Question

When a ball is thrown upwards, the time, $T$ seconds, during which the ball remains in the air is directly proportional to the square root of the height, $h$ metres, reached. We know $T=4.47sec$ when $h=25m$. Find the value of $T$ when $h=50$ give the answer correct to $3$ decimal places

Answer 1

The statement " $x$ varies directly as $y$ ," means that when $y$ increases, $x$ increases by the same factor. In other words, $y$ and $x$ always have the same ratio that is:

$x=ky$

where $k$ is the constant of variation.

Here, it is given that the time $T=4.47$ sec is directly proportional to the square root of the height $h=25$ m, therefore,

$T=k\sqrt{h}\Rightarrow 4.47=k\sqrt{25}\Rightarrow 4.47=5k\Rightarrow k=\frac{4.47}{5}=0.894$

Thus, the general equation is $T=0.894\sqrt{h}$.

Since $h=50$ m, therefore,

$T=0.894\sqrt{50}=0.894\times 5\sqrt{2}=4.47\sqrt{2}=6.321$

Hence, $T=6.321$ seconds when $h=50$ m.