Chapter 2 - Motion in One-Direction
Definitions of words you'll hear a lot…
· Frame of reference - before solving any physical problems, you need to consider the frame of reference. The frame of reference is a choice of coordinate axes that defines the starting point for measuring necessary quantities. I'll often say "pick where your origin will be" and "choose a positive direction"; each of these things make up your frame of reference.
· Displacement - the displacement of an object is defined as its change in position. We will use to represent the displacement of an object. This is not the same thing as the distance traveled.
displacement = the change in position
Displacement is measured in meters (m).
· Vector - a vector is a quantity that is characterized by both a magnitude and a direction. Vectors are denoted in bold font or as letters with an arrow over the letter. (Example: or )
· Scalar - a scalar is a quantity that is characterized by only magnitude, not direction. (Example: time or speed)
· Average Speed - the average speed of an object is defined as the total distance traveled divided by the total time elapsed.
average speed = total distance/total time
Average speed is a scalar quantity and it is measured in meters per second (m/s).
· Average Velocity - the average velocity of an object is defined as the displacement divided by the total time elapsed. We will use to represent average velocity.
average velocity = displacement/total time
Average velocity is a vector quantity and it is measured in meters per second (m/s).
· Instantaneous Velocity - the instantaneous velocity of an object is defined as the limit of the average velocity as the time interval goes to zero. We will use to represent instantaneous velocity. We will usually be lazy and refer to this as "the velocity".
Instantaneous velocity is a vector quantity and it is measured in meters per second (m/s).
You drive from here to Atlantic City. The total distance traveled is 70 miles. You drive the first 20 miles at 55 mph and the next 45 miles at 65mph and the last 5 miles at 45 mph. The total trip takes you 1 hour and 30 minutes.
What is your displacement?
What is your average speed?
What is your average velocity?
You're done in Atlantic City and you drive back home. You travel another 70 miles but this time you drive the first 10 miles at 40 mph, the next 50 miles at 70 mph, and the last 10 miles at 35 mph. This time, the trip takes you 1 hour and 15 minutes.
What is your total displacement for your entire trip?
What is your average speed?
What is your average velocity?
· Average Acceleration - the average acceleration of an object is defined as the change in velocity divided by the total time elapsed (the change in time). We will use to represent average acceleration.
average acceleration = change in velocity/change in time
Average acceleration is a vector quantity and it is measured in meters per second per second (m/s2).
· Instantaneous Acceleration - the instantaneous acceleration of an object is defined as the limit of the average acceleration as the time interval goes to zero. We will use to represent instantaneous velocity.
Instantaneous acceleration is a vector quantity and it is measured in meters per second per second (m/s2).
Graphical Representations of Position, Velocity, and Acceleration
This is called a Position vs. Time Graph
The average velocity of an object during the time interval is equal to the slope of the straight line joining the initial and final points on a graph of the object's position vs. time.
Slope = change in vertical axis/change in horizontal axis
Slope = rise/run
Slope =
The slope of the line tangent to the position vs. time curve at "a given time" is defined to be the instantaneous velocity at that time.
Similarly, the slope of a Velocity vs. Time Graph is defined as the average acceleration. And the slope of the line tangent to the velocity vs. time plot at a given time is defined as the instantaneous acceleration.
You can plot Velocity vs. Time and Acceleration vs. Time Graphs too.
Match the velocity vs. time graphs to the appropriate acceleration vs. time graphs.
Graph (a) has a constant slope, indicating a constant acceleration, which is represented by (c).
Graph (b) represents an object with increasing speed, but as time progresses, the lines drawn tangent to the curve have increasing slopes. Since the acceleration is equal to the slope of the tangent line, the acceleration must be increasing and the graph that best indicates this behavior is graph (d).
Graph (c) depicts an abject that first has a velocity that increases at a constant rate, which means the acceleration is constant. The velocity then stops changing, which means the acceleration of the object is zero. This behavior is best matched by graph (f).
NOTE:
When there is no change in the velocity of an object, the acceleration of the object is zero.
Is there such a thing as a negative acceleration? Yes. If the object is slowing down, then the object is said to have a negative acceleration (deceleration is not a word we use in physics).
Motion Diagrams - A motion diagram is a representation of a moving object at successive time periods, with velocity and acceleration vectors sketched at each position. A motion diagrams looks like a series of photographs of the objects. The time intervals between "photographs" in the motion diagram are assumed to be equal.
One-Dimensional Motion with Constant Acceleration
When an object moves with constant acceleration, the instantaneous acceleration at any point in a time interval is equal to the value of the average acceleration over the entire time interval.
So, the average acceleration equals the instantaneous acceleration when is constant.
We can, for convenience, decide that . We can also let (the initial velocity at t = 0) and let (the velocity at any time t)
or
The graphical representations of motion with constant acceleration show that the velocity varies linearly with time. Because this change is linear (or uniform, using physics terms) over time, we can express the average velocity in any time interval as the average of the initial and final velocity:
Since
or
We can substitute in the above equation to get a different form
We can solve for one more very useful equation by solving for and plugging the result into
or
Example: A car traveling at a constant speed of 24 m/s (not 45 m/s as indicated in the picture) passes a hidden state trooper on the side of the road. One second after the speeding car passes the trooper, he sets off in chase with a constant acceleration of 3 m/s2. (a) What is the acceleration of the speeding car? (b) How long does it take the trooper to overtake the speeding car? (c) How fast is the trooper going at that time?
Freely Falling Objects
A freely falling object is any object moving freely under the influence of gravity alone, regardless of its initial motion.
Objects thrown upward or dropped (or thrown) downward are all considered freely falling.
We will denote the magnitude of the free-fall acceleration (gravity) by the symbol . The value of varies with altitude and it varies slightly with latitude; however for our purposes, we will consider the value of as a constant.
m/s2
Since is a constant acceleration, we can use all of the equations we derived for motion in one direction with constant acceleration. It is conventional to define "up" as the +y-direction and to use as the position variable. In that case, the acceleration is
m/s2