A particle of mass  m is projected with a speed  u from the ground at an angle  θ  =  π/3 with respect to horizontal (x-axis). When it has reached its maximum height, it collides completely inelastically with another particle of the same mass and velocity uî. The horizontal distance covered by the combined mass before reaching the ground is:

To solve this problem, we need to break it down into several steps:

Step 1: Determine the initial velocities of the first particle in the x and y directions. The initial velocity in the x direction (u_x) is ucos(π/3) = u/2. The initial velocity in the y direction (u_y) is usin(π/3) = u*sqrt(3)/2.

Step 2: Determine the time it takes for the first particle to reach its maximum height. This is when its velocity in the y direction is zero. Using the equation v = u + at, where a is acceleration and t is time, we get 0 = usqrt(3)/2 - gt. Solving for t gives t = u*sqrt(3)/(2g).

Step 3: Determine the velocity of the first particle in the x direction when it reaches its maximum height. This is simply its initial velocity in the x direction, as there is no acceleration in the x direction. So, v_x = u/2.

Step 4: Determine the total momentum in the x direction after the collision. The momentum of the first particle is mv_x = mu/2. The momentum of the second particle is mu. So, the total momentum is mu/2 + mu = 3mu/2.

Step 5: Determine the velocity of the combined mass after the collision. The total momentum is the mass times the velocity, so 3mu/2 = 2mv, where v is the velocity of the combined mass. Solving for v gives v = 3u/4.

Step 6: Determine the time it takes for the combined mass to reach the ground. This is the same as the time it took for the first particle to reach its maximum height, as the problem is symmetric. So, t = u*sqrt(3)/(2g).

Step 7: Determine the horizontal distance covered by the combined mass before reaching the ground. This is the velocity times the time, so d = vt = (3u/4) * (usqrt(3)/(2g)) = 3u^2*sqrt(3)/(8g).

So, the horizontal distance covered by the combined mass before reaching the ground is 3u^2*sqrt(3)/(8g).