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>> In this video, we're going
to take a Closer Look
at Kinematic Equations.
Let's investigate
kinematic equations
and how to solve problems
involving one dimensional
motion with them.
Our first problem involves
the calculation of displacement,
while there is
a constant acceleration.
The problem states that a car
accelerates uniformly
from rest to a velocity
of 7.1 meters
per second in 8.1 seconds.
Find the displacement of
the car during this time.
When solving
kinematic equations,
the first thing we always
want to do is write down
the variables that are given,
as well as what we're
solving for.
In this question,
we're told that the car
accelerates uniformly.
This means the acceleration
is constant, although
we don't know its value
from the information given.
We can place that right here
in our data table.
We also know, since the question
indicated that the car
accelerated from rest,
that the initial velocity
is zero, so we can place that
in our data table right here.
The question asks us to solve
for displacement so we can
write down a "d"
with a question mark here,
since we know that's
the variable we're solving for.
We know the final velocity
is 7.1 meters per second,
so we can plug that in
for final velocity right here.
We also know that
the time elapsed was 8.1 meters
per second, so we can plug
that in for "t," right here.
Next, we want to consult
the kinematic equations,
and determine which equation
we should use to solve
the problem based on
the variables we are given.
Since we need to solve
for displacement,
and the only variables
we've been given are initial
and final velocities,
along with the time elapsed,
the equation we should use
would be displacement equals
one half initial velocity,
plus final velocity,
times time.
So when we plug in those values
and take 7.1 meters per second,
plus zero, then multiply that
times one half, times the time,
which was 8.1 seconds,
we find our answer to be
28.755, but if we will round
that to two significant figures,
that is 29 meters.
So the displacement of the car
was 29 meters.
Our next problem involves
a kinematic calculation
involving velocity
and displacement, while there
is a constant acceleration.
A car with an initial velocity
of 5.15 meters per second,
accelerates uniformly
at a rate of 2.75 meters
per second squared.
Find the final velocity
and the displacement
after 8.00 seconds.
As always, the first thing
we want to do is set up
a data table
with all of our variables.
The time was given to us
at 8.00 seconds, so let's
write that down here.
The initial velocity was given
to us at 5.15 meters per second,
so let's write that down here.
The acceleration was constant,
and it was given.
Its value was 2.57 meters
per second squared.
So let's write that down here.
We need to solve
for final velocity
as well as displacement.
When we look at
the kinematic equations
that we have available to us,
you may notice that if we use
the equation, final velocity
is equal to initial velocity,
plus acceleration, times time,
that we have enough information,
so we can solve
for final velocity
with that equation.
So let's do that first.
Let's plug in 5.15 meters
per second for initial velocity,
and we will add that to
the product of the acceleration,
which is 2.75 meters
per second squared,
times the time,
which is 8.00 seconds.
When we solve for
final velocity,
we find that the car
was traveling 27.2 meters
per second, after 8.00 seconds
have elapsed.
To solve for displacement,
we can now use the equation
that we used in
the first problem.
Displacement equals one half
of the product of the sum
of the initial and final
velocities, times the time.
So if we take one half
of the sum of 5.15 meters
per second, and 27.2 per second,
we get 16.2 meters per second,
and then if we multiply that
times the time elapsed,
which was 8.00 seconds,
we get 129.
So the displacement
was 129 meters for this car.
Now here's one for you
to try.
A car is traveling at
7.0 meters per second,
and accelerates at
a constant rate of 2.5 meters
per second squared to reach
a velocity of 12.0 meters
per second.
How long does it take
for the car to reach
its final velocity?
Go ahead, press pause.
Give it a try, and when
you think you have the answer,
come back and check your work.
Did you get 2.0 seconds?
If so, great job.
If not, let's take a look
at the solution together.
The first thing we wanted
to do was create a data table
with the variables
we were given, as well as
what we were solving for.
We knew that the car
was traveling at 7.0 meters
per second.
That was its initial velocity.
We also know from the problem
that the acceleration was
constant, and it was 2.5 meters
per second squared.
The question also tells us
that the final velocity
was 12.0 meters per second.
We needed to solve for the time.
So now that we know we need
to solve for time, and we were
given both initial
and final velocity,
as well as acceleration,
the kinematic equation to use
would be final velocity
is equal to initial velocity
plus the product of acceleration
and time.
If we rearrange that equation
for time by first subtracting
both sides of the equation
by the initial velocity,
then dividing both sides
of the equation
by the acceleration,
we would have the time isolated
on one side of the equation.
Now we can plug in our values
and find the difference in
final velocity
and initial velocity,
which would be 12.0 meters
per second, minus 7.0 meters
per second, which is 5.0 meters
per second.
Then, we can divide that
by the acceleration, which is
2.5 meters per second squared,
and we are left with the time,
2.0 seconds.
That's a closer look
at Kinematic Equations.
For more "Physics in Motion",
go to our home page,
where the entire series
is available to you.
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