Main content

### Course: High school physics - NGSS > Unit 4

Lesson 3: Predictions using energy# Calculating height using energy

Learn how you can calculate the maximum height of a launched object by using the total energy of a system. Energy that is conserved can be transferred within a system from one object to another changing the characteristics of each object, like position. Created by Sal Khan.

## Want to join the conversation?

- Okay so this is more of a math question than a physics question, but I'm hoping someone here can help me out.

For one of the quiz questions, I came up with the equation:

1/2*1200N/m*Δx^2=1/2*0.43kg*(15m/s)^2+0.43kg*9.8m/s^2*Δx

How would I solve for Δx with it being raised to the first power on one side and the second power on the other side of the equation?

I took Algebra 1 a while back so maybe I learned this and then forgot (I'm reviewing it now). I haven't had any trouble with this course yet. Do I need Algebra 2 or any other math classes to continue with it?(1 vote)- Howdy Ryan,

Remember that any if we have any variable x raised to a power b, multiplied by that same variable raised to different power c, then the result will be x raised to b + c.

e.g.: x^b * x^c = x^(b + c)

For division, you do exactly the opposite. Instead of adding the exponents, you subtract.

So if you divided Δx by both sides of the equation, you would be left with Δx^2/Δx = Δx on the left-hand side and Δx/Δx = 1 on the other side.

Does that answer your question?(2 votes)

- Ah yes, K stands for kstiffness?(1 vote)
- No, it stands for Constant of Proportionality because the letter C is often used for other things (it's also possible that scientists are bad at spelling).(1 vote)

- when sal says "the potential energy in scenario 1 plus the kinetic energy in scenario 1 needs to be equal to the potential energy in scenario 2 plus the kinetic energy in scenario 2" is this the same as saying " the mechanical energy needs to be the same in scenario 1 and scenario 2" ?(1 vote)

## Video transcript

- [Instructor] So I have
an uncompressed spring here and the spring has a spring constant of four Newtons per meter. Then I take a 10 gram
mass, a 10 gram ball, and I put it at the top of
the spring and I push down to compress that spring by 10 centimeters. And so let's call that
scenario one right over there, where our mass is on top
of this compressed spring. And then let's say, we let go. And then the spring launches
this mass into the air, and then this mass is gonna
hit some maximum height. And let's call that scenario two when we are hitting that maximum height. And my question to you is,
what is that maximum height based on all of the information
that I have given you here? And I'll give you a hint. What we have to think about is the idea that energy is conserved. The total energy in scenario
one is gonna be equal to the total energy in scenario two. So pause this video and
see if you can figure out what that maximum height is going to be. All right, now let's work
through this together. I told you that the total
energy is gonna be conserved, but what's that total energy
going to be made up of? Well, it's gonna have
some potential energy and it's going to have
some kinetic energy. And so another way to think about it is, our potential energy in scenario one plus our kinetic energy in scenario one needs to be equal to our
potential energy in scenario two, plus our kinetic energy in scenario two. And there might be other energies here that you could think
about heat due to friction with the air, but we're not going to, we're going to ignore those
to just simplify the problem. We can assume that this
is happening in a vacuum, that might help us a little bit. So if we think about
the potential energies, there's actually two types of
potential energies at play. There's gravitational potential energy, and there is elastic potential energy due to the fact that this mass is sitting on a compressed spring. So our gravitational potential energy is going to be our mass times the strength of our gravitational field, times our height in position one, and our elastic potential energy is going to be equal to one
half times our spring constant, times how much that spring
is compressed, squared. And so this is all of our
potential energy in scenario one, right over here. And then we add our kinetic energy. So plus, we have one half times our mass, times our velocity in
scenario one, squared. And so the sum of this is gonna be the sum of all of this in scenario two. And so that is going to
be equal to mass times G times height in scenario two, plus one half times our spring constant times how much that's spring is compressed in scenario two squared, plus one half times our mass, times our velocity in
scenario two, squared. And just as a reminder, what
we have right over here, this is our potential
energy in scenario two. And then our kinetic
energy is right over there. Now this might look
really hairy and daunting, but there's a lot of simplification here. We can define our starting
point right over here, H one as being equal to zero, which will simplify this dramatically. Cause if H one is zero, then
this term right over here, zero, we also know that
our velocity is zero in our starting scenario. So that would make our
kinetic energy zero. So the left-hand side is really all about our elastic potential energy. So it's gonna be one half
times our spring constant, times how much we have
compressed the spring in scenario one, squared. And then on the right-hand
side, what's going on? Well, in this scenario, our
spring is no longer compressed. So our elastic potential
energy is now zero. And what about our kinetic energy? Well, at maximum height right over here, your ball is actually
stationary for an instant, for a moment. It's right at that
moment where it is going from moving up to starting to move down. And so our velocity is
actually zero right over here. So V2 is actually zero,
just like V1 was zero. So that's gonna be zero. And so you have a scenario
where our initial elastic potential energy is going to
be equal to our scenario two, gravitational potential energy. And so we just need to solve
for this right over here. That is going to be our maximum height. To do that, we can divide both sides by mg and we get h2 is equal to
1/2 k delta x, one squared, all of that over mg. And we know what all of these things are, and I'll write it out with the units. This is going to be equal to one half times our spring constant, which is four Newtons per meter. Now our change in x is 10 centimeters, but we have to be very careful. We can't just put a 10 centimeters
here and then square it. We want the units to match up. So we're not dealing with
centimeters and grams. We're dealing with meters and kilograms. So I wanna convert this
10 centimeters to meters. Well, that's gonna be 0.1 meters. So I will write this as times 0.1. That's how much the spring is compressed. That's our delta x in terms of meters. And so that is gonna be squared. And then all of that
over, what is our mass? And once again, we want
to express our mass in terms of kilograms. So our mass is 0.01 kilograms. G is 9.8 meters per second squared. And when you calculate it, this
is approximately 0.2 meters. And the units actually all do work out. Because if you look at
Newtons as being the same as kilogram meters per second squared, this kilograms cancels with
that kilograms right over there. You can see that this per second squared is gonna cancel with
that per second squared right over there. And then this meter is going
to cancel out with that meter. And so what you're left
with is a meter squared divided by meters, which is just going to
leave you with meters, just like that. And we're done. We figured out the maximum
height using just our knowledge of the conservation of energy
is approximately 0.2 meters.