A body of mass 1.7 kilograms is projected vertically upward at 13.7 meters per second from the surface of Earth. Find, to the nearest two decimal places, its kinetic energy one second after it was projected. Consider the acceleration due to gravity to be 9.8 meters per second squared.
Well, what we can see straightaway from this question is that we’re looking to find the kinetic energy. So let’s remind ourselves of the formula for kinetic energy. And that is that kinetic energy is equal to a half 𝑚𝑣 squared, where 𝑚 is the mass and 𝑣 is the velocity. Well, if you take a look at this formula, we can see that we’ve got part of it because we know the mass because the mass is 1.7 kilograms. However, we don’t know the velocity at this point because we only know the velocity when it was launched. So it’s projected vertically upward. So therefore, this is what we want to find.
So to do that, what we’re gonna use is one of our constant acceleration equations. And these are sometimes known as SUVAT equations because of the different variables that we have involved within them. So let’s take a look at the information that we’ve got from our question. Well, 𝑠 is our distance or displacement, which we don’t know. Well, we know that our initial velocity is 13.7 meters per second. So that’s our 𝑢. Well, just like the 𝑠, so our distance or displacement, we don’t know what the final velocity is going to be. But in this case, this is in fact what we’re looking to find. So I’ve circled this in orange.
Well then, 𝑎 is the next thing we’re gonna have a look at. But let’s take a little look at this sketch. Well, we know that the initial velocity is moving upwards because the body is projected vertically. However, our acceleration is the acceleration due to gravity. So this can be 𝑎 or 𝑔. And in fact, this is gonna be working downwards, so against the direction of our initial velocity. So therefore, we can say that it’s going to be negative. So we can say that 𝑎 is equal to negative 9.8 meters per second squared. And as I already said, you might see this as 𝑔 or 𝑎 because it’s the acceleration due to gravity.
And then we know that 𝑡 is equal to one. So our time is equal to one second. And that’s because what we’re trying to find is the kinetic energy one second after the body was projected. So now that we’ve got our variables, what we need to do is decide which one of our constant acceleration formulae we’re going to use to find our velocity. Well, here are four of the most common. And in fact, there are other different equations that can be formed from the same equations we’ve got here. So first of all, we’ve got 𝑣 equals 𝑢 plus 𝑎𝑡, 𝑠 equals 𝑢𝑡 plus half 𝑎𝑡 squared, 𝑣 squared equals 𝑢 squared plus two 𝑎𝑠, and finally 𝑠 equals 𝑣 plus 𝑢 multiplied by 𝑡 over two.
Well, we can rule out the bottom three. And that’s because they all contain 𝑠, which is something we don’t know and we’re not interested in finding. So therefore, the equation we’re gonna use is 𝑣 equals 𝑢 plus 𝑎𝑡. And in fact, this is often the most commonly used one. So therefore, if we substitute in our values, we’re gonna get 𝑣 is equal to 13.7 plus negative 9.8 multiplied by one, which is gonna give us the answer 𝑣 is equal to 3.9 meters per second. Okay, great, so we’ve now got 𝑣. So now we’ve got 𝑣, what we can do is use our formula for kinetic energy to work out what the kinetic energy is going to be one second after the body was projected.
So we know our mass is 1.7 and our velocity is 3.9. So therefore, we can say the kinetic energy is gonna be equal to a half multiplied by 1.7 multiplied by 3.9 squared. So therefore, we can say the kinetic energy is gonna be equal to 12.9285 joules. And we know that the units are going to be joules because we’ve got our mass in kilograms and our velocity in meters per second. However, we haven’t quite finished there. And that’s because our question asks us to give our kinetic energy to two decimal places. So therefore, after the rounding, we can say that the kinetic energy of the body one second after it was projected is going to be 12.93 joules. And as we’ve said, that’s to two decimal places.