Video Transcript
A body of mass eight kilograms was projected vertically upwards at 34.3 meters per second. After a certain time 𝑡, its kinetic energy became 198.45 joules. Find 𝑡. Take 𝑔 equals 9.8 meters per second squared, where 𝑔 is the acceleration due to gravity.
To solve this problem, we’re going to use the formula for kinetic energy to calculate the velocity at time 𝑡. And that’s really helpful, as we will then have enough information to substitute into an equation of constant acceleration to find 𝑡. So first, what we need to do is recall the formula for kinetic energy. And that is a half 𝑚𝑣 squared. And what we can see from the question is that we know that after a certain time 𝑡 the body of mass has a kinetic energy of 198.45 joules. And we can see that the kinetic energy is in joules.
We can also see that we’ve got a mass. And that’s eight kilograms. So our mass is in kilograms. And because the energy is in joules, we know that the velocity must be equal to meters per second. Okay, great, so now what are we going to do? Well, what we want to do is find out what 𝑣 is. And we can do that because we know what 𝑚 and KE are cause we know the mass and the kinetic energy. So what we need to do is substitute them into our formula. And when we do that, we get 198.45 is equal to a half multiplied by eight multiplied by 𝑣 squared. So then what we’re gonna do is divide both sides by four. And that’s cause if you got a half multiplied by eight, it’s four. So when we do that, we get 198.45 over four equals 𝑣 squared. So therefore, 49.6125 is equal to 𝑣 squared.
Okay, great, all we need to do now is take the square root of both sides to find 𝑣. So therefore, we can say that 𝑣 is equal to positive or negative root 49.6125. Now what we’re gonna do is keep it in this form at the moment and not actually work out what the values are. And that’s because what it will do is maintain accuracy in the next part of the question.
So now, for the next part the question because what we want to do is find our time, what we’re gonna use is something called the equations of constant acceleration. Or sometimes they’re known as SUVAT equations because of the variable letters that we use to represent the variables in the problem. So first of all, let’s see which ones of these we actually have or want to find.
Well, first of all, we start with 𝑠, which is our distance or displacement. Well, we don’t know what this is. And actually, we don’t want to find it because it’s not useful for us in this question. So we can remove this. So next, we move down to 𝑢. And this is our initial velocity. Well, we can see that initially the body of mass was projected vertically upwards at 34.3 meters per second. And then our final velocity or our velocity at the time 𝑡 is gonna be equal to positive or negative root 49.6125 because we calculated that just now.
And next, we have 𝑎. Well, we could have 𝑔 because in fact it is our acceleration due to gravity in this particular type of problem. But it means acceleration. And we can see that the acceleration will be 9.8 meters per second squared. However, we’re gonna take this to be negative since we define the upwards direction to be positive. And the reason that we’re saying it’s negative is because gravity acts downwards. And we’re taking upwards to be positive in this problem. And that’s because the body of mass was projected vertically upwards.
And then finally, we have 𝑡, the time, which we don’t know and in fact is what we’re trying to find in this problem. And as we said earlier, what we want to do is use one of our five equations of constant acceleration or SUVAT equations to find our 𝑡, time. Well, here are five, and there are the most common. We’ve got 𝑣 equals 𝑢 plus 𝑎𝑡, 𝑣 squared equals 𝑢 squared plus two 𝑎𝑠, 𝑠 equals 𝑢𝑡 plus half 𝑎𝑡 squared, 𝑠 equals 𝑣𝑡 plus a half 𝑎𝑡 squared, and 𝑠 equals 𝑢 plus 𝑣 over two 𝑡. And in fact, you might see these written in slightly different ways. Or actually, we could find different ones using these equations or formulae that we have here.
And when we use these, then what we need to do is decide which one we want to use by looking at the variables that we have or want to know. When we do that, we can see that we’re gonna use 𝑣 equals 𝑢 plus 𝑎𝑡. And that’s because we have 𝑢𝑣 and 𝑎 and we want to find 𝑡. And in fact, we could’ve ruled all the others out because they all involve 𝑠. Well, we don’t know what 𝑠 is and we’re not looking to find 𝑠. So we could rule them out. Okay, great, so now let’s use this to find out our 𝑡.
Well, because we’ve got two values of 𝑣, the easiest way to do this first of all was take our 𝑣 equals 𝑢 plus 𝑎𝑡 and rearrange it to make 𝑡 the subject. And to do that, what we’re gonna do is subtract 𝑢 and divide by 𝑎. Then what I’ve done is actually flipped it so we got 𝑡 on the left-hand side. So we’ve got 𝑡 equals 𝑣 minus 𝑢 over 𝑎. And in fact, this is one of the other forms that we sometimes see this equation. Okay, great, so now let’s substitute in our values.
So first of all, if we substitute in the positive root 49.6125 for our 𝑣, we’re gonna get 𝑡 is equal to root 49.6125 minus 34.3 divided by negative 9.8, which is gonna be equal to 2.78 seconds. And that’s to two decimal places. Okay, great, so now let’s substitute in our other value for 𝑣. And to enable us to do that, what I’m gonna do is clear some space. And when we do that, we’re gonna get 𝑡 is equal to negative root 49.6125 minus 34.3 over negative 9.8. So therefore, we’re gonna get 𝑡 is equal to 4.22 seconds. And again, this is to two decimal places.
So therefore, we can say that if a body of mass eight kilograms was projected vertically upwards at 34.3 meters per second, then if its kinetic energy became 198.45 joules at a certain time 𝑡, then that certain time 𝑡 would be either 2.78 seconds or 4.22 seconds.
