A 124-kg balloon carrying a 22-kg basket is descending with a constant downward velocity of 20.0 m/s. A 1.0-kg stone is thrown from the basket with an initial velocity of 15.0 m/s perpendicular to the path of the descending balloon, as measured relative to a person at rest in the basket. That person sees the stone hit the ground 5.00 s after it was thrown. Assume that the balloon continues its downward descent with the same constant speed of 20.0 m/s. (a) How high is the balloon when the rock is thrown?

Verified step by step guidance

1. Identify the initial conditions for the stone and the balloon. The balloon and the stone are moving downward at 20.0 m/s when the stone is thrown. The stone is also given an initial horizontal velocity of 15.0 m/s relative to the basket.
2. Analyze the vertical motion of the stone. Since the stone is thrown from a moving balloon, its initial vertical velocity is the same as the balloon's, which is 20.0 m/s downward. Use the kinematic equation for vertical displacement, $y = v_{0y}t + \frac{1}{2}gt^2$, where $g$ is the acceleration due to gravity (approximately 9.8 m/s^2), $v_{0y}$ is the initial vertical velocity, and $t$ is the time of flight.
3. Substitute the known values into the vertical motion equation. Here, $v_{0y} = -20.0$ m/s (negative because it's downward), $g = 9.8$ m/s^2, and $t = 5.00$ s. Solve for $y$, the vertical displacement of the stone from the point it was thrown to when it hits the ground.
4. Determine the initial height of the balloon. The vertical displacement $y$ found in the previous step represents how far the stone traveled downward from the point of release to the ground. Since the balloon continues to move downward at the same speed, add the absolute value of $y$ to the distance the balloon travels in the same time to find the total height of the balloon when the stone was thrown.
5. Calculate the total height of the balloon above the ground at the moment the stone was thrown by adding the distance the balloon descends during the stone's flight to the vertical displacement of the stone.

Key Concepts

Constant Velocity

Constant velocity refers to an object moving at a steady speed in a straight line, meaning its speed and direction do not change over time. In this scenario, the balloon descends at a constant velocity of 20.0 m/s, indicating that the forces acting on it are balanced, resulting in no acceleration. Understanding this concept is crucial for analyzing the motion of the balloon and the stone relative to each other.

Relative Motion

Relative motion describes how the position and velocity of an object are observed from a particular frame of reference. In this case, the stone is thrown from the basket and its motion is analyzed from the perspective of a person in the basket. This perspective is essential for determining the stone's trajectory and the time it takes to hit the ground, as it allows us to separate the stone's motion from that of the descending balloon.

Projectile Motion

Projectile motion refers to the motion of an object that is launched into the air and is subject to gravitational acceleration. The stone, thrown perpendicular to the balloon's path, follows a parabolic trajectory influenced by gravity. Analyzing the stone's vertical and horizontal components of motion is key to calculating its height and the time it takes to reach the ground, which is critical for solving the problem.