♪♪
How fast are these cars going?
Really, really fast.
The cars on this track have some incredible acceleration.
We're gonna be learning about acceleration and how it works,
and the major role that it plays in physics.
♪♪
Compared to the real thing,
this Porsche may not look all that impressive,
but it can still accelerate even if it's not as fast
as the one we just saw.
In physics, we define acceleration
as the rate of change of velocity over time.
The SI unit for acceleration is meters per second squared.
It's a vector quantity, so it has a magnitude
and a direction.
The formula for finding acceleration goes like this.
The acceleration of an object
measured in meters per second squared
is equal to the final velocity
of the object minus the initial velocity
of the object,
both in meters per second,
divided by the time taken in seconds.
And it doesn't matter if we're talking about
this little Porsche or a real one on a track.
The initial velocity of this car
as it's standing still
is zero meters per second.
The driver hits the gas
and it starts to accelerate.
1.5 seconds later,
the final velocity of the car
is 10.5 meters per second.
So, the acceleration will be equal to
10.5 meters per second
minus zero meters per second,
divided by 1.5 seconds.
Hence, our acceleration will be seven meters per second squared.
So acceleration is simply a change in the velocity
of an object over time.
And it doesn't matter if the object
is speeding up or slowing down.
For example, if your initial velocity and acceleration
have the same sine, you're speeding up.
Both sines could be positive, in which case,
you are accelerating in the positive direction,
or both sines could be negative,
in which case, you are accelerating
in the negative direction.
If your initial velocity and acceleration
have opposite sines,
then you're slowing down, or decelerating.
Remember from the previous segment that velocity
doesn't just have a magnitude.
It also has a direction.
So when an object is changing direction,
even if its velocity has the same magnitude,
it's still accelerating.
Acceleration doesn't just mean going faster.
It can mean slowing down,
or even going around a corner.
Now that we've spent some time on acceleration,
let's talk about kinematics.
Kinematics is the science of describing the motion
of an object.
There are a few ways to work through kinematic problems.
In this segment, we're going to use equations.
Kinematic equations give us the big picture
when we're talking about motion.
They show us the relationship between displacement,
velocity, acceleration, and time.
We can use kinematic equations to solve problems for scenarios
that have constant acceleration,
when the acceleration isn't changing,
instantaneous acceleration,
which is the acceleration
at a specific, given moment in time,
and average acceleration,
the overall change in velocity over time,
over the course of an entire journey.
In this physics course, all of our examples
will have acceleration that is constant.
There are four main kinematic equations,
and they can help us navigate most kinematic problems.
Each one of the equations has three or more of the variables
that we mentioned earlier, displacement, velocity,
acceleration, and time.
Let's take a look at the equations themselves.
The first one states that the final velocity of an object
is equal to the initial velocity of an object,
plus its acceleration multiplied by the time.
The second equation says that the displacement
of an object is equal to half times the sum of the initial velocity
and the final velocity, multiplied by the time.
The third states that displacement
is equal to the initial velocity
multiplied by the time,
plus half times the acceleration
multiplied by the times squared.
The fourth and last kinematic equation
states that the square of the final velocity
is equal to the square of the initial velocity
plus two times the acceleration
multiplied by the displacement.
So, when we want to solve a kinematic problem,
the first thing we want to do
is identify all the known variables.
Then we pick an equation that allows us to solve
for the variable that we don't know.
Let's go back to the race track for an example.
The car is moving along at 70 miles an hour,
which is about the same as 30 meters per second.
Let's measure the time it takes to get to a complete stop
once the brake is applied,
bringing the car to a gentle stop.
Okay, so exactly two seconds pass between hitting the brakes
and the car coming to a complete stand still.
Let's use this information to calculate our acceleration
as the car was slowing down.
The first thing we do is identify all the variables
that we know.
Our initial velocity was 30 meters per second.
Our final velocity was zero meters per second.
The time it took us to stop was two seconds.
The unknown we will be solving for is acceleration.
Out of the four kinematic equations,
this one best matches what we need.
It has initial velocity, final velocity, and time,
which we have the values for, and it has acceleration,
which will be our one unknown variable.
Starting with the equation, we can replace the variables
with known values.
Our final velocity of zero meters per second
equals the initial velocity of 30 meters per second
plus A,
which is our variable for acceleration,
times our time of two seconds.
We subtract 30 meters per second on both sides
to get rid of it on the right,
which gives us negative 30 meters per second
equals acceleration times two seconds.
We divide both sides by two seconds,
and find our acceleration to be negative fifteen meters
per second, per second,
or negative fifteen meters per second squared.
Because our values for initial velocity and acceleration
have opposite sines,
we know our car is slowing down.
Now, let's do an example where we solve
for displacement.
In this case, the initial velocity's
twenty meters per second,
final velocity is zero meters per second,
time is four seconds,
and our rate of deceleration is negative five meters
per second squared.
What equation can we use to find our displacement?
We actually have three choices.
They all contain our unknown variable displacement,
and we have enough information to plug into them to solve.
So we can use any of the three,
and we will get the same answer.
Let's use the simplest looking one.
Displacement equals one half times
our initial and final velocities added together,
multiplied by the time.
Let's plug in the values we know into the equation.
Displacement equals one half of twenty meters per second,
plus zero meters per second, times four seconds.
Twenty plus zero is twenty times one half,
which leaves us with ten meters per second
times four seconds.
Our seconds and our units cancel out,
leaving us with a displacement
of 40 meters.
Now, what do you think would happen to the stopping displacement
if the initial velocity of the car was doubled,
and our rate of deceleration
remained the same?
The stopping displacement
when the car is going 20 meters per second
was 40 meters.
The initial velocity of the car is now
40 meters per second, double what it was before.
The driver applies the brakes at the same spot as before,
and decelerates at the same rate as before.
Of the four kinematic equations,
we can use this one to calculate
the stopping distance.
Our final velocity is zero meters per second.
Our initial velocity is 40 meters per second,
and our rate of acceleration
is negative five meters per second squared.
When we work through the equation,
we find out that our stopping distance
is 160 meters,
more than three times what it was before.
So, as the initial velocity increased,
so did the stopping displacement.
This matters when looking at
how fast we're going,
and how far we have to go to stop.
Let's move on to how we can use graphs
to represent the acceleration of an object.
If we have a velocity versus time graph,
we can use it to learn about an object's acceleration.
In a velocity versus time graph,
the time elapsed will be shown on the X-axis,
and its velocity shown on the Y-axis.
This graph shows constant acceleration,
which on a velocity versus time graph
looks like a straight line.
By looking at the slope of a line
in a velocity time graph,
we can learn about an object's acceleration.
It makes sense that this object
is moving at a constant acceleration,
because the slope of the line
is not changing.
The velocity's increasing
by the same amount per unit time.
We can calculate the constant acceleration
by finding the slope of the slide,
but we can also have graphs that look like this
with a line going up and down.
This part indicates I have a velocity
that is increasing a constant amount.
I am speeding up in the positive direction.
This would be me moving at a constant deceleration
with my object continuing to move
in a positive direction
but with a velocity decreasing in magnitude,
slowing down in the positive direction.
It doesn't mean that I'm moving back
towards the origin.
The negative slope
just means that I'm slowing down.
But I'm still moving in the positive direction
since my velocities have a positive value.
Remember, the slope of the line
on a velocity versus time graph
shows the acceleration of the object.
In this segment, we talked about what acceleration is
and how we can calculate it.
We also went over kinematics
and the four equations
that can help us solve through
most problems of motion.
We've also looked at how we can deduce the magnitude
and direction of acceleration
by looking at the slope of the line
in a velocity time graph.
That's it for this segment of "Physics in Motion,"
and we'll see you next time.
(announcer) For more practice problems,
lab activities, and note-taking guides,
check out the "Physics in Motion" toolkit.