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A particle $A$ moves along the line, $y=30 \mathrm{~m}$ with a constant velocity, $\mathbf{v}$ parallel to $x$-axis. At the moment particle $A$ passes the $y$-axis, a particle $B$ starts from the origin with zero initial speed and a constant acceleration, $\mathbf{a}=0.40 \mathrm{~m} / \mathrm{sec}^2$. The angle between $\mathbf{a}$ and $y$-axis is $60^{\circ}$. If the particles $A$ and $B$ collide after sometimes, then the value of $|\mathbf{v}|$ will be
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Verified Answer
The correct answer is:
$3 \mathrm{~m} / \mathrm{s}$
As shown in the figure, two particles are moving with velocity $v$ and acceleration a.
Let the time of collision of two particles be $t$, then the equation of motion,
$$
s=u t+\frac{1}{2} a t^2
$$
For particle $A$,
$$
\begin{aligned}
30 \mathrm{~m} & =(0) t+\frac{a \cos 60^{\circ} t^2}{2} \\
\Rightarrow \quad 30 & =\frac{0.4}{2} \times \frac{1}{2} \times t^2 \\
t & =\sqrt{300}
\end{aligned}
$$
Hence, in time $t=\sqrt{300}$, both travelled equal distance in horizontal direction, so that the collision takes place.
$$
\Rightarrow u t=\frac{1}{2} a \cos 30^{\circ} t^2\left(\because 30^{\circ}=90^{\circ}-60^{\circ}\right)
$$
$$
\begin{aligned}
& \Rightarrow \quad u=\frac{1}{2} \times 0.4 \times \frac{\sqrt{3}}{2} \times \sqrt{300} \\
& \Rightarrow \quad u=3 \mathrm{~m} / \mathrm{s} \\
&
\end{aligned}
$$
Hence, the correct option is (2).
Let the time of collision of two particles be $t$, then the equation of motion,
$$
s=u t+\frac{1}{2} a t^2
$$
For particle $A$,
$$
\begin{aligned}
30 \mathrm{~m} & =(0) t+\frac{a \cos 60^{\circ} t^2}{2} \\
\Rightarrow \quad 30 & =\frac{0.4}{2} \times \frac{1}{2} \times t^2 \\
t & =\sqrt{300}
\end{aligned}
$$
Hence, in time $t=\sqrt{300}$, both travelled equal distance in horizontal direction, so that the collision takes place.
$$
\Rightarrow u t=\frac{1}{2} a \cos 30^{\circ} t^2\left(\because 30^{\circ}=90^{\circ}-60^{\circ}\right)
$$
$$
\begin{aligned}
& \Rightarrow \quad u=\frac{1}{2} \times 0.4 \times \frac{\sqrt{3}}{2} \times \sqrt{300} \\
& \Rightarrow \quad u=3 \mathrm{~m} / \mathrm{s} \\
&
\end{aligned}
$$
Hence, the correct option is (2).
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