A particle of mass 500 grams fell vertically from a height of 17.6 meters above the ground. Determine its kinetic energy just before it hit the ground. Consider the acceleration due to gravity 𝑔 is equal to 9.8 meters per second squared.
Okay, so the first thing we’ve done is made a little sketch, and this sketch includes all the information we’ve been given. So we know that there’s a particle and its mass is 500 grams. And we know that it’s fallen from a vertical height of 17.6 meters above the ground. Then what we also know is that it was released. So that means the initial velocity, 𝑢, is equal to zero meters per second. And we also know that the acceleration due to gravity is 9.8 meters per second squared.
Well, in this question, what we’re asked to find is the kinetic energy. So let’s think about our formula for kinetic energy. And that is that kinetic energy is equal to a half 𝑚𝑣 squared, so a half multiplied by the mass multiplied by the velocity squared. Well, we know the mass because we were told that at the beginning of the question. And that’s 500 grams. However, we don’t know the velocity because the velocity we want to find is the velocity of the particle just as it’s about to hit the floor. So how can we work out the velocity?
Well, what we do to work out velocity is use one of our equations of constant acceleration. And these are sometimes known as SUVAT. And that is because of the letters we use to represent each of the variables. Well, 𝑠 is our distance or displacement. And this is 17.6 meters because that’s the distance or height of the particle above the ground. And we’re gonna assume that 𝑢 is equal to zero meters per second. 𝑣 we don’t know. But in fact, 𝑣 is what we’re looking for.
Well, 𝑎 is our 𝑔 because 𝑎 is our acceleration. And it’s our acceleration in this case due to gravity. And that’s gonna be 9.8 meters per second squared. And the reason it’s 9.8 meters per second squared not negative 9.8 meters per second squared is that we are taking down as being the positive direction in this scenario. And that’s because our particle is moving downwards. And we don’t know 𝑡, but we don’t need to find 𝑡 as part of this question. So we can ignore that variable.
Well, now what we need to do is decide which one of our equations of constant acceleration we’re going to use. Now the main ones we often see are shown here. So we’ve got 𝑣 equals 𝑢 plus 𝑎𝑡, 𝑣 squared equals 𝑢 squared plus two 𝑎𝑠, 𝑠 equals 𝑢𝑡 plus a half 𝑎𝑡 squared, 𝑠 equals 𝑣𝑡 minus a half 𝑎𝑡 squared, or 𝑠 equals then you got 𝑢 plus 𝑣 multiplied by 𝑡 over two. And in fact, we can have adaptations of these. And we see them in lots of different forms.
Well, if we take a look at the variables we’ve got and the one that we’re looking for, we could see that the one that we’re wanting to use out of our equations we’ve got here is 𝑣 squared equals 𝑢 squared plus two 𝑎𝑠. And that’s because we’ve got 𝑠, 𝑢, 𝑣, and 𝑎. So therefore, when we substitute our values in, we’re gonna get 𝑣 squared equals zero squared plus two multiplied by 9.8 multiplied by 17.6. So 𝑣 squared is gonna be equal to 344.96. So now if we wanted to find out 𝑣, what we could do is take the square root of both sides. And we could say that 𝑣 is equal to the square root of 344.96.
However, this actually isn’t necessary in this problem. And that’s because if we take a look at kinetic energy, we can see that the formula tells us that kinetic energy is equal to a half 𝑚𝑣 squared. So, in fact, we’re interested in 𝑣 squared. And that’s also why we don’t have to worry about the positive or negative results for 𝑣 because we’d square it, which would give us a positive result anyway.
So now what we’re gonna have is the kinetic energy is equal to 0.5, and that’s because a half is the same as 0.5, multiplied by 0.5. And we’ve got 0.5 because 500 grams is 0.5 kilograms, and we need it to be in kilograms if we’re going to give our answer in joules, and then multiply it by 344.96. And we’re gonna get the 344.96. And that’s because that is 𝑣 squared. And if we think about 𝑣, what this would be in would be meters per second. And for kinetic energy, if we have the mass in kilograms and the velocity in meters per second, then the units will be joules. So this is gonna give us a result of 86.24 joules. So we can say that the kinetic energy just before the particle hit the ground was 86.24 joules.