Video: Solving World Problems Involving Direct Variation of a Quantity with Two Others

The kinetic energy ๐พ of a moving object varies jointly with its mass ๐‘š 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.

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Video Transcript

The kinetic energy ๐พ of a moving object varies jointly with its mass ๐‘š 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 weโ€™re dealing with kinetic energy. Now in the question, it says kinetic energy and it gives us the letter for it ๐พ. Iโ€™m actually gonna use ๐พ๐ธ in our answer, and thatโ€™s just so that we donโ€™t get mixed up with our proportionality constant.

So therefore, using the information that weโ€™ve got, we can say that the kinetic energy is gonna be equal to ๐พ, which is our proportionality constant, multiplied by ๐‘š multiplied by ๐‘‰ squared. And we know that because it says that it varies jointly with this mass ๐‘š and the square of its velocity ๐‘‰. So therefore, if itโ€™s varying jointly, itโ€™s going to be proportional to. So we can say that ๐พ is proportional to ๐‘š๐‘‰ squared. And then, the way that weโ€™ve actually brought in the ๐พ means that we can say that ๐พ is equal to ๐พ, our proportionality constant, multiplied by ๐‘š๐‘‰ squared.

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

Well, from the question, we can see that actually when weโ€™ve got a kinetic energy of 1000, weโ€™ve got a mass of 40 kilograms and weโ€™ve got a velocity of 15 meters per second. So what we can actually do is actually substitute this back in to ๐พ equals ๐พ๐‘š๐‘‰ squared. And when we do that, weโ€™re gonna get 1000 is equal to ๐พ multiplied by 40 multiplied by 15 squared. So therefore, if we actually calculate this, weโ€™re gonna get 1000 is equal to 9000๐พ.

So now what we want to do is actually find out what ๐พ is, cause that was the point of the first stage, to find out what our proportionality constant is. And to do that, weโ€™re gonna divide each side of our equation by 9000. So we can say that ๐พ is equal to 1000 over 9000. Well, if we divide the numerator and denominator by 1000, well this gives us that ๐พ is equal to one over nine, or one-ninth. So great! Weโ€™ve actually found ๐พ. 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 ๐‘š๐‘‰ squared, so one over nine multiplied by ๐‘š๐‘‰ squared. So now what we need to do, and this is invariably the same with any question like this, is actually use some information weโ€™ve got to find a value we donโ€™t 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 donโ€™t know, and thatโ€™s what we want to find. So again, what weโ€™re gonna do is actually substitute our values into our formula.

So therefore, when we do that, we get that ๐พ๐ธ, 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 thatโ€™s because we have 40 multiplied by 20 squared โ€” itโ€™s 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 thatโ€™s to two decimal places.

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