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##### Projectile motion (part 3) : An example of solving for the final velocity when you know the change in distance, time, initial velocity, and acceleration

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- In the last video, I said that we started off with the change
- in distance, so we said that we know
- the change in distance.
- These are the things that we are given.
- We're given the acceleration, we're given the initial
- velocity, and I asked you how do we figure out what the
- final velocity is?
- In the last video-- if you don't remember it, go watch
- that last video again-- we derived the formula that vf
- squared, the final velocity squared, is equal to the
- initial velocity squared plus 2 times
- the change in distance.
- You'll sometimes just see it written as 2 times distance,
- because we assume that the initial distance is at point
- 0, so the change in distance would just
- be the final distance.
- We could write it either way, and hopefully, at this point,
- you see why I keep switching between change in
- distance and distance.
- It's just so you're comfortable when you see it
- either way.
- This is for the situation when we didn't
- know what the vf was.
- Let's say we want to solve for time instead.
- Once we solve for the final velocity, we could actually
- solve for time, and I'll show you how to do that, but let's
- say we didn't want to go through this step-- how can we
- solve for time directly, given the change in distance, the
- acceleration, and the initial velocity?
- Let's go back once again to the most basic distance
- formula-- not the distance formula, but how distance
- relates to velocity.
- We know that-- I'll write it slightly different this time--
- the change in distance over the change in time is equal to
- the average velocity.
- We could have rewritten this as the change in distance is
- equal to the average velocity times the change in time.
- This is change in time and change in distance.
- Sometimes we'll just see this written as d equals-- let me
- write this in a different color, so we have some
- variety-- velocity times time, or d equals rate times time.
- The reason why I have change in distance here, or change in
- time, is that I'm not assuming necessarily that we're
- starting off at the point 0 or at time 0.
- If we do, then it just turns out to the final distance is
- equal to the average velocity times the final time, but
- let's stick to this.
- We want to figure out time given this set of inputs.
- Let's go from this equation.
- If we want to solve for time, or the change in time, we
- could just could divide both sides by the average
- velocity-- actually, no, let's not do that.
- Let's just stay in terms of change in distance.
- I've wasted space too fast, so let me clear
- this and start again.
- We're given change in distance, initial velocity,
- and acceleration, and we want to figure out what the time
- is-- it's really the change in time, but let's just assume
- that we start time 0, so it's kind of the final time.
- Let's just start with the simple formula: distance, or
- change in distance-- I'll use them interchangeably, with a
- lower case d this time-- is equal to the average velocity
- times time.
- What's the average velocity?
- The average velocity is just the initial velocity plus the
- final velocity over 2.
- The only reason why we can just average the initial and
- the final is because we're assuming constant
- acceleration, and that's very important, but in most
- projectile problems, we do have constant acceleration--
- downwards-- and that's gravity.
- We can assume, and we can do this-- we can say that the
- average of the initial and the final velocity is the average
- velocity, and then we multiply that times time.
- Can we use this equation directly?
- No. we know acceleration, but don't know final velocity.
- If we can write this final velocity in terms of the other
- things in this equation, then maybe we can solve for time.
- Let's try to do that: distance is equal to-- let me take a
- little side here.
- What do we know about final velocity?
- We know that the change in velocity is equal to
- acceleration times time, assuming that time
- starts a t equals 0.
- The change in velocity is the same thing is vf minus vi is
- equal to acceleration times time.
- We know that the final velocity is equal to the
- initial velocity plus acceleration times time.
- Let's substitute that back into what I was
- writing right here.
- We have distance is equal to the initial velocity plus the
- final velocity, so let's substitute this expression
- right here.
- The initial velocity, plus, now the final velocity is now
- the initial velocity, plus acceleration times time, and
- then we divide all of that by 2 times time.
- We get d is equal to-- we have 2 in the numerator, we have 2
- initial velocity, 2vi's plus at over 2, and all
- of that times t.
- Then we can simplify this.
- This equals d is equal to-- this 2 cancels out this 2, and
- then we distribute this t across both terms-- so d is
- equal to vit plus-- this term is at over 2, but then you
- multiply the t times here, too-- so it's at squared over
- 2 plus at squared over 2.
- We could use this formula if we know the change in
- distance, or the distance-- this actually should be the
- change in distance, and the change in time-- is equal to
- the initial velocity times time plus acceleration times
- squared divided by 2.
- Let me summarize all of the equations we have, because we
- really now have in our arsenal every equation that you really
- need to solve one dimensional projectile problems-- things
- going either just left, right, east, west, or north, south,
- but not both.
- I will do that in the next video.
- Let's summarize everything we know.
- We know the change in distance divided by the change in time
- is equal to velocity-- average velocity, and it would equal
- velocity if velocity's not changing, but average when
- velocity does change-- and we have constant acceleration,
- which is an important assumption.
- We know that the change in velocity divided by the change
- in time is equal to acceleration.
- We know the average velocity is equal to the final velocity
- plus the initial velocity over 2, and this assumes
- acceleration is constant.
- If we know the initial velocity, acceleration, and
- the distance, and we want to figure out the final velocity,
- we could use this formula: vf squared equals vi squared plus
- 2a times-- really the change in distance, so I'm going to
- write the change in distance, because that sometimes matters
- when we're dealing with direction-- change in
- distance, but so you'll sometimes just
- write this as distance.
- Then we just did the equation-- I think I did this
- in the third video, as well, early on-- but we also learned
- that distance is equal to the initial velocity times time
- plus at squared over 2.
- In that example that I did a couple of videos ago, where we
- had a cliff-- actually, I only have a minute
- left in this video.
- I will do that in the next presentation.
- I'll see you soon.