Yup i find the answer...the ratio of the height of the 2 balls is 1:1...😊👍

Introduction:
In this scenario, we have two balls, one with mass m and the other with mass 2m. The first ball is thrown vertically upwards, while the second ball is thrown at an angle θ with the vertical. Both balls stay in the air for the same period of time. We need to determine the ratio of the heights attained by the two balls.

Analysis:
To solve this problem, we can consider the motion of each ball separately and then compare their heights.

Ball thrown vertically:
When the ball with mass m is thrown vertically upwards, it experiences a downward force due to gravity. The only force acting on the ball is its weight, which can be calculated as W = mg, where g is the acceleration due to gravity. The net force acting on the ball is given by F = ma, where a is the acceleration of the ball.

Using Newton's second law, we have F = ma = -mg, where the negative sign indicates the opposite direction to the upward motion. Solving for acceleration, we get a = -g. Since the ball stays in the air for a certain time, the total change in velocity is zero when it reaches its maximum height. Therefore, the initial velocity u is equal to the final velocity v at the maximum height.

Using the equation v = u + at, we can rewrite it as 0 = u - gt. Solving for u, we get u = gt. Now, using the equation v^2 = u^2 + 2as, where s represents the height, we can substitute the values to find the height h.

v^2 = (gt)^2 + 2(-g)s
0 = g^2t^2 - 2gs
2gs = g^2t^2
s = (g^2t^2)/(2g)
s = (gt^2)/2

Ball thrown at an angle:
When the ball with mass 2m is thrown at an angle θ with the vertical, it experiences two forces - its weight (2mg) in the downward direction and an upward force due to the projection. The net force acting on the ball is given by F = ma, where a is the acceleration of the ball.

Using Newton's second law, we have F = ma = 2mg - Fp, where Fp is the upward force due to the projection. We can decompose this force into two components - one in the vertical direction and another in the horizontal direction.

The vertical component of the force is given by Fp_vertical = Fp * sin(θ), where θ is the angle with the vertical. The horizontal component of the force is given by Fp_horizontal = Fp * cos(θ).

The vertical component of the force is responsible for the vertical motion of the ball. The horizontal component does not affect the height attained by the ball.

Using the equation v = u + at, we can calculate the time taken for the ball to reach its maximum height. The initial velocity u is given by Fp_vertical = mu, where m is the mass of the ball.

Using the equation v^2 = u^2 + 2as, we can substitute the values to find the height H attained by the ball.

v^2 = u^2 + 2as
0 = u^2 - 2g