- [Instructor] What does distance mean?
In physics, distance is the total length traveled
for a particular trip.
In other words, if an object went forward then backward
then forward again, to find the total distance traveled
you would take this positive path length add it
to this positive path length
and then add this positive path length which would give you
the total distance traveled.
That's why a formula for distance can be thought of
as the summation of all the individual
positive path lengths.
The units for distance are meters, it is not a vector,
and its always positive or zero, it can never be negative.
What's an example problem for distance look like?
Let's say you have this graph and the question was
what's the distance traveled for the object in the graph
between zero and six seconds.
So between zero and two seconds,
the object moved forward four meters.
Between two and four seconds,
the object moved backward four meters.
Between four and six seconds,
the object moved backward another four meters
which means the total distance would be four meters
plus four meters plus four meters which would give you
12 meters of distance traveled.
What does displacement mean?
In physics, displacement is the difference in position.
If an object forward then backward then forward again,
the displacement could be represented with an arrow
that points from the initial position all the way
to the final position.
That's why a formula for displacement could be thought of
as the final position minus the initial position.
This delta here represents the change in
which means final minus initial.
The SI units for displacement are meters.
It's a vector which means if the displacement is leftward
or downward the displacement can be negative.
In other words, if you end up further left
from where you started,
your horizontal displacement's negative,
and if you end up further down from where you started
your vertical displacement is negative.
What's an example problem fro displacement look like?
If you were given this graph and you were asked
to find the displacement of the object
between zero and six seconds, you wouldn't care
that the object moved forward and backward
and changed directions.
All that you would focus on is that the initial position
was zero, the final position at six seconds
was negative four.
That means the displacement was negative four meters
since you ended four meters behind where you started.
What does speed mean?
Speed is a way to measure how fast something's moving.
It's a scalar.
For our purposes is really just another way
of saying not a vector.
Speed is defined to be the distance per time.
You could define average speed as the distance per time
over a large time interval,
or you could define instantaneous speed
as the rate of distance per time
at a particular moment in time.
In other words, if an object is taking a long winding path,
you could find the average speed for the entire trip
by taking the total distance for the entire trip
and dividing by the time it took to travel that distance,
or to determine the instantaneous speed,
you could look at an infinitesimally small distance
and divide by the time it took to travel
that infinitesimally small distance.
The units for speed are meters per second.
It is not a vector, and speed can only be positive or zero.
Speed can never be negative.
What's an example problem for speed look like?
If you had this graph and you wanted
to determine the average speed between zero and six seconds,
you'd have to remember that average speed
is the distance per time.
The distance between zero and six seconds would be
four meters plus four meters plus another four meters
would be 12 meters of distance, and the time it took
to travel that 12 meters of distance was six seconds
which gives us an average speed of two meters per second.
What does velocity mean?
Velocity's another way to determine
how fast something's moving,
but this one's a vector which means velocity
can be positive or negative.
Velocity's defined to be the displacement per time.
You could define the average velocity
as the displacement per time
over a large time interval, or you could define
the instantaneous velocity as the rate of displacement
per time at a particular moment in time.
In other words, if an object took a long winding path
and you wanted to determine the average velocity
for the entire trip, you would take the displacement
for the entire trip and divide by the time it took
for that displacement,
or if you wanted the instantaneous velocity
at a particular moment, you would take
an infinitesimal displacement at that moment
and divide by the time it took for that displacement.
The units of velocity are meters per second,
and it is a vector.
That means it can be negative.
If the velocity is directed leftward or downward,
we typically consider the velocity to be a negative value.
What's an example problem for velocity look like?
If you have this graph and you wanted
to determine the average velocity of the object
between zero and six seconds, you'd have to remember
that average velocity is the displacement per time.
Since the object starts at zero meters and ends
at six seconds at negative four meters, the displacement
is negative four and the time it took to travel
that displacement was six seconds which gives
us an average velocity
of negative two-thirds meters per second
for the time interval between zero and six.
What does acceleration mean?
Acceleration is the rate at which velocity changes.
You could change your velocity by speeding up,
slowing down, or by changing directions by turning.
It's good to remember that the acceleration of an object
always points in the same direction as the net force
on that object.
In other words, if you had a ball that had
an initial velocity v i to the right and the ball speeds
up to a larger v final to the right,
there must have been acceleration since the velocity changed
and since acceleration always points in the direction
of the change in velocity since this object gained velocity
to the right the acceleration points to the right
which also means there must have been a net force
on this ball to the right as well.
The formula for acceleration is the change in velocity
over the time it took for that velocity to change.
Since delta means final minus initial,
you could write this as the final velocity
minus the initial velocity over the time it took
for that velocity to change.
The units of acceleration are meters per second squared
since acceleration's telling you how many meters per second
the velocity is changing by per second.
Acceleration is a vector which means it can be negative
if the acceleration points left or down.
We typically consider that acceleration
to have a negative value.
What's an example problem for acceleration look like?
Let's determine the two cases where the acceleration
of this ball points to the right.
If a ball's moving to the right and it's slowing down,
there's gotta be a net force to the left that slows it down.
If the net force points to the left, the acceleration
also points to the left 'cause the net force
and the acceleration always point in the same direction.
If a ball's moving to the right and it's speeding up,
there's gotta be a net force to the right that's speeding
it up that also means the acceleration points
to the right as well.
If the ball's moving leftward and it's slowing down,
there's gotta be a net force to the right slowing it down
which means the acceleration also points to the right.
If a ball's moving leftward and it's speeding up,
there's gotta be a net force to the left
speeding the ball up which means the acceleration
also points to the left.
Since acceleration is always pointing in the same direction
as the net force, we can see that B and C are the two cases
where acceleration points to the right.
How do you interpret a position versus time graph?
Well the value of the graph is giving you the position X.
In other words, at some time t, if you measure the value
of the graph on the vertical axis that value is giving
you the position at that time t.
Why do we care about position versus time graphs?
One reason we care is that the slope
is gonna equal the velocity of the object.
If you can determine the slope, you can find the velocity.
How do you find the slope?
You find the region you want to determine the slope in
and then you take the rise over the run which is
the change in x over the change in time for that portion
of the graph.
Since the rise over run here is displacement per time,
the slope is equal to the velocity.
Also you can tell if there's acceleration on a position
versus time graph depending on
whether there's curvature or not.
Curvature that looks like a smiley face
represents positive acceleration.
Curvature that looks like a frowny face
represents negative acceleration.
Graphs that have constant slope would have constant velocity
and zero acceleration.
What would an example problem with position
versus time graphs look like?
Let's say we have this graph and we wanted
to determine the instantaneous velocity at seven seconds.
We locate seven seconds.
To find the velocity in this region,
we'll just find the slope in this region.
Since the slope is constant, we can find the slope
between any two points in this region
choosing six seconds and eight seconds
will be the most convenient.
We'll use rise over run.
The rise in this case would be the change in position.
The run would be the change in time.
Since between six seconds and eight seconds,
the graph drops by four meters the displacement
would be negative four meters.
The time it took for that to happen i.e. the run
would be two seconds.
Since the slope in this region is negative two meters
per second that also equals the velocity in this region.
How do you interpret a velocity versus time graph?
Well, the first thing you need to know is that the value
of the graph on the vertical axis is giving you the value
of the velocity at that particular moment in time.
Why do we care about velocity versus time graphs?
One reason we care is that the value of the slope
at any given moment is gonna be equal to the acceleration
of the object at that moment.
You find the slope using the definition
which is rise over run.
In this case, the rise is gonna be the change in v.
The run is gonna be the change in t and since the change
in velocity over the change in time is the definition
of acceleration, the slope on a velocity graph
is equal to the acceleration.
Also the area under any particular section of the graph
between two times is gonna equal the displacement
of the object between those two times.
What would an example problem involving velocity
versus time graphs look like?
Say you had this graph and you wanted to determine
the acceleration of the object at two seconds.
The acceleration is gonna be equal to the slope
at two seconds, so we need to find the slope
in this region.
Since the slope is constant, we can find the slope
between any two points between zero and four seconds.
I'm just gonna pick zero seconds and four seconds
because they're convenient, and we'll find the slope
which is always the rise over the run.
The rise in this case is the change in velocity.
The run is the change in time.
The change in velocity between zero seconds
and four seconds would be negative 60 meters per second.
The change in time between zero and four seconds
would be four seconds.
This gives us slope of negative 15 meters per second squared
which is also the acceleration in that time period.
If we wanted to determine the displacement of the object
between four and six seconds, we can determine the area
under the graph between four and six seconds,
the area under a rectangle is gonna be equal to the height
times the width.
The height in this case is gonna be equal to negative 30
and the width in this case is gonna be equal to two seconds
which gives us an area of negative 60 meters
which means the displacement between four and six seconds
was negative 60 meters.
How do you interpret an acceleration versus time graph?
Well the value of the vertical axis
is gonna give you the acceleration.
So if you read the graph at a particular moment in time,
the value of that vertical axis is giving
you the acceleration of the object at that moment in time.
The slope in this case is gonna be equal to the jerk
which really isn't asked about all that much,
but you do need to know that the change in velocity
is gonna be represented by the area under this graph.
If you can find the area under the graph between two times,
that area is gonna equal the change in velocity
experienced by that object between those two times.
What would an example problem involving
acceleration versus time graphs look like?
Let's say an object started from rest at t equals zero,
and you want to determine what this acceleration graph
what the velocity was at six seconds.
We know that the area is gonna represent the change
in velocity.
Since I'm going from zero seconds to six seconds,
I want to determine the area under the graph
between zero and six seconds this first area
would be a positive area.
This second area would be a negative area.
This third area would also be a negative area.
Note that this first triangular area will cancel
with this second triangular area.
Since one is positive and the other is equally negative.
The only important area in this case is gonna be
this rectangle which the height is negative 10
and the width is two seconds which gives us
negative 20 meters per second for the area.
That's also gonna equal the change in velocity.
Now since the object started from rest at t equals zero
and it changed its velocity by negative 20,
the final velocity at six seconds is gonna equal
negative 20 meters per second.
What are the kinematic formulas?
The kinematic formulas are formulas
that relate the five kinematic variables.
Those variables are displacement, initial velocity,
final velocity, acceleration, and time,
but these kinematic formulas only give true relationships
if the acceleration is constant.
These three kinematic formulas are given on the AP test,
but this last one is not.
One way to remember this formula is that the left hand side
is v final plus v initial over two which is the average
of the velocities, and the right hand side
is displacement per time which is the definition
of average velocity so it says that the average
of the velocities is equal to the average velocity.
In other words, during some motion it'll
take a certain time t for an object to change its velocity
from v initial to v final, and during that time
the object will have a certain acceleration and displacement
which will all be related by these formulas
if that acceleration is constant.
What would an example problem
involving kinematic formulas look like?
Let's say a confused chipmunk started from rest
and sped up with constant acceleration for three seconds
traveling nine meters in the process,
we could figure out the acceleration of the chipmunk
using the second kinematic formula.
The displacement would be nine meters.
The initial velocity would be zero since the chipmunk
started from rest.
The acceleration is the unknown,
and the time was three seconds which solving
for the acceleration gives us two meters per second squared.
We could also solve for the final speed of the chipmunk
using the first kinematic formula.
The final speed would be our unknown.
The initial speed is still zero.
We know the acceleration is two,
and the time was three seconds which gives a final velocity
of six meters per second.
The reason we could use these formulas
is because we knew the acceleration was constant.
A freely falling or flying object is any object
that has been dropped or thrown through the air
that only has the force of gravity acting upon it.
This means we'll typically ignore air resistance
on these objects unless otherwise stated.
We care about these objects because the acceleration
in the vertical direction for all freely falling
and flying objects near the earth
will be negative 9.8 meters per second squared.
Since this acceleration is gonna be constant
we can use the kinematic formulas to describe the motion
of freely flying or falling objects.
We just have to plug negative 9.8 in
for the vertical acceleration.
Watch out for code words.
Dropped is code word for initial velocity is zero.
The final velocity at the maximum height will be zero.
The acceleration due to gravity in the vertical direction
for a freely flying object is always
negative 9.8 meters per seconds squared.
Be careful.
If you analyze the motion of an object flying upward
the velocity gets smaller on the way up
until the velocity is zero at the top.
After which velocity gets more and more negative
on the way down, however, the acceleration
remains a constant 9.8 the entire trip on the way up,
on the way down, and even at the top of the motion
the acceleration and the vertical direction
is still negative 9.8 meters per second squared.
What would an example problem
involving freely flying objects look like?
Let's say a student drops a book from a height h,
and we want to know how long it takes for the book
to hit the ground, but we need to represent it in terms
of given quantities and fundamental constants.
That means we're gonna
have to solve this problem symbolically.
We still use a kinematic formula, but we'll plug
in symbolic values instead of numbers.
Delta y is not gonna be h.
It's gonna be negative h since the book dropped down
which is a negative vertical displacement.
The book was dropped.
That means the initial velocity was zero.
The acceleration due to gravity and the y direction
is negative 9.8, but since we're solving
this problem symbolically, we're not gonna use
an actual number.
We're gonna represent this as negative little g.
What we mean by little g is the magnitude
of the acceleration due to gravity which means little g
is positive 9.8.
Since the acceleration due to gravity is negative 9.8,
we can represent the acceleration due to gravity
as negative g.
Now we can solve for our time and we get the square root
of two h over g which is our symbolic answer
in terms of given quantities and fundamental constants.