Video Transcript
Kinetic energy is given by the
formula 𝐸 equals a half 𝑚𝑣 squared. The weight, 𝑊, 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) 𝐸 over 𝑊 equals two 𝑣 squared
over 𝑔. B) 𝐸 over 𝑊 equals 𝑔𝑣 squared
over two. C) 𝐸 over 𝑊 equals two 𝑔 over 𝑣
squared. D) 𝐸 over 𝑊 equals 𝑣 squared
over two 𝑔. Or E) 𝐸 over 𝑊 equals 𝑔 over two
𝑣 squared.
Since this question is asking us to
find the ratio between two quantities, let’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 𝐴 to
𝐵 is just 𝐴 divided by 𝐵. Since this question asks us to find
the ratio of the kinetic energy of an object to its weight, that means we’re looking
for kinetic energy divided by weight. The question tells us that kinetic
energy is represented by the symbol 𝐸. And weight is represented by the
symbol 𝑊. So kinetic energy divided by weight
is 𝐸 over 𝑊. And sure enough, we can see that
all of our available answers start with 𝐸 over 𝑊.
So, we’re looking to find a formula
that has a subject equal to 𝐸 over 𝑊. That’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, 𝐸, can be expressed as a half 𝑚𝑣 squared. And it also tells us that weight,
𝑊, can be expressed 𝑚𝑔. That means we can rewrite a ratio
𝐸 divided by 𝑊 by replacing 𝐸 with a half 𝑚𝑣 squared and replacing 𝑊 with
𝑚𝑔. Now, what we have here is actually
what the question is asking for. It’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 hasn’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 it’s always possible to rewrite
fractions within fractions as a single fraction. So whenever you see a fraction
inside another fraction, it’s a good sign that the expression can be simplified
further. The second thing that shows us that
we haven’t fully simplified our expression is that we can see a factor of 𝑚 on the
numerator and on the denominator. And if we look at our possible
answers again, we’ll see that none of them actually have an 𝑚 in them at all.
Let’s deal with these 𝑚’s
first. And then, we can take a look at the
fraction inside a fraction issue. Because we have a factor of 𝑚 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, we’ll divide them
both by 𝑚. So the numerator of this fraction
is a half 𝑚𝑣 squared divided by 𝑚, which just leaves us with a half 𝑣
squared. And the denominator of the fraction
is 𝑚𝑔 divided by 𝑚, which just leaves us with 𝑔. So we’ve now simplified our
expression so that it no longer contains an 𝑚. When simplifying equations in this
way, it’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 we’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, let’s use a slightly
simpler example. Let’s say we have the expression 𝑎
times 𝑏 divided by 𝑐. There are a few different ways we
can write this. For example, we can remove 𝑎 from
the numerator of the fraction and write it like this. So we can take any factor from the
numerator, in this case 𝑎 or 𝑏, and write it as a factor of the entire
fraction. Similarly, we could take 𝑏 out of
the numerator of the fraction and write the expression as 𝑏 times 𝑎 over 𝑐. 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 𝑣 squared over 𝑔.
All we have to do now is multiply
these two fractions together. And we’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 𝑣 squared. And in the denominator, we have two
times 𝑔, which we can write as 𝑣 squared over two 𝑔. 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 𝐸 over 𝑊
equals 𝑣 squared over two 𝑔.
