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. (c) At the instant the rock hits the ground, how far is it from the basket?

Verified step by step guidance

1. Identify the initial velocities of the stone in both the horizontal and vertical directions. The stone has an initial horizontal velocity of 15.0 m/s (as seen from the basket). Since the basket is moving downward at 20.0 m/s, the initial vertical velocity of the stone relative to the ground is also 20.0 m/s downward.
2. Calculate the horizontal displacement of the stone. Since the horizontal motion is not affected by gravity and there is no horizontal acceleration, use the formula for horizontal displacement: $x=v_x\times t$, where $v_x$ is the horizontal velocity and $t$ is the time.
3. Determine the vertical displacement of the stone. Use the formula for vertical displacement under constant acceleration due to gravity: $y=v_{y}t+\frac{1}{2}g{t}^2$, where $v_y$ is the initial vertical velocity, $g$ is the acceleration due to gravity (approximately 9.8 m/s^2), and $t$ is the time.
4. Calculate the total distance the stone travels from the basket. Since the basket continues to move downward at a constant velocity, its position after 5.00 s is directly below its starting position but 100 m lower (since 20.0 m/s $\times$ 5.00 s = 100 m).
5. Use the Pythagorean theorem to find the distance between the stone and the basket at the moment the stone hits the ground. The horizontal displacement represents one leg of a right triangle, and the difference in vertical positions of the stone and the basket represents the other leg.

Key Concepts

Relative Velocity

Relative velocity refers to the velocity of an object as observed from a particular reference frame. In this scenario, the stone's velocity is measured relative to the basket, which is descending at a constant speed. Understanding relative velocity is crucial for analyzing the motion of the stone as it is thrown perpendicular to the balloon's path, allowing us to determine its trajectory and position when it hits the ground.

Projectile Motion

Projectile motion describes the motion of an object that is thrown into the air and is subject to gravitational acceleration. The stone, once thrown, follows a parabolic trajectory influenced by gravity. Analyzing the stone's vertical and horizontal components of motion separately helps in calculating its position at any given time, including the distance from the basket when it hits the ground.

Constant Velocity

Constant velocity indicates that an object moves at a steady speed in a straight line without acceleration. In this problem, the balloon descends at a constant velocity of 20.0 m/s, which simplifies the analysis of the stone's motion. Since the balloon's speed does not change, we can easily calculate the distance it travels during the time the stone is in the air, aiding in determining how far the stone is from the basket upon impact.