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Physics Notes "1-D Kinematics"

· Concepts:
o Kinematics is the study of motion.  1-D kinematics is the study of motion in one direction only.

o A Motion Diagram is an abstract visual representation of an object’s motion.  The object is simplified for our purposes by treating the center of mass of an object as a point particle, & diagramming the motion of that, while ignoring the movement of any extraneous portions of the object.

o All motion is relative (i.e. object A moved relative to object B).  That is why you must define an origin in your motion diagram.  The origin serves as a fixed reference point the object moves in relation to.

o A coordinate system with a defined origin is used to draw an accurate motion diagram.  In a coordinate system you must maintain a consistent scale (i.e. each regularly spaced line must be the same scaled distance from any other line).

o Magnitude & Vector vs. Scalar
- Magnitude simply means “how much”.
- Scalar is a measure of magnitude.  Examples are temperature, time and mass (to help you remember this, think a scale measures scalar mass). Symbolically, scalar measurements (quantities) are represented as an ordinary typed or written variable symbol. I.e. d is the symbol for distance.
- Vector is a measure of magnitude with a direction.  Examples are displacements, velocity, and acceleration. Symbolically, vector measurements (quantities) are represented by bold typed letter symbols in print and hand written on paper using the printed symbols accompanied by an arrow drawn above them. I.e. d is the symbol for displacement.

o Displacement is the relative distance between two places or objects.  Note that not only is there a scalar distance component, but each place or object also has a direction you must head that distance in order to reach the other place or object (i.e. Western Avenue is 1 kilometer (magnitude) West (direction) of school).

o When an object experiences displacement (moves) it takes a certain amount of time to do so.

o Velocity is the word for when displacement changes over time.  The scalar component of velocity is speed.

o Average vs. Instantaneous:
- Average Velocity is displacement over time mathematically described as:`v = d / t
wherein`v = just means average velocity (a bar over a vector variable means average) and /\ means change of or change in a variable value.
- Instantaneous Velocity is the velocity at any given point.  The speedometer on a car measures the magnitude of the car’s instantaneous velocity.
- There is also Average and Instantaneous Acceleration.

o Acceleration is the change in velocity over time, just as velocity is the change in displacement over time.  Acceleration is also a vector quantity.
 

· Graphs:
o Graphs are one of our most powerful methods for visualizing and understanding physics.

o How to graph:
1. Take your equation and plug in several values for the independent variable and solve for the dependent variable.
2. Put these in a table.
3. Draw a coordinate plane with origin and a consistent and appropriate scale with the dependent variable on the left (y) axis and the independent variable on the bottom (x) axis.
4. Draw points on the coordinate plane corresponding to the variables in your table.
5. Draw a connecting line through these points. 

o Displacement-Time Graph:
Sometimes called a Position-Time graph, this graphs the displacement between two points/objects over time.

o Uniform Motion:
Is when an object experiences constant velocity.  The Displacement-time graph of a uniform motion situation is a straight line.  If the graph of a Displacement-Time graph is curved or bent at any point, then motion is not constant and the object is undergoing acceleration.

o Instantaneous Velocity:
In a Displacement-Time Graph instantaneous velocity can be found by graphing the straight tangent line to the point where the instantaneous velocity to be found is located.

o Velocity-Time Graph:
This graphs the velocity of an object over time.  You can get the displacement from a Velocity-Time graph by finding the area under the curve.  (Note: the area under the curve is not a graph of the displacement, even though it allows you to find the displacement).

o Acceleration-Time Graph:
This graphs the Acceleration of an object over time.

o Gravity and Free-Fall:
When an object in a vacuum experiences the force of earth’s gravity, it accelerates at ±9.8m/s^2 (read as meters per second squared).  For our purposes we will ignore wind resistance when doing falling body problems, and thus falling body problems are graphed like any other acceleration-time graph, but with the distinction that a always = ±9.8m/s^2.

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The Graphs below exemplify 3 different scenarios A, B, and C, and shows their Displacement-Time, Velocity-Time, and Acceleration-Time graphs.  This should help you in recognizing what conceptually is going on when you see a graph of a given situation.
 
 

Scenario A (blue) graphs an object that undergoes constant acceleration.  In the Displacement-Time graph you can see it travels further and further every second.  The shape of the graph is a parabola. In the Velocity-Time graph you can see that it speeds up over time. The shape of the graph is an ascending straight line. The Acceleration-Time graph shows a constant acceleration over time. The shape of this graph is a straight horizontal line, parallel to, but not on the x-axis.

Scenario B (pink) shows an object that is undergoing Uniform Motion.  You can see in the Displacement-Time graph that the object is moving steadily. The shape of the graph is an ascending straight line.  The Velocity-Time graph shows it has constant velocity. The shape of the graph is a straight line, parallel to, but not on the x-axis. The Acceleration-Time graph shows zero acceleration. It is a straight line on the x-axis.

Scenario C (yellow) shows an object at rest.  Its displacement is constant. The shape of the graph is an ascending straight line, parallel to, but not on the x-axis. The Velocity-Time and Acceleration-Time graph show zero motion. They are both straight lines on the on the x-axis.

NOTE:
The info below this line -----
 is in the process of being edited and shoud not be used in your notes yet.
 
 
 

· Equations
o In order to solve any equation follow these steps:
1.  List everything that you know (velocity? distance? Time?)
2. Convert all measurements and values to the right units (they all have to be the same units, for example don’t mix cm and m.  Usually you will want to put them into m & s, unless the answer specifically calls for another unit).
3. Figure out exactly what you need to find.
4. Find an equation that has the unknown variable you must find, and only has other variables that you know the values of.
5. If necessary, rearrange the equation to isolate the variable that you must find on one side of the equals sign.
6. Insert your known values into the equation and solve for the unknown value.
o List of 1-D Kinematics Equations
§ The Equations
· d = d0 + vt

· d = d0 + ½(v + v0)t

· d = d0 + v0t + ½at^2

· v = v0 + at

· v^2 = v0^2 + 2a(d – d0)

· `v = Dd / Dt = (d1 – d0) / (t1 – t0)

· `a = Dv / Dt = (v1 – v0) / (t1 – t0)

· Dd = vDt