To solve the problem of the balloon's height as a function of time, we need to analyze the motion of both the balloon and the stone. The balloon is moving upward vertically, while the stone is projected horizontally with a velocity \( u \) relative to the balloon. The stone always lands at a fixed horizontal distance of \( \frac{2u^2}{g} \) from the balloon. Let's break this down step by step.
Understanding the Motion of the Balloon and the Stone
First, let's establish some variables:
- h(t): height of the balloon at time \( t \)
- u: horizontal velocity of the stone relative to the balloon
- g: acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \))
- t: time elapsed since the stone was projected
Vertical Motion of the Balloon
Assuming the balloon starts from an initial height \( h_0 \) and moves upward with a constant velocity \( v_b \), the height of the balloon at any time \( t \) can be expressed as:
h(t) = h_0 + v_b t
Horizontal Motion of the Stone
The stone is projected horizontally with a velocity \( u \). The horizontal distance \( x \) that the stone travels in time \( t \) is given by:
x = u t
Since the stone lands at a fixed distance of \( \frac{2u^2}{g} \) from the balloon, we can set up the equation:
u t = \frac{2u^2}{g}
Finding Time of Flight
From the equation above, we can solve for time \( t \):
t = \frac{2u}{g}
Vertical Motion of the Stone
During the time \( t \), the stone also falls vertically under the influence of gravity. The vertical displacement \( y \) of the stone can be described by the equation:
y = \frac{1}{2} g t^2
Setting Up the Height Equation
At the moment the stone hits the ground, its vertical position is equal to the height of the balloon minus the distance it has fallen:
0 = h(t) - \frac{1}{2} g t^2
Substituting \( h(t) \) into the equation gives:
0 = (h_0 + v_b t) - \frac{1}{2} g t^2
Solving for the Height of the Balloon
Rearranging this equation, we find:
h_0 + v_b t = \frac{1}{2} g t^2
Now, we can express the height of the balloon as a function of time:
h(t) = \frac{1}{2} g t^2 - v_b t
Final Expression for Height
Thus, the height of the balloon as a function of time is:
h(t) = \frac{1}{2} g t^2 - v_b t + h_0
This equation shows how the height of the balloon changes over time, considering both its initial height and the effects of gravity on the stone. The term \( \frac{1}{2} g t^2 \) represents the vertical displacement due to gravity, while \( -v_b t \) accounts for the upward motion of the balloon. This relationship allows us to predict the balloon's height at any given moment during the stone's flight.
