a ball is thrown vertically upwards with a velocity 'u' from the balloon descending with a constant velocity 'v'.. the ball will pass by the balloon after what time?

The answer will depend upon whether the velocity of the thrown ball is with respect to the ground or with respect to the balloon. After you have figured out which it is supposed to be, write equations for the height of the ball and the balloon relative to the altitude when the ball is thrown.

Set them equal and solve for t.

To determine the time it takes for the ball to pass by the descending balloon, we need to consider the motion of both the ball and the balloon.

Let's assume that the initial velocity of the ball when thrown upwards is 'u', and the velocity of the descending balloon is 'v'. Since the ball is thrown vertically upwards, the initial velocity will have a positive direction. On the other hand, the balloon is descending, so its velocity will have a negative direction.

To find the time it takes for the ball to pass the balloon, we need to determine the point where their positions are equal. Since the ball is moving upwards and the balloon is moving downwards, the ball will eventually reach a maximum height and then start descending. At this point, the position of the ball will be equal to the position of the balloon.

To find this point, we can use the equations of motion. The position of the ball can be given by the equation:

s_ball = u*t - (1/2)*g*t^2

where 's_ball' is the position of the ball at time 't', 'u' is the initial velocity, 'g' is the acceleration due to gravity, and 't' is the time.

Similarly, the position of the balloon can be given by:

s_balloon = v*t

where 's_balloon' is the position of the balloon at time 't' and 'v' is the velocity of the balloon.

Now, we can set these two equations equal to each other and solve for 't':

u*t - (1/2)*g*t^2 = v*t

Rearranging the equation, we get:

(1/2)*g*t^2 - u*t + v*t = 0

Now, we can solve this quadratic equation for 't' using the quadratic formula:

t = [-(-u) ± √((-u)^2 - 4*(1/2)*g*v)] / (2*(1/2)*g)

Simplifying further:

t = (u ± √(u^2 + 2*g*v)) / g

Depending on the signs of 'u' and 'v', the positive or negative solution of 't' will give you the time it takes for the ball to pass by the balloon.