Calculating velocity using energy | Modeling Energy | High School Physics | Khan Academy

So we have a spring here that has a spring constant of 4 newtons per meter. What we then do is take a 10 gram mass and we put it on top of the spring, and we push down to compress the spring by 10 centimeters. We then let go, and what I'm curious about is: what is going to be the magnitude of the velocity of our ball here, of our 10 gram mass, right as the spring is no longer compressed or stretched, or essentially when the ball is being launched? Pause this video and see if you can figure that out. I'll give you a hint: the energy in this first state, the total energy, has got to be equal to the total energy of the second state. We can't create or destroy energy.

Alright, now let's work through this together. So let's call this first scenario state one. So in state one, what is the total energy going to be? Well, it's going to be the sum of the gravitational potential energy, so that's mg times the height in state one, plus our elastic potential energy, that's one-half times the spring constant times how much we've compressed that spring in state one squared, plus our kinetic energy, so it's one-half times our mass times the magnitude of our velocity in state one squared. That has got to be equal to, as we just talked about, the total energy in state two.

Well, what's that going to be? Well, that's your gravitational potential energy in state two, plus your elastic potential energy in state two, plus your kinetic energy in state two. Now let's think about which of these variables we know and which ones we need to solve for.

So first of all, mass, your spring constant, your gravit, your the strength of your gravitational field. Well, we know what these are going to be. These are going to be our mass is equal to 10 grams; the strength of the gravitational field, also the acceleration due to gravity near the surface of the earth, is 9.8 meters per second squared; our spring constant is 4 newtons per meter.

I like to remind myself what a newton is. A newton is kilogram meter per second squared, so this is also equal to 4 kilogram meter per second squared. Then we also have a meter over there, and actually these meters will cancel out. That's useful because it's reminding us that we want everything to be in kilograms and meters.

With that in mind, actually, let me rewrite our mass right over here as 0.01 kilograms. Then let's think about what's going on specifically in each of those states. So what is going to be our h1, our initial height? Well, I didn't give it to you, but what really matters is the difference between h1 and h2.

So we could just define h1 right over here to be equal to zero. So let me write that down: h1 is equal to zero. If we say that, then what is h2 going to be? h2 would then be equal to 10 centimeters. But remember, we want everything in kilograms and meters, so 10 centimeters is the same thing as 0.1 meters.

What is our spring compression in scenario 1 going to be? Well, that is going to be 10 centimeters; but once again, we want to write that in terms of meters, so I'll write that as 0.1 meters. Then, what is our spring compression in scenario 2 going to be? Well, we're completely uncompressed and unstretched, so that is going to be zero meters.

And then, what is going to be our velocity, or at least the magnitude of our velocity in state one? Well, we're stationary, so it's zero meters per second. What is going to be the magnitude of our velocity in state two? Well, that's exactly what we want to solve for; that is our launch velocity.

So let’s see if we can simplify this and then solve for v2. So we know that h1 is equal to 0, so that simplifies that right over there; that term is 0. We know that v1 is 0, so that simplifies that term there. We know that delta x in scenario 2 is equal to 0, so that simplifies that term right over there.

We can now rewrite all of this, and I'll switch to one color just to speed things up a little bit: one-half k times delta x1 squared is equal to mg h sub 2 plus one-half m v sub 2 squared. Now let's just try to solve for this character.

So let's subtract mgh sub 2 from both sides. We're going to have one-half k times delta x sub 1 squared minus m g h sub 2 is equal to one-half m v sub 2 squared. Now let's see: if we multiply both sides by 2 over m, then that will get rid of this one-half m over here.

So let me multiply that. I'm kind of doing two steps at the same time. One way to think about it is I'm multiplying times the reciprocal of the coefficient on the v squared. So right over here, it's m over two, so the reciprocal is two over m, and we have to be sure that I'm going to multiply times that whole side.

And so what do we now have? This is going to be equal to, let’s see, it's going to be k delta x sub 1 squared over m minus 2 g h sub 2 is going to be equal to v sub 2 squared. To solve for our launch velocity, we just take the principal root, the square root of both sides.

So I could say v sub 2, which is equal to our launch velocity, is equal to the square root of our spring constant times delta x sub 1 squared over our mass minus 2 times the gravitational field times h2. And so we just have to plug in the numbers now.

This is going to be equal to, and I'll switch colors just to ease the monotony. Let me use a new color right over here. This is going to be equal to the square root of— and I want to make sure all the units work out, so I'm actually going to write this version of my spring constant so I can work with all the units.

So it's going to be 4 kilograms per second squared, and now delta x1, we know, is 0.1 meters squared. And then all of that is going to be over our mass, which we know is 0.01 kilograms, 0.01 kilograms, and then minus 2 times 9.8 meters per second squared times height in the second scenario, which we already know is 0.1 meters.

So times 0.1 meters. Let me extend this radical right over here and let's look at the units first to just make sure we're getting the right units. So this kilogram is going to cancel with this kilogram, and then we have over here we're going to have something that's in terms of meters squared per second squared.

Then over here, we're going to have something in terms of meters squared per second squared. So that makes sense: we're going to have a difference of 2 meters squared per second squared, and then when you take the principal root, you're going to get meters per second, which is the unit for the magnitude of velocity.

So now we just have to get our calculator out and calculate this. So the 0.01s will cancel out, so this part right over here is going to be 4. And then from that, I am going to subtract 2 times 9.8 times 0.1. Close the parentheses; that's going to be equal to that, and then I need to take the square root of all of that business, and I get this right over here.

So this is approximately 1.43 meters per second, 1.43 meters per second, and we're done.