Question Video: Finding a Formula for the Ratio of Two Quantities | Nagwa Question Video: Finding a Formula for the Ratio of Two Quantities | Nagwa

# Question Video: Finding a Formula for the Ratio of Two Quantities Physics

Kinetic energy is given by the formula πΈ = 1/2 ππ£Β². The weight, π, of an object is mg. Which of the following is a correct formula that has a subject equal to the ratio of the kinetic energy of an object to its weight? [A] πΈ/π = 2π£Β²/π [B] πΈ/π = ππ£Β²/2 [C] πΈ/π = 2π/π£Β² [D] πΈ/π = π£Β²/2π [E] πΈ/π = π/2π£Β²

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### Video Transcript

Kinetic energy is given by the formula πΈ equals a half ππ£ squared. The weight, π, of an object is ππ. Which of the following is a correct formula that has a subject equal to the ratio of the kinetic energy of an object to its weight? A) πΈ over π equals two π£ squared over π. B) πΈ over π equals ππ£ squared over two. C) πΈ over π equals two π over π£ squared. D) πΈ over π equals π£ squared over two π. Or E) πΈ over π equals π over two π£ squared.

Since this question is asking us to find the ratio between two quantities, letβs start by recalling that the ratio of two quantities is the first quantity divided by the second quantity. So, for example, the ratio of π΄ to π΅ is just π΄ divided by π΅. Since this question asks us to find the ratio of the kinetic energy of an object to its weight, that means weβre looking for kinetic energy divided by weight. The question tells us that kinetic energy is represented by the symbol πΈ. And weight is represented by the symbol π. So kinetic energy divided by weight is πΈ over π. And sure enough, we can see that all of our available answers start with πΈ over π.

So, weβre looking to find a formula that has a subject equal to πΈ over π. Thatβs the ratio of kinetic energy to weight. And the other side of the formula should look like one of these expressions. The question tells us that kinetic energy, πΈ, can be expressed as a half ππ£ squared. And it also tells us that weight, π, can be expressed ππ. That means we can rewrite a ratio πΈ divided by π by replacing πΈ with a half ππ£ squared and replacing π with ππ. Now, what we have here is actually what the question is asking for. Itβs a correct formula. And the subject is equal to the ratio of the kinetic energy of an object to its weight. However, we have a bit more work to do before it matches one of the available answers.

If we look at our expression, there are a couple of signs that it hasnβt been fully simplified. Firstly, on the right-hand side of the equation, we have a fraction with a fraction inside it. Now, this can look a bit confusing. And itβs always possible to rewrite fractions within fractions as a single fraction. So whenever you see a fraction inside another fraction, itβs a good sign that the expression can be simplified further. The second thing that shows us that we havenβt fully simplified our expression is that we can see a factor of π on the numerator and on the denominator. And if we look at our possible answers again, weβll see that none of them actually have an π in them at all.

Letβs deal with these πβs first. And then, we can take a look at the fraction inside a fraction issue. Because we have a factor of π in the numerator and the denominator of this fraction, they actually cancel each other out. To show this, recall that you can rewrite a fraction by multiplying or dividing the numerator and the denominator by the same thing. In this case, weβll divide them both by π. So the numerator of this fraction is a half ππ£ squared divided by π, which just leaves us with a half π£ squared. And the denominator of the fraction is ππ divided by π, which just leaves us with π. So weβve now simplified our expression so that it no longer contains an π. When simplifying equations in this way, itβs common to just draw a line through any factors that appear on the numerator and the denominator, to show how the quantities cancel out.

Okay, so weβve simplified our expression. But we still need to write the right-hand side as a single fraction so that it matches one of the possible answers. To do this, letβs use a slightly simpler example. Letβs say we have the expression π times π divided by π. There are a few different ways we can write this. For example, we can remove π from the numerator of the fraction and write it like this. So we can take any factor from the numerator, in this case π or π, and write it as a factor of the entire fraction. Similarly, we could take π out of the numerator of the fraction and write the expression as π times π over π. We can apply this method to the right-hand side of our expression by taking out the factor of a half from the numerator, giving us a half times π£ squared over π.

All we have to do now is multiply these two fractions together. And weβll get a single fraction as a result. Recall that when multiplying fractions together, we can just multiply the numerators and the denominators separately. So in our numerator, we have one times π£ squared. And in the denominator, we have two times π, which we can write as π£ squared over two π. This is the simplest form that the right-hand side of our formula can be written in. And it now matches one of the available answers.

The formula that has a subject equal to the ratio of the kinetic energy of an object to its weight is πΈ over π equals π£ squared over two π.

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