Question

Fill in the Blank : Airplanes A and B are flying with constant velocity in the same vertical plane at angles $30^{\circ}$ $60^{\circ}$◦ with respect to the horizontal respectively as shown in the figure. The speed of A $100\sqrt{3} m/s$ At time t = 0 s, an observer in A finds B at a distance of 500 m. This observer sees B moving with a constant velocity perpendicular to the line of motion of A. If at $t=t_0$ A just escapes being hit by B, $t_0$ in seconds is . . . . . . .

Answer 1

Answer:

54

Explanation:

Answer 2

Velocity of the airplane A will be given by,

$\longrightarrow\bf{v_A}=\sf{100\sqrt3\left(\cos30^o\,\hat i+\sin30^o\,\hat j\right)}$

$\longrightarrow\bf{v_A}=\sf{100\sqrt3\left(\dfrac{\sqrt3}{2}\,\hat i+\dfrac{1}{2}\,\hat j\right)}$

$\longrightarrow\bf{v_A}=\sf{150\,\hat i+50\sqrt3\,\hat j}$

Let speed of the airplane B be $\sf{v_B}.$ Then velocity is given by,

$\longrightarrow\bf{v_B}=\sf{v_B\left(\cos60^o\,\hat i+\sin60^o\,\hat j\right)}$

$\longrightarrow\bf{v_B}=\sf{v_B\left(\dfrac{1}{2}\,\hat i+\dfrac{\sqrt3}{2}\,\hat j\right)}$

$\longrightarrow\bf{v_B}=\sf{\dfrac{v_B}{2}\,\hat i+\dfrac{v_B\sqrt3}{2}\,\hat j}$

Relative motion of B wrt A, or the observer in A will be,

$\bf{\longrightarrow v_{BA}= v_B -v_A}$

$\longrightarrow\bf{v_{BA}}=\sf{\left(\dfrac{v_B}{2} -150\right)\,\hat i+\left(\dfrac{v_B\sqrt3}{2}-5 0\sqrt3\right)\,\hat j\quad\quad\dots(1)}$

This velocity is perpendicular to motion of A, so this velocity should be inclined at 30° with the vertical, or 120° with the horizontal.

So,

$\sf{\longrightarrow\dfrac{\left( \dfrac{v_B\sqrt3}{2}-50\sqrt3\right)}{\left(\dfrac{v_B}{2}-150\right)} =\tan120^o}$

$\sf{\longrightarrow\dfrac{v_B\sqrt3- 100\sqrt3 }{v_B-300}=-\sqrt3}$

$\sf{\longrightarrow v_B=200\ m\,s^{-1}}$

Then (1) becomes,

$\longrightarrow\bf{v_{B A}}=\sf{-50\,\hat i+50\sqrt3\, \hat j}$

so the relative speed is,

$\longrightarrow\sf{v_{BA}=\sqrt{(-50)^2+(50\sqrt3)^2}}$

$\longrightarrow\sf{v_{BA}= 100\ m\,s^{-1}}$

Hence the time $\sf{t_0}$ will be,

$\sf{\longrightarrow t_0=\dfrac{500}{100}}$

$\sf{\longrightarrow\underline{\underline{t_0=5\ s}}}$