### Video Transcript

The kinetic energy ๐พ of a moving object varies jointly with its mass ๐ and the square of its velocity ๐. If an object weighing 40 kilograms with a velocity of 15 meters per second has a kinetic energy of 1000 joules, find the kinetic energy if the velocity is increased to 20 meters per second.

First of all, we can see that weโre dealing with kinetic energy. Now in the question, it says kinetic energy and it gives us the letter for it ๐พ. Iโm actually gonna use ๐พ๐ธ in our answer, and thatโs just so that we donโt get mixed up with our proportionality constant.

So therefore, using the information that weโve got, we can say that the kinetic energy is gonna be equal to ๐พ, which is our proportionality constant, multiplied by ๐ multiplied by ๐ squared. And we know that because it says that it varies jointly with this mass ๐ and the square of its velocity ๐. So therefore, if itโs varying jointly, itโs going to be proportional to. So we can say that ๐พ is proportional to ๐๐ squared. And then, the way that weโve actually brought in the ๐พ means that we can say that ๐พ is equal to ๐พ, our proportionality constant, multiplied by ๐๐ squared.

Well, now with this type of question, the first thing we gonna do is always find ๐พ, so find our proportionality constant. And the way we do that is by using some information weโve got, which tells us the values of a certain situation.

Well, from the question, we can see that actually when weโve got a kinetic energy of 1000, weโve got a mass of 40 kilograms and weโve got a velocity of 15 meters per second. So what we can actually do is actually substitute this back in to ๐พ equals ๐พ๐๐ squared. And when we do that, weโre gonna get 1000 is equal to ๐พ multiplied by 40 multiplied by 15 squared. So therefore, if we actually calculate this, weโre gonna get 1000 is equal to 9000๐พ.

So now what we want to do is actually find out what ๐พ is, cause that was the point of the first stage, to find out what our proportionality constant is. And to do that, weโre gonna divide each side of our equation by 9000. So we can say that ๐พ is equal to 1000 over 9000. Well, if we divide the numerator and denominator by 1000, well this gives us that ๐พ is equal to one over nine, or one-ninth. So great! Weโve actually found ๐พ. So what we can do is actually substitute this back in to our formula to give us a new formula.

So now we can say that the kinetic energy is equal to a ninth ๐๐ squared, so one over nine multiplied by ๐๐ squared. So now what we need to do, and this is invariably the same with any question like this, is actually use some information weโve got to find a value we donโt have.

So now we know that the mass is again gonna be equal to 40 kilograms, but the velocity has actually been increased to 20 meters per second. However, the kinetic energy we donโt know, and thatโs what we want to find. So again, what weโre gonna do is actually substitute our values into our formula.

So therefore, when we do that, we get that ๐พ๐ธ, our kinetic energy, is equal to one over nine, or one-ninth, multiplied by 40 multiplied by 20 squared. So therefore, this is gonna give us 16000 over nine, and thatโs because we have 40 multiplied by 20 squared โ itโs 40 multiplied by 400, which gives 16000 โ and then divided by nine, cause we had it multiplied by a ninth.

So therefore, we can say that when the velocity has actually been increased to 20 meters per second, the kinetic energy is gonna be equal to 1777.78 joules. And thatโs to two decimal places.