Given that an object was projected vertically upward at 619.92 kilometres per hour from the ground, what is the maximum height it can reach if the acceleration due to gravity is 9.8 metres per square second?
So we have an object being projected vertically upward at 619.92 kilometres per hour from the ground. So its starting height is zero. Acceleration due to gravity acts in the opposite direction, and that’s 9.8 metres per square second. To work out the maximum height the object can reach, we’re going to refer to the equations of motion. This is sometimes called the SUVAT equations, where SUVAT is an acronym from the variables. 𝑠 is displacement, 𝑢 is initial velocity, 𝑣 is final velocity, 𝑎 is acceleration, and 𝑡 is equal to time.
It’s really helpful to list out what we know about our object. And that will help us decide which of the equations we’re going to use. 𝑠, displacement, is what we’re looking to find. We’re told that the initial speed, 𝑢, was 619.92 kilometres per hour. And when the object reaches its maximum height, it will momentarily have a speed of zero. So 𝑣 is equal to zero. The final velocity is equal to zero. We know that acceleration acts in the opposite direction from the initial velocity. And that’s negative 9.8 metres per square second. We don’t know the value of 𝑡, but we’re not really interested in it. So we know 𝑠, 𝑢, 𝑣, and 𝑎. Meaning that the equation we’re interested in is this fourth one, 𝑣 squared equals 𝑢 squared plus two 𝑎𝑠.
Now, you might notice there’s nothing we can do with this until we make sure that our units are all equal. We have kilometres per hour for 𝑢 and 𝑣. And we have metres per second for acceleration. We generally prefer to give displacement in metres. So we’re going to convert 619.92 kilometres per hour into metres per second. And it’s quite clear we don’t need to repeat this process for 𝑣, zero kilometres per hour. It’s, of course, the same as zero metres per second.
Now, we can begin by converting from kilometres to metres. We do this by multiplying through by 1000. And when we do, we find that the initial velocity of the object is 619920 metres per hour. Then, we recall that to convert from hours to seconds, we multiply by 60 and then multiply by 60 again. Now, of course, this is kilometres per hour and metres per second. So we do the opposite. We divide by 60 and then divide by 60 again. That is the same as dividing by 3600. And when we do, we find that the speed is 172.2 metres per second.
And we can now substitute everything we know into the formula 𝑣 squared equals 𝑢 squared plus two 𝑎𝑠. 𝑣 is zero; 𝑢 is 172.2. And if we define the displacement we’re looking for as 𝑥, purely because the letter 𝑠 looks a little bit like the number five, we find that our equation becomes zero squared equals 172.2 squared plus two times negative 9.8 times 𝑥. We square 172.2 to get 29652.84. And two times negative 9.8 is negative 19.6. And now we can solve for 𝑥 by adding 19.6𝑥 to both sides and then dividing through by 19.6, to give us 1512.9.
Now, of course, we were working in metres per second and metres per square second. So our displacement will be in metres. And we find the maximum height the object can reach is 1512.9 metres.