Understanding the Trajectory of a Ball Thrown Vertically from a Descending Balloon
In a scenario where a ball is thrown vertically upward from a descending balloon, understanding the motion and time taken for the ball to pass by the balloon requires a detailed analysis of the physics involved. This article explores this problem using the principles of kinematics.
The Problem Statement
A ball is thrown vertically upward with a velocity (u) from the balloon, which is descending with a constant velocity (v).
Given Data
Initial velocity of the ball: (u) (upward) Velocity of the balloon: (v) (downward) Acceleration due to gravity: (g) (downward)Analysis of Motion
Position of the Balloon
The balloon is descending with a constant velocity (v). Its position (y_{balloon}(t)) as a function of time (t) can be described by:
[ y_{balloon}(t) -vt ]The negative sign indicates that the balloon is moving downward.
Position of the Ball
The ball is thrown upward with an initial velocity (u) and is subject to gravitational acceleration (g). Its position (y_{ball}(t)) as a function of time (t) is given by:
[ y_{ball}(t) ut - frac{1}{2}gt^2 ]Equating the Positions
To find when the ball passes the balloon, we need to equate their positions:
[ ut - frac{1}{2}gt^2 -vt ]Rearranging the equation, we get:
[ ut vt - frac{1}{2}gt^2 0 ]This can be rewritten as:
[ -frac{1}{2}gt^2 (u v)t 0 ]Factoring out (t) gives:
[ tleft(-frac{1}{2}gt (u v)right) 0 ]From the equation, we have two solutions:
(t 0) (initial throw) ( -frac{1}{2}gt (u v) 0 )From the second solution:
[ -frac{1}{2}gt - (u v) ]Thus:
[ t frac{2(u v)}{g} ]Conclusion
The time (t) when the ball passes by the balloon is:
[ t frac{2(u v)}{g} ]This matches the provided answer and completes our analysis.
Additional Insights
When considering the stone descending with a constant velocity (v), its net acceleration is effectively zero. If the ball is thrown from a stationary balloon (balloon at rest) with an initial velocity of (u), the problem becomes simpler as the initial velocity of the ball relative to the ground is (u). The total time of flight for a ball thrown upward with an initial velocity (X) is given by (2X/g). Therefore, the time for the ball to return to its initial position from a stationary balloon would be:
[ frac{2u}{g} ]