Total Final Velocity for Projectile

Physics

Two-dimensional projectile motion

Visualizing Vectors in 2 Dimensions
Projectile at an Angle
Different Way to Determine Time in Air
Launching and Landing on Different Elevations
Total Displacement for Projectile
Total Final Velocity for Projectile
Correction to Total Final Velocity for Projectile
Projectile on an Incline
Unit Vectors and Engineering Notation
Clearing the Green Monster at Fenway
Green Monster at Fenway Part 2
Unit Vector Notation
Unit Vector Notation (part 2)
Projectile Motion with Ordered Set Notation

Total Final Velocity for Projectile Calculating the total final velocity for a projectile landing at a different altitude (mistake near end: I write 29.03 when it should be 26.03 m/s and the final total magnitude should be 26.55 m/s 78.7 degrees below horizontal

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Total Final Velocity for Projectile

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In the last video, I told you that we would figure out the final velocity of when this thing lands.
So, lets do that. I forgot to do that in the last video.
So, lets figure out the final velocity, the vertical and horizontal components of that final velocity,
and then we can reconstruct the total final velocity.
So the horizontal component is easy, because we already know that
the horizontal component of the velocity is this value over here.
This 30 cos of 80 degreees, and that is not going to change at any point in time.
So this is going to be the horizontal component of the projectile's velocity, when it lands.
but what we need to do is figure out the vertical component of its velocity.
But one thing we did figure out in the last video;
We figured out what the time in the air is going to be.
And we know a way of figuring out our final velocity from the initial velocity,
given our time in the air.
We know that a change in velocity
(only dealing with the vertical component, not the horizontal,
because the horizontal velocity is not going to change, as we have assumed that air resistance is negligible)
We know that the change in velocity, or the vertical component of the change in velocity,
is equal to the vertical component of the acceleration
multiplied by time.
Now we know what the change in time is...
And what is our, change in velocity?
It is our final vertical velocity, minus our initial vertical velocity.
And we know what our initial vertical velocity is, we solved for it...
We figured out that our initial vertical velocity was 29.54m/s, thats 30 sin 80 degrees.
So this is going to be minus 29.54 meters per second (m/s) is equal to...
our acceleration in the vertical direction, as it is accelerating us downward, negative 9.8 m/s squared
and our time in the air is 5.67 seconds, so times 5.67 seconds.
And so we can solve for the vertical component of our final velocity.
Once again, this is the vertical component, not the total one.
I wrote vertical up here so you know this is the vertical component.
So lets solve for this: if you add 29.54 to both sides,
you get the vertical component of your final velocity,
It is not marked up here properly.
Vertical component of final velocity is equal to 29.54m/s minus 9.8 meters per seconds squared
times 5.67 seconds, and the seconds cancel out with one of the seconds.
so everything is meters per second.
And so ill bring the calculator out again.
we have 29.54 minus 9.8, times 5.67.
So we get, our change, our final velocity is -26.03 ill say.
So this is, negative 26.03 m/s.
You might say, oh wait Sal! What does this negative 26.03 m/s mean?
Remember, when were dealing in the vertical dimension...
Positive means up, and negative means down.
So it means that we are going 26.03 m/s downwards, right when we land.
So what is our total velocity when we fall back to that landing?
The vertical component of our velocity is negative 29.03 in the downward direction.
And the horizontal component of our velocity, we know, hadn't changed the entire time.
That we figured it out, was 30 cosine of 80 degrees.
Here is 30 cosine of 80 degrees. Ill get the calculator out to caluclate it.
30 cosine of 80 degrees, which is equal to 5.21.
So this is 5.21 m/s, and these are both in meters per second.
So what is the total velocity?
Well I can do the heads to tails, and i can shift this guy over.
So that it's tail is at the head of the blue vector. So, it would look like that.
The length of this, the magnitude of our vertical component is 29.03.
And then we can just use the pythagorean theorem
to figure out the magnitute of the total velocity upon impact.
For the length of that, we can just use the pythagorean theorem!
So the magnitude of our total velocity, that's this length right here,
or our total final velocity, is going to be equal to
(let me write it this way), this is just straight from the pythagorean theorem.
(The square root of) 5.21 squared, plus 29.03 squared.
And we get it as being,
the square root of 5.21 squared plus 29.03 squared....
gives us, 29.49 m/s.
That is the magnitude of our final velocity, but we also need to figure out its direction.
So we need to figure out this angle, and now we are talking about an angle below the horizontal.
Or if you wanted to view it in pure terms, it would be a negative angle.
It is an angle below the horizontal. So what is this angle right over here?
If we view it as a positive angle in the traditional trigonometric way...
We could use any of the trig functions.
Let's use tangent, we could say that the tangent of the angle...
is equal to the opposite over the adjacent.
29.03 over 5.21
Or, that theta is equal to the inverse tangent, or the arc tangent
of 29.03 over 5.21, and that gives us...
the inverse tangent of 29.03 divided by 5.21 and we get...
79.8 degrees. But it is going to be 79.8 degrees south, or, below the horizontal.
Or you could view this as an angle of negative 79.8 degrees above the horizontal.
Either one of those work.
What is neat about this, is that we have our final velocity vector.
The entire vector, and we know what that vector is.
It is, 29.49 meters per second, at 79.8 degrees below the horizontal!

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When naming a variable, it is okay to use most letters, but some are reserved, like 'e', which represents the value 2.7831...

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