An object is thrown vertically upward with some speed. It crosses… - Vidyadip

Question

An object is thrown vertically upward with some speed. It crosses two points $p$ and $q$ which are separated by $h$ metre. If $t_p$ is the time between $p$ and the highest point and coming back and $t_q$ is the time between $q$ and the highest point and coming back, relate acceleration due to gravity, $\mathrm{t}_{\mathrm{p}}, \mathrm{t}_{\mathrm{q}}$, and h .

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Answer

Let H = distance between point p and the highest point A. This distance H travelled while falling from A top is given by

$\therefore \text{H}=\frac{1}{2}\text{g}\Big(\frac{\text{t}_\text{p}}{2}\Big)^2$
$=\frac{\text{g t}^2_\text{p}}{8}\ \dots(\text{i})$ Also, $\text{H}-\text{h}=\frac{1}{2}\text{g}\Big(\frac{\text{t}_\text{q}}{2}\Big)^2$
$=\frac{\text{g t}^2_\text{q}}{8}\ \dots(\text{ii})$
Now, From eqns. (i) and (ii), we get $\text{h}=\frac{\text{g t}^2_\text{p}}{8}-\frac{\text{g t}^2_\text{q}}{8}$ $=\frac{\text{g}}{8}(\text{t}^2_\text{p}-\text{t}^2_\text{q})$

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