Mr. P: Good morning.
Today we are going to introduce you to
and help you understand the range equation
of projectile motion.
Bo: Hey guys.
Billy: Hey Bo! -Bobby: Hi Bo!
♫ Flipping physics ♫
Mr. P: The range of a projectile in physics
means the horizontal displacement of the projectile
when the overall displacement in the Y direction
of the projectile equals zero.
Again, the range of a projectile
is defined as the displacement of the object
in the X direction when the total displacement of the object
in the Y direction is equal to zero.
It might be better to call it the horizontal range,
because it is the horizontal displacement,
however it is usually just called the range of a projectile.
In just a moment, we are going to walk through
the equation for range.
However, first I do want to clarify something.
Looking at the video that shows the range of the projectile.
Class, in the Y direction,
does the ball move?
Billy and Bo: Yes. -Bobby: Yes?
Mr. P: Bobby, how can the total displacement of the ball
in the Y direction be equal to zero
if it moves in the Y direciton.
Bobby: Oh right.
Because it starts and ends at the same height,
therefore the straight line distance between
where it starts and ends in the Y direction is zero.
Mr. P: Correct.
Now let's look at the variables in the equation.
Capital R is the range of the projectile.
V sub i is actually the magnitude
of the initial velocity vector,
these double vertical lines mean
the magnitude of the initial velocity.
Theta sub i is the initial angle,
or the launch angle of the projectile.
And notice because you have both the magnitude
and the direction of the initial velocity vector
that you do not have to resolve the initial velocity
into its components to use the range equation.
So the range equation should be read as,
the range of a projectile equals
the magnitude of the initial velocity squared
times the sine of two times the launch angle
all divided by the acceleration due to gravity.
Bobby: Good job holding the ball in the shot pocket, Mr. P.
Mr. P: Thanks Bobby.
Bo: The what?
Bobby: The shot pocket.
Bo: The shot pocket?
Bobby: Yeah, the shot pocket.
Mr. P: Bo, what is the value of little g,
the acceleration due to gravity,
here on planet Earth?
Bo: Negative 9.81 meters per second squared.
Mr. P: Which is exactly why I asked the question.
Because your answer is not correct.
The acceleration due to gravity here on planet Earth
is positive 9.81 meters per second squared.
I think the confusion arises from the fact that
the first time we use little g
is when we define the acceleration due to gravity
here on planet Earth in projectile motion.
Remember, in projectile motion...
the acceleration in the Y direction
is equal to negative g,
where g on planet Earth is a positive 9.81
meters per second squared.
Therefore the acceleration in the Y direciton
is equal to negative 9.81 meters per second squared.
Again, because g is a positive number.
Please remember that:
g is postive.
Billy, what is the maximum value
for the sine of any angle?
Billy: The sine of any angle can never be greater than one.
Mr. P: Correct, the sine of any angle
can not be greater than one.
Now we're gonna use this fact
to figure out the angle that will give
the maximum range given the same initial velocity magnitude.
In order to maximize the range equation,
we need to set the sine of two theta initial
equal to its maximum value of one.
So one equals the sine of two times the initial angle,
and we're solving for that initial angle,
so we take the inverse sine of both sides.
Which gives us two times the launch angle
is equal to the inverse sine of one.
The inverse sine of one is 90 degrees,
so then we divide both sides by two
and we get that the launch angle is equal to
90 degrees divided by two,
or 45 degrees.
In other words, a launch angle of 45 degrees
will give us the maximum range
given the same magnitude initial velocity.
Billy: So any launch angle other than 45 degrees
will give us smaller range, right?
Mr. P: Correct, Billy, that's why we just went through this.
Something else.
Bobby, what are the dimensions
on the acceleration due to gravity?
Bobby: Meters per second squared.
Why?
Mr. P: Because that tells you what dimension we need to use
on the range and on the initial speed
or the magnitude of the initial velocity.
Bobby: Oh right,
the dimensions need to match,
so the magnitude of the initial velocity
should be in meters per second,
and the range should be in meters.
But what about theta initial?
Should theta initial be in degrees or radians?
Mr. P: Correct, range needs to be in meters,
and the initial speed needs to be in meters per second.
As far as the initial angle,
it can be in either degrees or radians,
really usually we use degrees.
And you need to make sure that your calculator mode
matches the dimensions you're using for your angle.
Billy: Yeah, I'm always in radian mode after Calculus,
and I have to switch my calculator
to degree mode for Physics.
This one time, I took an entire quiz in radian mode
and I didn't notice til--
Mr. P: Billy, do you mind?
Billy: Oh sorry.
Mr. P: So we do have more to learn.
So one other thing about the range equation,
and it starts by remembering
the shape of the sine theta graph.
Remember the sine curve starts at zero,
and at 90 degrees, or pi over two radians,
it hits its maximum value of one.
Then it returns to zero at 180 degrees,
or pi radians.
Negative one at 270 degrees,
or three pi over two radians,
and then at 360 degrees,
or two pi radians,
it returns back to zero and starts the cycle over.
In other words, one full cycle is 360 degrees.
However, in the range equation,
we have the sine of two times theta,
which changes the graph slightly.
Now instead of having one full cycle in 360 degrees,
we have two full cycles.
That means that it takes 180 degrees for one full cycle
or just 45 degrees for a quarter cycle.
And you can see the maximum value
is at 45 degrees.
But that's not the main reason we're looking at this graph.
Watch what happens when I put a horizontal line
that intersects the sine two theta curve
at 30 degrees.
Notice that at 60 degrees,
it also intersects the sine two theta curve.
Bo: Ok, but what does that mean?
Mr. P: It means that the sine of two times theta
will the same for both 30 and 60 degrees.
Billy: Which means that the range
from 30 and 60 degrees will be the same as well.
Bo: Hm, that's pretty cool. - Billy: Yeah.
Bobby: Agreed.
Bo: Hold up.
Aren't 30 and 60 degrees called complementary angles?
Bobby: Oh yeah, complementary angles,
because they add up to 90 degrees.
Mr. P: That is correct.
Complementary angles are going to have the same range.
Mathematically, it looks like this.
The sine of two times the initial angle
equals the sine of two times the quantity
90 minus the initial angle.
For example, if we have a first launch angle
of 30 degrees,
a second launch angle that will give the same value
for sine of two theta initial,
and therefore the same range
given the same magnitude initial velocity,
is going to be 90 minus that first launch angle,
or 90 minus 30,
which is 60 degrees.
Which is exactly what we got
using the graph of sine two theta initial.
Now this is a graph that shows
the projectile motion parabolas
for various launch angles
all with the same initial velocity magnitude
of 30 meters per second.
Of course, an initial angle of 45 degrees
gives the largest range.
And you can see that launch angles
of 30 and 60 degrees have the same range.
The same is true for launch angles
of 75 and 15 degrees.
Now while 75 and 15 degrees
may have the same range,
they do not follow the same projectile motion path.
You can see that the larger initial angle
will have a higher maximum height,
and therefore will spend more time in projectile motion
before it returns back to the height it started from.
Let's do a quick review.
Capital R stands for the range,
which is the displacement in the X direction
when the overall displacement in the Y direction
is equal to zero
V sub i is the magnitude of the initial velocity
or the initial speed,
theta i is the initial angle,
or the launch angle,
g, the acceleration due to gravity here on planet Earth,
is a positive 9.81 meters per second squared.
Complementary angles will give you the same range,
and you get the maximum range
when the initial angle is 45 degrees.
Thank you very much for learning with me today.
I enjoyed learning with you.