SSCE HSC Mathematics Extension 1 Binomial distribution: Two particles are fired simultan

Question

Two particles are fired simultaneously from the ground at time $t = 0$.

Particle 1 is projected from the origin at an angle $	heta$, $0 < 	heta < \frac{\pi}{2}$, with an initial velocity $V$.
Particle 2 is projected vertically upward from the point $A$, at a distance $a$ to the right of the origin, also with an initial velocity $V$.

It can be shown that while both particles are in flight, Particle 1 has equations of motion:

$x = V \cos 	heta \cdot t$

$y = V \sin 	heta \cdot t - \frac{1}{2} g t^2$

and Particle 2 has equations of motion:

$\dot{x} = a$

y = $V t - \frac{1}{2} g t^2$.

Let $L$ be the distance between the particles at time $t$.

(i) Show that, while both particles are in flight,

$L^2 = 2V^2t^2(1 - \sin 	heta) - 2aV\cos 	heta + a^2$.

(ii) An observer notices that the distance between the particles in flight first decreases, then increases.
Show that the distance between the particles in flight is smallest when

t = \frac{a \cos 	heta}{2V(1 - \sin 	heta)}

and that this smallest distance is $\sqrt{a \cdot \frac{1 - \sin 	heta}{2}}$.

(iii) Show that the smallest distance between the two particles in flight occurs while Particle 1 is ascending if $V > \frac{a \cos 	heta}{\sqrt{2} \sin 	heta(1 - \sin 	heta)}$.

Two particles are fired simultaneously from the ground at time $t = 0$ - HSC - SSCE Mathematics Extension 1 - Question 6 - 2006 - Paper 1

Worked Solution & Example Answer:Two particles are fired simultaneously from the ground at time $t = 0$ - HSC - SSCE Mathematics Extension 1 - Question 6 - 2006 - Paper 1

Show that, while both particles are in flight, $L^2 = 2V^2t^2(1 - \sin \theta) - 2aV\cos \theta + a^2$.

Show that the distance between the particles in flight is smallest when t = \frac{a \cos \theta}{2V(1 - \sin \theta)}

Show that the smallest distance between the two particles in flight occurs while Particle 1 is ascending if $V > \frac{a \cos \theta}{\sqrt{2} \sin \theta(1 - \sin \theta)}$.

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