yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Analyzing motion problems: position | AP Calculus AB | Khan Academy


2m read
·Nov 11, 2024

Divya received the following problem: A particle moves in a straight line with velocity ( v(t) ) is equal to the square root of ( 3t - 1 ) meters per second, where ( t ) is time in seconds. At ( t = 2 ), the particle's distance from the starting point was eight meters in the positive direction. What is the particle's position at ( t = 7 ) seconds? Which expression should Divya use to solve the problem?

So pause this video and have a go at it all right now.

Let's do this together! So we want to know the particle's position at ( t = 7 ). They tell us what our position is at ( t = 2 ). Thus, the position at ( t = 7 ) would be your position at ( t = 2 ) plus your change in position from ( t = 2 ) to ( t = 7 ).

There's another word for this; you could also call this your displacement from ( t = 2 ) to ( t = 7 ). We know how to think about displacement: velocity is your rate of change of displacement. If you want to figure out your displacement between two times, you would integrate the velocity function.

So this is going to be the integral from ( t = 2 ) to ( t = 7 ) of our velocity function ( v(t) , dt ). This would be our displacement from time ( 2 ) to time ( 7 ). If they asked what our change in position from time ( 2 ) to time ( 7 ) is, it would be just this expression.

But they want us, or they want Divya, to figure out what the particle's position is at ( t = 7 ) seconds. So what you would want to do is take your position at ( t = 2 ). We know what our position at ( t = 2 ) is; it was 8 meters in the positive direction, so we could just call that positive 8 meters.

Therefore, it’s going to be ( 8 ) plus your change in position, which is going to be your displacement. We can see this choice right over there, and that’s what we would pick.

The first option, ( v(7) ), just gives us our velocity at time ( 7 ) or, exactly at ( 7 ) seconds, or in other words, our rate of change of displacement at ( 7 ) seconds. So that’s not what we want.

The second option shows your position at ( t = 2 ), but then you have your change in position from ( t = 0 ) to ( t = 7 ), so this doesn’t seem right.

Lastly, this is your position at time ( 2 ) plus ( v' ), the derivative of velocity, which is the acceleration, plus your acceleration at time ( 7 ). So that's definitely not going to give you the particle’s position. We like that second choice.

More Articles

View All
Valley of the Boom: Trailer #1 | National Geographic
That little A and At? See, that’s what I said. Mm-hm. Um, Katie said she thought it was “about.” Yeah. Oh. MAN: But I’d never heard it. KATIE COURIC: Or around or about. MAN: I’d never heard it said. I’d always seen the mark but never heard it said. Y…
Geometric constructions: parallel line | Congruence | High school geometry | Khan Academy
Let’s say that we have a line. I’m drawing it right over there, and our goal is to construct another line that is parallel to this line that goes through this point. How would we do that? Well, the way that we can approach it is by creating what will even…
Worked example: Lewis diagram of the cyanide ion (CN⁻) | AP Chemistry | Khan Academy
In this video, we’re going to try to get more practice constructing Lewis diagrams, and we’re going to try to do that for a cyanide anion. So, this is interesting; this is the first time we’re constructing a Lewis diagram for an ion. So, pause this video …
Growing Greens (Deleted Scene) | Life Below Zero
[Music] [Music] Well, I’m about out of water for water in my greenhouse, so I got to pump some water up from the river to fill up my tank. I go through a lot of water on hot sunny days. If I have a hot week, I’ll go through almost two of these tanks in on…
Warren Buffett's Advice for Investors for 2024
I don’t know if you guys have noticed, but Warren Buffett has kept very quiet over the past six months. No media interviews, very few changes to his portfolio. The guy has been keeping well out of the spotlight. So much so that when his longtime business …
15 Questions to Unlock Your Potential
Hey, we’re going to have a heart-to-heart today, all right? This is a secret tool that we’ve used for decades. Every couple of years, we go through these exact questions to have a reality check with ourselves, and today we’re going to do it together. Save…