A particle $R$ moves along the line $y = 30 m$ with velocity of magnitude $3 m / s$ . At the instant particle $R$ passes the $y$ -axis, particle $S$ leaves with zero initial speed and constant acceleration of magnitude $0.40 m / s^{2}$ . What is the value of $sec θ$ for which a collision will take place?

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Solution

Draw a labelled diagram.

Find the value of $sec θ$ .
$_{1} = 30^j$
$_{2} = 3 t^i$
$_{3} = \frac{1}{2} [(0.4) sin θ^i + (0.4) cos θ^j] t^{2}$
For collision,
${\to r}_{1} + {\to r}_{2} = {\to r}_{3}$
$\Rightarrow (30)^j + (3 t)^i = 0.2 t^{2} [sin θ^i + cos θ^j]$
$\Rightarrow 30 = 0.2 t^{2} cos θ and 3 t = 0.2 t^{2} sin θ$
$\Rightarrow 150 = t^{2} cos θ$ and $15 = t sin θ$
Dividing both the equations,
$\Rightarrow \frac{150}{225} = \frac{cos θ}{{sin}^{2} θ} \Rightarrow \frac{2}{3} = \frac{cos θ}{(1 - {cos}^{2} θ)}$
$\Rightarrow cos θ = \frac{1}{2}$
$sec θ = 2$
Final Answer: (2)

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