In this problem, we have to compare the motion of two particles of different masses thrown simultaneously with the same speed from the same point on the ground under gravity. The first particle is projected upwards and the second at an angle theta with the horizontal.




Let us assume that the mass of the first particle is m1 and the mass of the second particle is m2. Both the particles are thrown simultaneously with the same speed u. The velocity of the first particle at the maximum height is zero. Using the equation of motion,
v² = u² - 2gh
where v is the final velocity, u is the initial velocity, g is the acceleration due to gravity and h is the maximum height attained.

For the first particle, the initial velocity is u and the final velocity is zero. Therefore, the time taken to reach maximum height is given by,
t = u/g

For the second particle, the initial velocity is u cos(theta) and the final velocity is zero at the maximum height. Therefore, the time taken to reach maximum height is given by,
t = u sin(theta)/g

At the maximum height, the velocity of both particles is zero. Therefore, the time taken for both particles to return to the ground is given by,
T = 2t = 2u/g

The final velocity of both particles when they reach the ground is given by,
v = u + gt

Therefore, the final velocity of the first particle is v1 = u - gt and the final velocity of the second particle is v2 = u cos(theta) - u sin(theta)

From the above equations, we can see that the final velocity of both particles is different. Hence, option A is incorrect.


From the above calculations, we can see that the time taken for both particles to return to the ground is the same. Therefore, both particles return in a group at the same time. Hence, option B is correct.


Height attained by the first particle is given by,
h1 = u²/2g

Height attained by the second particle is given by,
h2 = u² sin²(theta)/2g

From the above equations, we can see that the height attained by both particles is different. Hence, option C is incorrect.


The velocity of the first particle is always in the upward direction and the velocity of the second particle is always inclined at an angle theta with the horizontal. Therefore, the velocities of both particles are never perpendicular to each other. Hence, option D is correct.


Hence, we can conclude that options B and D are correct and options A and C are incorrect.

I think return on the ground after same time