Video Transcript
A motorboat at rest on a lake has a mass of 665 kilograms. The motorboat then accelerates until it has a kinetic energy of 340480 joules. When the motorboat has accelerated, at what rate does it move toward a stationary boat that is directly in front of it?
Let’s start by sketching a diagram. Here’s the motorboat in question. We’re told it has a mass represented by 𝑚 of 665 kilograms. The boat starts at rest and then accelerates for some time until it has a kinetic energy of 340480 joules. It’s now moving toward a stationary boat that’s directly in front of it. And we wanna know the rate at which it’s moving or its velocity represented by 𝑣. Let’s recall that we can calculate the kinetic energy of an object using the formula one-half 𝑚𝑣 squared, where 𝑚 is its mass and 𝑣 is its velocity. In this question, we want to find the velocity of the motorboat, so we’ll copy the formula below and rearrange it to make 𝑣 the subject.
First, we’ll multiply both sides by two over 𝑚. This way, on the right-hand side, two cancels out the factor of one-half, and we can also cancel out mass. Now, writing this a bit more neatly, we have two times the kinetic energy divided by mass equals velocity squared. All we have to do now to get 𝑣 by itself is take the square root of both sides of the formula and writing it the other way so that 𝑣 is on the left-hand side. We have that velocity equals the square root of two times the kinetic energy divided by mass. Since we already have values for both the variables on the right-hand side of the formula expressed in base SI units, we’re ready to substitute them in.
But before we calculate, let’s make sure the units are gonna work out properly. Recall that the joule is equivalent to a kilogram meter squared per second squared. So, if we write it like this in our formula, notice that units of kilograms cancel out of the numerator and denominator, leaving only meters squared per second squared under the radical or just meters per second for our final answer. Now, it’s time to calculate. The square root of the quantity two times 340480 divided by 665 comes out to exactly 32, and so we have our answer.
When the motorboat has accelerated, it moves toward the stationary boat at a rate of 32 meters per second.
