### Video Transcript

A body was projected vertically upwards at 53.9 metres per second. Given that at a certain time ๐ก its height was 49 metres, find all the possible values of ๐ก. Take ๐ equals 9.8 metres per second squared.

In this question, weโre given that a body or an object is projected upwards. Weโre asked to find all the possible values of ๐ก when the height is 49 metres. We might wonder why thereโs more than one possible value of ๐ก. So letโs have a look at the path of this object. Weโre told that itโs projected upwards with a velocity of 53.9 metres per second. But at some point, the velocity will reach zero. And the body will return to the starting point. Therefore, our body could potentially be at a height of 49 metres on two separate times.

To answer this question, weโre going to use the kinematic equations of motion. So letโs start by seeing if we can assign values to any of these variables. Weโre asked to find the value of time ๐ก. So that variable will need to be in the equation of motion that we select. Weโre told that the body is projected upwards at 53.9 metres per second. So our ๐ฃ sub zero โ thatโs our initial velocity โ is 53.9 metres per second. You may also know initial velocity written with the letter ๐ข instead.

The height or displacement of this object is at 49 metres. So we can use the letter ๐ to indicate this variable although it can often be seen with the letter ๐ . And finally, our acceleration ๐ is taken as negative gravity, which we can write as negative 9.8 metres per second squared. An equation of motion including these four variables will be ๐ equals ๐ฃ sub zero ๐ก plus a half ๐๐ก squared. In terms of the letters ๐ข and ๐ , the equation would be ๐ equals ๐ข๐ก plus a half ๐๐ก squared. Substituting the values of our variables into this equation then will give us 49 equals 53.9๐ก plus a half times negative 9.8๐ก squared. Simplifying our right-hand side then, we have 49 equals 53.9๐ก minus 4.9๐ก squared.

Noticing that we now have a term in ๐ก and a term in ๐ก squared, weโre going to rearrange this quadratic into a form where it can easily be factored. So adding 4.9๐ก squared and subtracting 53.9๐ก from both sides of the equation will give us 4.9๐ก squared subtract 53.9๐ก plus 49 equals zero. Now we have our quadratic in a form where itโs easier to factor. But letโs make life a little bit easier by removing the decimals. And we can do this by multiplying our entire equation by 10.

We now have quite a few large coefficients. But we notice that we have 49 and 490. So letโs see if we can take out 49 as a common factor. Since 539 is also divisible by 49, then we can write our equation as ๐ก squared minus 11๐ก plus 10 equals zero. So now, to factor our equation, weโre looking for two values which multiply to give 10 and add to give negative 11. Since the values of negative 10 and negative one would fit, then our factored form is ๐ก minus 10 ๐ก minus one equals zero.

To find a solution for ๐ก then, we put each of the expressions in parentheses equal to zero and solve, giving us that ๐ก minus 10 equals zero or ๐ก minus one equals zero. In our first option then, when ๐ก minus 10 is zero, then ๐ก must be equal to 10 seconds. And in our second option, when ๐ก minus one equals zero, then ๐ก is equal to one second.

So our final answer for our time ๐ก then is ๐ก equals one second or ๐ก equals 10 seconds.