A body of mass 500 grams is moving at a constant velocity 𝑣 equals two 𝐢 minus three 𝐣 centimeters per second, where 𝐢 and 𝐣 are two perpendicular unit vectors. Find its kinetic energy.
So, first of all, what we can see is that we’re trying to find the kinetic energy. So let’s remind ourselves of the formula for kinetic energy. Well, we know that kinetic energy is equal to a half 𝑚𝑣 squared, where 𝑚 is the mass and 𝑣 is the velocity. But what we want to do is consider our units.
Well, if the units for mass were kilograms and units for velocity were meters per second, then this means that our kinetic energy would be in joules. However, if our mass was in grams and our velocity was in centimeters per second, then our kinetic energy is measured in ergs. Well, let’s take a look at our question to see what our units are going to be. Well, we can see that the mass is in grams and the velocity is in centimeters per second. So we know that our answer is going to be in ergs.
So now what we want to do is substitute in our values into the formula to find our kinetic energy. So as we already said, our mass is 500 grams and our velocity is equal to two 𝐢 minus three 𝐣 centimeters per second. However, we can’t just pop these values into our formula. And that’s because what we’ve got here is our velocity in vector form. And what we need to do is find the magnitude.
Well, if we want to find the magnitude of a vector, if you think of our vector as 𝑎𝐢 plus 𝑏𝐣, then this is gonna be equal to the square root of 𝑎 squared plus 𝑏 squared. And this comes from the Pythagorean theorem. And we can use that because we know that 𝐢 and 𝐣 are two perpendicular unit vectors. So in that case, they are at right angles to each other. So therefore, we can adapt our Pythagorean theorem to give us this result for our magnitude.
Okay, so what we want to do now is find the magnitude of two 𝐢 minus three 𝐣. So therefore, we can say that the velocity is going to be equal to the square root of two squared plus negative three squared. So therefore, we can say that the velocity is gonna be equal to root 13 centimeters per second. And we’ll keep it in this form because actually we want to maintain the accuracy for the next part of the question. And as it’s magnitude, we’re only interested in the positive result.
So then we can substitute our values into the kinetic energy formula. When we do that, we get kinetic energy is equal to a half multiplied by 500 multiplied by root 13 squared, which is gonna be equal to 250 multiplied by 13. That’s because root 13 squared is just 13. So therefore, we can say that if a body of mass 500 grams is moving at a constant velocity 𝑣 equals two 𝐢 minus three 𝐣 centimeters per second, where 𝐢 and 𝐣 are two perpendicular unit vectors, then the kinetic energy is going to be equal to 3,250 ergs.