposite direction of the velocity (different signs)

o When you see a graph of motion, relate it to actual motion o Be able to go from actual motion to a graph as well o If the slope of x(t) is negative, you can only derive the velocity. You cannot find the speed since it cannot be negative. Speed could = |m| o Slope of a position vs. time graph corresponds to the velocity (or y-axis) values of a velocity vs. time graph o Also, the slope of a velocity vs. time graph corresponds to the y-axis values of the acceleration vs. time graph
 * __Graphing Position, Velocity, and Acceleration vs. Time__ **

__Position vs. Time__

Slope= D x / Dt = dx/dt

The blue lines all depict motion at a constant rate (same velocity because of same slope). The only difference is that they have different initial positions.  Example scenario: At a constant speed, Jill is walking back home from school. (Velocity would be constant but negative.) <span style="font-family: 'Arial','sans-serif'; font-size: 10pt; line-height: 115%; mso-ansi-language: EN-US; mso-bidi-language: AR-SA; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin;"> <span style="color: red; font-family: 'Calibri','sans-serif'; font-size: 10pt; line-height: 115%; mso-ansi-language: EN-US; mso-bidi-font-family: Arial; mso-bidi-language: AR-SA; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin;">General Form: y=mx²+b In this case: x=x0+v0t+½at² The slope is constantly increasing so velocity is not constant <span style="font-family: 'Arial','sans-serif'; font-size: 10pt; line-height: 115%; mso-ansi-language: EN-US; mso-bidi-language: AR-SA; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin;"> <span style="color: #76923c; font-family: 'Calibri','sans-serif'; font-size: 10pt; line-height: 115%; mso-ansi-language: EN-US; mso-bidi-font-family: Arial; mso-bidi-language: AR-SA; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin; mso-themecolor: accent3; mso-themeshade: 191;">The slope of the line (velocity) is 0, meaning that the position never changes. Someone or something is motionless for t seconds at a certain distance away from the origin.

__Velocity vs. Time__ Slope= D v/ Dt = dv/dt

<span style="font-family: 'Calibri','sans-serif'; font-size: 11pt; line-height: 115%; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; msoansilanguage: EN-US; msoasciithemefont: minor-latin; msobidifontfamily: 'Times New Roman'; msobidilanguage: AR-SA; msobidithemefont: minor-bidi; msofareastfontfamily: Calibri; msofareastlanguage: EN-US; msofareastthemefont: minor-latin; msohansithemefont: minor-latin; msospacerun: yes;"> __Acceleration vs. Time__

Slope= D a/ D t = da/dt

The red line is an example of a constant, positive acceleration

__General steps to solve constant acceleration problems__

1. Draw a sketch of the motion, including a coordinate axis 2. List all of the knowns and unknowns 3. Choose an equation 4. Solve the equation symbolically 5. "Plug n' Chug" (plug in numbers) =__Projectile Motion: 2-D Motion__=

__Kinematic Equations__: v=velocity (m/s) v0=initial velocity (m/s) a=acceleration (m/s²) t=time interval/ a moment in time-instantaneous (s) D x=change in the position=x-x0 (m) (note: direction is important-- pay attention to the signs of vector quantities)
 * only for use in the case of __constant__ acceleration


 * 1) v=v0+at
 * 2) D x=v0t+½a D t
 * 3) ​v²=v0²+2a D x

__Derivation of Kinematic Equations__ It all starts with Newton's 2nd law of motion:



__Freefall Motion__
 * Objects in motion solely under the influence of gravity are said to be in free-fall
 * It is assumed that there is no air resistance
 * If any object is falling, its displacement is negative

__Newton's Law of Gravitation media type="youtube" key="hZi8TXtRRYg" height="344" width="425"__ r2 G= universal gravitational constant (determined experimentally) = 6.67x10-11 (Nm2)/(kg2) m= masses involved with the attraction r= radial distance between the centers of the 2 objects Therefore: Fg= __GM__E__M__object RE = __6.67x10__-11 __(Nm__2__)/(kg__2__) * 5.98x10__24__kg * M__object 6.38x106m2 = 9.8 N/kg (Mobject) 9.8 m/s2 (Mobject) [and since 9.8 m/s2 g]….= Mobject* g
 * Every object with mass will attract any other object with mass
 * The force between objects would generally be weak (because the small constant G would be outweighed by gravity) UNLESS the mass of the object is VERY large- like the mass of the earth
 * Fg= __Gm__1__m__2

<span style="font-family: arial,helvetica,sans-serif; font-size: 13px; line-height: 19px;"> Stone is dropped from height h (initial velocity is 0 since it is "dropped") Stone is thrown downward Stone is thrown upward
 * All objects in free-fall will undergo the same acceleration
 * · ag= -g = -9.8 m/s2 (the rate at which the velocity changes)
 * Even when v=0 the magnitude and direction of a remain the same the __entire__ time that the object is in free-fall

Tips:
 * write subscripts relative to x and y
 * x and y movement are independent of each other

__**Conceptual Questions**__ 1. Figure 2-15 shows four paths along which objects move from a starting point to a final point, all in the same time interval. The paths pass over a grid of equally spaced straight lines. Rank the paths according to (a) the average velocity of the objects and (b) the average speed of the objects, greatest first. <span style="font-family: Verdana,Helvetica,Arial,sans-serif; font-size: 12px; line-height: normal;">
 * [[image:http://edugen.wiley.com/edugen/courses/crs1650/art/images/halliday8019c02/image_t/tfg015.gif align="center" caption="Figure 2-15"]] ||
 * Answer: a) They are all the same because the displacement for all is one unit and the time interval is the same. (b) 4, (1 and 2), 3

<span style="font-family: Verdana,Helvetica,Arial,sans-serif; font-size: 12px; line-height: normal;">2. (a) Is it possible to be accelerating while traveling at constant speed? Is it possible to round a curve with (b) zero acceleration and (c) a constant magnitude of acceleration? Answer: (a) Yes, if you round a turn. (b) a=(v^2)/r so if a=0 then v=0 and you wouldn't be moving. (c) Yes

__**Links:**__ [] Just watch this video. Good times. Shows a crazy video of an "impossible" basketball shot. Is this possible? Look at the analysis from "Popular Science" columnist Adam Weiner who is also the author of the book: Don't Try This at Home! The Physics of Hollywood Movies. Make your own decision!

[|Gravity Tractor could deflect killer asteroids]Article shows real-life application of gravity beams. [|Japanese Human Bottle Rockets] This article also has a video that shows the launch of a human via bottle rocket jetpack. [|Earthquake Scars Earth's Gravity] Article shows the effects of an earthquake on Earth's gravity. [|What Causes Gravity?] This article takes it a bit further than Newton's law and probes what actually causes gravity.



This picture shows the motion of a basketball in two dimensions.

media type="youtube" key="xZYF4fnAs3c" height="344" width="425" Gravity by John Mayer - self explanatory http://www.youtube.com/watch?v=xZYF4fnAs3c Enjoy!

Basketball: [|www.aapt.com] Sheep: [|www.wunderground.com] Cat: **[] ||
 * __SOURCES__