A particle of mass m is projected with a speed u from the
Question:

A particle of mass $m$ is projected with a speed $u$ from the

ground at an angle $\theta=\frac{\pi}{3}$ w.r.t. horizontal (x-axis). When

it has reached its maximum height, it collides completely inelastically with another particle of the same mass and velocity $u \hat{i}$. The horizontal distance covered by the combined mass before reaching the ground is:

1. (1) $\frac{3 \sqrt{3}}{8} \frac{u^{2}}{g}$

2. (2) $\frac{3 \sqrt{2}}{4} \frac{u^{2}}{g}$

3. (3) $\frac{5}{8} \frac{u^{2}}{g}$

4. (4) $2 \sqrt{2} \frac{u^{2}}{g}$

Correct Option: 1

Solution:

(1) Using principal of conservation of linear momentum for horizontal motion, we have

2 m v_{x}=m u+m u \cos 60^{\circ}

$v_{x}=\frac{3 u}{4}$

For vertical motion

$h=0+\frac{1}{2} g T^{2} \Rightarrow T=\sqrt{\frac{2 h}{g}}$

Let $R$ is the horizontal distance travelled by the body.

$R=v_{x} T+\frac{1}{2}(0)(T)^{2}$ (For horizontal motion)

$R=v_{x} T=\frac{3 u}{4} \times \sqrt{\frac{2 h}{g}}$

$\Rightarrow \quad R=\frac{3 \sqrt{3} u^{2}}{8 g}$