Video Transcript
The quantity 𝑣, the velocity of an object, is how much the object changes its displacement from a given position each second. The quantity 𝐸 is the kinetic energy of an object. Which of the following formulas correctly shows the relationship between 𝑣, 𝐸, and the mass of an object? (A) 𝐸 equals 𝑚𝑣. (B) 𝐸 equals 𝑚𝑣 squared. (C) 𝐸 equals 𝑚 squared 𝑣 squared. (D) 𝐸 equals one-half 𝑚𝑣 squared. (E) 𝐸 equals one-half 𝑚 squared 𝑣 squared.
Here, we’re trying to mathematically relate three different quantities mass 𝑚, velocity 𝑣, and kinetic energy 𝐸. It will be helpful to think about their associated SI units: mass in kilograms; velocity, the rate of change of displacement in meters per second; and kinetic energy, which, like all categories of energy, has SI units of joules. Recall that the joule is considered a compound unit because it comprises a combination of base SI units. Specifically, a joule is equal to a kilogram meter squared per second squared. So, when we express it like this, it’s easier to see how units of kilograms and meters per second can combine to result in the joule.
Now for each of these answer choices, let’s try substituting in the base SI units for mass and velocity on the right-hand side of the formula and see if it results in the correct units for energy on the left-hand side of the formula. Let’s start with (A), and we have mass times velocity or kilogram-meters per second. It’s close but not quite. Remember, what we’re looking for is kilogram meters squared per second squared. Because this formula does not have the correct units associated with it, we know that it can’t be the correct formula for kinetic energy. So, let’s eliminate answer choice (A).
Moving on to (B), we have mass times velocity squared, which gives us kilograms times meters per second quantity squared or kilogram meters squared per second squared. This doesn’t mean that it’s the correct formula exactly, but at least it does result in units of joules. So, it’s by far a better option than answer choice (A). Let’s keep (B) around while we check out the other options.
Next, (C) has mass squared times velocity squared, which gives kilograms squared times meters squared per second squared. Notice that there’s one factor of kilograms too many. So, we can eliminate (C) and move on to (D). Notice though that (D) is very similar to (B); the only difference is this factor of one-half. One-half is just a plain, unitless number, so it doesn’t have any impact on how the formula as a whole handles the SI units. Thus, we know one-half 𝑚𝑣 squared gives kilogram meters squared per second squared, which corresponds to a joule. So, we’ll leave answer choice (D) around as well.
Next, notice that answer choice (E) is very similar to (C) but with an extra factor of one-half. Therefore, we know that answer choice (E) gives the same units as answer choice (C), which we’ve already mentioned is incorrect. After assessing the units, we have it narrowed down to options (B) and (D). We just need to recall that there is indeed a factor of one-half in the kinetic energy formula. We should eliminate answer choice (B), and we know that (D) is the correct answer. Kinetic energy equals one-half times mass times velocity squared.
