Physics – Kinematics
I. Introduction
v Motion à the change in position and/or orientation of an object.
§ All motion is relative à that is all objects move w/r/t other objects.
à in order to describe the motion of an object it must be compared to another object.
v Kinematics à the study of HOW things move.
v Dynamics à the study of WHY things move.
Distance vs. Displacement
v Position à the separation between an object and some reference point.
v Distance (d, s, r) à a scalar quantity describing the total path length.
à “how far did you actually travel?”
à think about miles added to the odometer
v Displacement (x, y, d, s, r) à a vector quantity describing the straight line path length between two points.
à “how far are you from where you started”
Examples: running around a track, road trip (total mileage vs. straight line distance), football
II. Speed vs. Velocity
v Speed (s, v) à the average speed of an object is the rate at which a (total) distance is covered in a (total) time period.
à speed is a scalar quantity.
v Velocity (v) à the average velocity of an object is the rate at which an object covers a given displacement.
à velocity is a vector quantity
Instantaneous Speed vs Instantaneous Velocity
Average speed/velocity considers the TOTAL distance/displacement traveled in a TOTAL TIME frame. Instantaneous speed/velocity is the rate at ONE SPECIFIC MOMENT.
Specifically the instantaneous speed/velocity is the rate of change in distance/displacement as the time frame approaches zero ()
Example: speedometer question
Velocity Big topics: narrative of each line segment, calculations off of graph (velocity), what does +/- slope indicate, what does above/below x axis indicate, how to figure out distance and displacement, where is east/west (positive/negative velocity), what does x axis indicate,
Graphing :
60 /50
40
30 / /
20
10
-10
-20
-30
-40
-50
****look at axis first to determine graph type*****
Line A:
Line B:
Line C:
Line D:
Line E:
Line F:
Line G:
What is the difference between line segments A, B and F?
X –axis (in this case) indicates:
Finding distance traveled on a velocity graph:
0-2 sec? 0-6 sec? 0-12 sec?
Finding displacement on a velocity graph:
0-2 sec? 0-6 sec? 0-12 sec?
What is the velocity from 0 - 2.0 seconds?
What is the velocity from 4.0-6.0 seconds?
What is the velocity at 4.5 seconds?
III. Graphing Acceleration
6 /5
4 / / /
3
2
1
-1
-2
-3
-4 /
-5
Line A:
Line B:
Line C:
Line D:
Line E:
Line F:
Line G:
What is velocity at 3 seconds?
What is velocity at 11seconds?
Define acceleration:
EQUATION:
What is the acceleration from 4-8 seconds?
Is the object speeding up or slowing down at 4-8 seconds? ______
What is the acceleration at 6.8 seconds?
What is the acceleration from 8-12seconds?
Is the object speeding up or slowing down at 8-12 seconds? ______
Speeding up or slowing down (Sign Big Picture)
You cannot figure out speeding up or slowing down off the sign on the acceleration alone!!!!
You must look at sign on both velocity and acceleration to figure it out.
SAME SIGN = speed up
OPPOSITE SIGNS = slowing down
Why is gray line segment from 4-8 seconds constant acceleration?
Video explanation link: https://www.youtube.com/watch?v=JoWwZa-HoGA
Acceleration Big topics: narrative of each line segment, calculations off of graph (acceleration), what does +/- slope indicate, what does above/below x axis indicate, how to figure out distance and displacement, what does x axis indicate,
Distance on an acceleration graph:
6 /5
4
3 /
2
1
-1
-2 /
-3
-4
-5
What is the distance from 6 – 11 seconds?
The area contained by the curve gives the distance or displacement of the motion
Distance = ∑ of the absolute value of all areas
· Remember distance is always goes up. You are always adding to your total distance
· │area 1│ + │area 2│ = distance
What is the displacement from 6 – 11 seconds?
The area contained by the curve gives the distance or displacement of the motion
.
Displacement = ∑ of all the areas
· Remember displacement is how far you are from where you started
· area 1 + area 2 = displacement
IV. Calculating Acceleration
v Acceleration à the average acceleration is the rate of change in an object’s VELOCITY.
v Fast acceleration video
v 5m/s2 means an object in moving at 5 m/s at one second and then increasing its velocity by 5m/s EVERY second. So the second second the object is moving at 10 m/s
When are you accelerating?
1. Gaining or losing speed
2. Change in direction
à acceleration is a vector quantity (it is defined using a vector quantity therefore it is also a vector quantity).
à DIRECTION! DIRECTION! DIRECTION
à an object whose velocity is changing the same amount over a period of time is experiencing Uniform Acceleration.
****** if a = 0 this could indicate the object is stopped BUT it can also indicate CONSTANT VELOCITY
à zero acceleration does NOT mean zero velocity!
Sample Problems:
A car takes 3.9 seconds to accelerate from 0 to 80.0 km/h. What is the magnitude of the average acceleration of the car?
Assuming the same acceleration how fast is the car traveling 2.0 seconds after it started the motion?
A car initially traveling at a speed of 30.0m/s slows down at a rate of 5.0m/s2 for 3.5 seconds. What is the car’s speed at the end of this interval?
A softball is initially traveling 12.0m/s is struck so it travels in the opposite direction at 17.0m/s. If the impact between the bat and the ball lasted 0.26 seconds, what was the ball’s acceleration?
IV. Uniform Acceleration and “The Big Five”
Find Average velocity
(m/s) / (m/s)5 / 5
4 / / 4
3 / 3
2 / 2 / /
1 / 1
Before we begin what can we say about the acceleration during 1-4s on graph one vs graph two?
Graph 1 =
Graph 2=
Remember to find average velocity we use
* remember the d means displacement in this case because we are solving for velocity
That’s a pain there is an easier equation.
Lets try it on both equations.
What happened? Why are the answers not the same as before????
What was the difference between the graphs????
Conclusion.
can ONLY be used when there is CONSTANT ACCELERATION!
This is #3 of the big five. You already know the first two
We know:
#1 #2 #3*
Derivation #1
Example Problem: A car traveling at 35 mph is to stop along a 35-m long stretch of road. What minimum acceleration is required?
vo = 35 mph = 15.6 m/s d = 35 m
vf = 0 a = ?
If we use what we currently know (the equations listed above), we CAN solve this problem, but it will take multiple steps. The more steps, the more likely you are to make a mistake. We can derive a new equation to help us solve this in one or two steps.
Derivation:
1) Average velocity:
2) Definition of an average:
3) Solve 1 for d:
4) Sub 2 into 3:
5) Average Accel:
6) Solve 5 for t:
7) Sub 6 into 4:
8) Simplify:
2ad = vf2-vo2
vf2 = vo2 + 2ad
Derivation #2
Example Problem: A car initially at rest, rolls down a hill such that it has a uniform acceleration of 5.0m/s2. How long will it take the car to travel 20.0-m down the hill?
vo = 0 d = 20 m t = ?
vf = ? a = 5 m/s2
If we use what we currently know (the equations listed above), we CAN solve this problem, but it will take multiple steps. The more steps, the more likely you are to make a mistake. We can derive a new equation to help us solve this in one or two steps.
Derivation:
1) Average velocity:
2) Definition of an average:
3) Solve 1 for d:
4) Sub 2 into 3:
5) Average Accel:
6) Solve 5 for vf: vf = vo + at
7) Sub 5 into 4:
8) Simplify:
d = vot + ½at2
The Big Five!
Always True (these are definitions) / True for uniform Acceleration/ d = vot + ½at2
vf2 = vo2 + 2ad
V. Free Fall
Example Problem:
A student drops a stone into a well. If the stone splashes into the water 1.0-second after it was dropped, how deep is the well?
The Law of Falling Bodies
NEGLECTING AIR RESTISTANCE, all objects fall with the same rate of acceleration. At sea-level (on Earth) this acceleration is 9.8 m/s2 DOWN (= 32.2 ft/s2).
· The acceleration due to gravity is uniform (for a given location) NO MATTER WHAT THE MASS OF THE OBJECT!!!
· This is NOT the force of gravity…it is the effect of the force of gravity!
History
à early “scientists” (like Aristotle) thought that heavier objects fall at a greater rate than light objects (based on observation of things like a stick verses a leaf). The problem was nobody ever TESTED anything to see if this was always true.
Demo: drop a book and a piece of paper
à GALILEO questioned this idea and designed an experiment to test it. This was the birth of the scientific method!
He HYPOTHESIZED that all bodies would demonstrate the same uniform acceleration when air (drag) was not a factor.
His METHOD was indirect. He couldn’t just time the fall for a falling body because there were no convenient timing methods (like stop watches) in the early 1600s.
He rolled various size/mass balls down a slight incline and would LISTEN for bells to chime.
Demos/examples:
Drop a book and a piece of paper (open) THEN crumple the paper and drop again à without the air resistance, they fall together.
Drop a tennis ball and a bowling ball
Video: Misconceptions about free fall
Web Clip: Moon Hammer and feather (David Scott August 2, 1971)
Video: Mechanical Universe (Galileo’s experiment)
Calculations: Free fall
You may recall in math:
H = -16t2 when Vo= 0
Where did this equation come from????
H = -16t2 when Vo= 0
d = V0t + ½ at2
Example Problem:
A stone dropped in to a well splashes into the water 1.5 seconds after it is released. How deep is the well?
Strategy – use chart for free fall, throw up, projectiles!!
#1 LIST WHAT YOU KNOW AND WHAT YOURE SOLVING FOR
VARIABLE / X / Y (UP) / Y (DOWN)Vo
Vf
a
d
t
Activity/Demo: Falling Dollar – Reaction Time Demo
Video Link
Dollar bill math explanation:
d = v0t + 1/2at2
since v0 = 0 (starting from rest) d = v0t + 1/2at2
resolve for t t = √(d/a) or t = √(2d/a)
since the catching fingers are at midway point, the falling distance of the dollar bill is about 8cm or 0.08m
t = √(2d/a)
t = √(2x0.08)/9.8
t = 0.128s
So what’s the big deal.
Most people’s reaction times are between 0.15sec and 0.20 sec, because it takes at least 0.143 seconds for nerve impulses to travel from the eye to the brain to the fingers. So MOST people will JUST miss the dollar!
Remember: Since acceleration is avector quantity, it has a direction associated with it. The direction of the acceleration vector depends on two things:
· whether the object is speeding up or slowing down
· whether the object is moving in the + or - direction
The general principle for determining the acceleationis:
If an object is slowing down, then its acceleration is in the opposite direction of its motion.
“Throw Up” Problems
Neglecting Air resistance the motion of an object projected upward is symmetrical!
GOING UP / TOP / GOING DOWNObject slows
a = 9.8m/s2 DOWN
mathematically we write this as:
a = -9.8m/s2
(acceleration is in opposite direction of gravity)
symmetry to way down / V= 0
a = 9.8m/s2 DOWN / Object gains speed
a = 9.8m/s2 DOWN
mathematically we write this as:
a = +9.8m/s2
(acceleration is in opposite direction of gravity)
symmetry to way up
OVERALL MOTION
dup = ddown
tup = tdown
*vup = vdown*
Example Problem: A boy throws a ball up with an initial speed of 12.0m/s. He catches it when it falls back down.
#1 what is the max height of the ball?
#2 What is the time to reach this elevation?
#3 what is the total time of the flight?
VARIABLE / X / Y (UP) / Y (DOWN)Vo
Vf
a
d
t
Sample #2 A clown throws a ball vertically upward with an initial speed of 5.5m/s. Determine the following
#1 max height
#2 time to reach the max height
#3 total flight time
#4 speed when the ball reaches his hand