Kinematic formulas and projectile motion
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Average Velocity for Constant Acceleration
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Acceleration of Aircraft Carrier Takeoff
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Deriving Displacement as a Function of Time, Acceleration and Initial Velocity
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Plotting Projectile Displacement, Acceleration, and Velocity
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Projectile Height Given Time
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Deriving Max Projectile Displacement Given Time
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Impact Velocity From Given Height
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Viewing g as the value of Earth's Gravitational Field Near the Surface
Deriving Displacement as a Function of Time, Acceleration and Initial Velocity Deriving displacement as a function of time, constant acceleration and initial velocity
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- What I want to do with this video
- is think about what happens to some type of projectile,
- maybe a ball or rock, if I were to throw it
- straight up into the air.
- To do that I want to plot distance relative to time.
- There are a few things I am going to tell you about
- my throwing the rock into the air.
- The rock will have an initial velocity (Vi) of 19.6 meters per second (19.6m/s)
- I picked this initial velocity because
- it will make the math a little bit easier.
- We also know the acceleration near the surface of the earth.
- We know the force of gravity
- near the surface of the earth is the mass of the obeject times the acceleration.
- (let me write this down)
- The force of gravity is going to be the mass of the object times little g.
- little g is gravity near the surface of the earth
- g is 9.8 meters per second squared (9.8m/s^2)
- Now if you want the acceleration on earth
- you just take the force divided by the mass
- Because we have the general equation
- Force equals mass times acceleration (F=ma)
- If you wan acceleration divide both sides by mass
- so you get force over mass
- So, lets just divide this by mass
- If you divide both sides by mass,
- on the left had side you will get acceleration
- and on the right hand side you will get the quantity little g.
- The whole reason why I did this is
- when we look at the g it really comes from
- the universal law of gravitation.
- You can really view g as
- measuring the gravitational field strength near the surface of the earth.
- Then that helps us figure out the force
- when you multiply mass times g.
- Then you use F=ma, the second law,
- to come up with g again
- which is actually the acceleration.
- This is accelerating you towards the center of the earth.
- The other thing I want to make clear:
- when you talk about the Force of gravity
- generally the force of gravity is equal to big G
- Big G (which is different than little g) times
- the product of the masses of the two things
- over the square of the distance between the two things.
- You might be saying "Wait, clearly the force of gravity is dependent on the distance.
- So if I were to throw something up into the air,
- won't the distance change."
- And you would be right!
- That is technically right, but
- the reality is that when you
- throw something up into the air
- that change in distance is so small
- relative to the distance between the object and the center of the earth
- that to make the math simple,
- When we are at or near the surface of the earth (including in our atmosphere)
- we can assume that it is constant.
- Remember that little g over there is
- all of these terms combined.
- If we assume that mass one (m1) is
- the mass of the earth, and
- r is the radius of the earth (the distance from the center of the earth)
- So you would be correct in thinking that it changes a little bit.
- The force of gravity changes a little bit, but
- for the sake of throwing things up into our atmosphere
- we can assume that it is constant.
- And if we were to calculate it
- it is 9.8 meters per second squared
- and I have rounded here to the nearest tenth.
- I want to be clear these are vector quantities.
- When we start throwing things up into the air
- the convention is
- if something is moving up it is given a positive value,
- and if it is moving down we give it a negative value.
- Well, for an object that is in free fall
- gravity would be accelerating it downwards, or
- the force of gravity is downwards.
- So, little g over here,
- if you want to give it its direction,
- is negative. Little g is -9.8m/s2.
- So, we have the acceleration due to gravity.
- The acceleration due to gravity (ag) is
- negative 9.8 meters per second squared (9.8m/s^2).
- Now I want to plot distance relative to time.
- Let's think about how we can set up a formula,
- derive a formula that, if we input time as a variable,
- we can get distance.
- We can assume these values right over here.
- Well actually I want to plot displacement over time because that will be more interesting.
- We know that displacement is
- the same thing as average velocity times change in time (displacement=Vavg*(t1-t2)).
- Right now we have
- something in terms of time, distance, and average velocity
- but not in terms of initial velocity and acceleration.
- We know that average velocity is the same thing as
- initial velocity (vi) plus final velocity (vf) over 2. (Vavg=(vi+vf)/2)
- If we assume constant acceleration.
- We can only calculate Vavg this way assuming constant acceleration.
- Once again when were are dealing
- with objects not too far from the center of the earth
- we can make that assumption.
- Assuming that we have a constant acceleration
- Once again we don't have what our final velocity is.
- So, we need to think about this a little more.
- We can express our final velocity in terms of our initial velocity and time.
- Just dealing with this part, the average velocity.
- So we can rewrite this expression
- as the initial velocity plus something over 2.
- and what is final velocity?
- Well the final velocity is going to be
- your initial velocity plus your acceleration times change in time.
- If you are starting at 10m/s
- and you are accelerated at 1m/s^2
- then after 1 second you will be going 1 second faster than that. (11m/s)
- So this right here is your final velocity.
- Let me make sure that these are all vector quantities...(draws vector arrows)
- All of these are vector quantities.
- Hopefully it is ingrained in you that these are all vector quantities, direction matters.
- And let's see how we can simplify this
- Well these two terms
- (remember we are just dealing with the average velocity here)
- These two terms if you combine them become 2 times initial velocity (2vi).
- two times my initial velocity
- and then divided by this 2
- plus all of this business divided by this 2.
- which is my acceleration times my change in time divided by 2.
- All of this was another way to write average velocity.
- the whole reason why I did this is because we don't have final velocity
- but we have acceleration
- and we are going to use change in time as our independent variable.
- We still have to multiply this by this green change in time here.
- multiply all of this times the green change in time.
- All of this is what displacement is going to be.
- This is displacement, and lets see...
- we can multiply the change in time times all this
- actually these 2s cancel out
- and we get (continued over here)
- We get: displacement is equal to
- initial velocity times
- change in time
- Some physics classes or textbooks put time there but it is really change in time.
- change in time is a little more accurate
- plus 1/2 (which is the same as dividing by 2)
- plus one half times the acceleration
- times the acceleration
- times (we have a delta t times delta t)
- change in time times change in time
- the triangle is delta and it just means "change in"
- so change in time times change in time
- is just change in times squared.
- In some classes you will see this written as
- d is equal to vi times t plus 1/2 a t squared
- this is the same exact thing
- they are just using d for displacement
- and t in place of delta t.
- The one thing I want you to realize with this video
- is that this is a very straight forward thing to derive.
- Maybe if you were under time pressure you would want to be able to whip this out,
- but the important thing, so you remember how to do this when you are 30 or 40 or 50
- or when you are an engineer and you are trying
- to send a rocket into space and you don't have a physics book to look it up,
- is that it comes from the simple displacement is equal to average velocity times change in time
- and we assume constant acceleration,
- and you can just derive the rest of this.
- I am going to leave you there in this video.
- Let me erase this part right over here.
- We are going to leave it right over here.
- In the next video we are going to use this
- formula we just derived.
- We are going to use this to actually
- plot the displacement vs time
- because that is interesting and we are going to be thinking about
- what happens to the velocity and the acceleration
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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