91Ó°ÊÓ

A 124 \(\mathrm{kg}\) balloon carrying a 22 \(\mathrm{kg}\) basket is descending with a constant downward velocity of 20.0 \(\mathrm{m} / \mathrm{s}\) . A 1.0 \(\mathrm{kg}\) stone is thrown from the basket with an initial velocity of 15.0 \(\mathrm{m} / \mathrm{s}\) perpendicular to the path of the descending balloon, as measured relative to a person at rest in the basket. The person in the basket sees the stone hit the ground 6.00 s after being thrown. Assume that the balloon continues its downward descent with the same constant speed of 20.0 \(\mathrm{m} / \mathrm{s}\) . (a) How high was the balloon when the rock was thrown out? How high is the balloon when the rock hits the vground? (c) At the instant the rock hits the ground, how far is it from the basket? (d) Just before the rock hits the ground, find its horizontal and vertical velocity components as measured by an observer (i) at rest in the basket and (ii) at rest on the ground.

Step by step solution

01

Calculate Initial Height of the Balloon

To find the initial height of the balloon when the stone is released, we use the equation of motion. As the balloon descends at a constant velocity of 20 m/s and the stone hits the ground 6 seconds after being thrown, the stone is in free fall for 6 seconds with an initial vertical velocity of -20 m/s (relative to the ground). The equation is given by: \[ h = v_i t + \frac{1}{2} a t^2 \] Substituting the known values: \[ h = (-20 \, \text{m/s}) \cdot 6 \, \text{s} + \frac{1}{2} \cdot (-9.8 \, \text{m/s}^2) \cdot (6 \, \text{s})^2 \] Solve to find \( h \).

02

Calculate Height of Balloon When Stone Hits the Ground

Determine the balloon's height at the moment the stone impacts the ground. The balloon descends at a constant speed of 20 m/s for 6 seconds, so: \[ \Delta h = v_b \times t = 20 \, \text{m/s} \times 6 \, \text{s} \] Subtract this change in height from the initial height found in the previous step.

03

Determine Horizontal Distance from Basket to Rock at Impact

The stone is projected horizontally at 15 m/s. The horizontal distance covered in 6 seconds is: \[ d = v_{horizontal} \times t = 15 \, \text{m/s} \times 6 \, \text{s} \] Solving for \( d \) gives the horizontal distance.

04

Calculate Rock's Velocity Components just Before Hitting Ground From Basket's View

From the basket's perspective (descending with 20 m/s), the stone's initial horizontal velocity is 15 m/s. The vertical motion of the stone relative to the basket includes the downward acceleration by gravity: \[ v_{vertical} = v_{i} + a t = -20 \, \text{m/s} + (-9.8 \, \text{m/s}^2) \cdot 6 \, \text{s} \] Calculate \( v_{vertical} \) and \( v_{horizontal} \).

05

Calculate Rock's Velocity Components just Before Hitting Ground From Ground's View

For an observer on the ground, the stone's initial vertical velocity is the sum of the basket's speed and the stone's speed relative to the basket: \(-20 \, \text{m/s}\). The horizontal velocity is 15 m/s. Thus, for vertical: \( v_{vertical} = v_i + a t = -20 \, \text{m/s} + (-9.8 \, \text{m/s}^2) \cdot 6 \, \text{s} \). Calculate both vertical and horizontal velocities.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinematics

Kinematics is the branch of physics that deals with motion without considering the forces that cause the motion. In this exercise, we explore a real-world scenario involving a balloon, a basket, and a stone. The stone is thrown from a basket that is descending with a balloon. To solve this, we use the basic equations of motion to understand the motion of the stone both in the horizontal and vertical directions.

• We start by considering the velocity of the stone as it leaves the basket. The stone is projected horizontally at 15.0 m/s.
• Since the balloon is moving downwards at 20.0 m/s, this influences the vertical velocity of the stone relative to the ground.

By understanding these initial conditions, we can predict how the stone moves through the air, using equations that relate initial velocity, time, acceleration, and displacement.

Free Fall

Free fall is a specific case of motion where an object moves under the influence of gravitational force only. In the problem, once the stone is thrown, it moves in a free-fall motion vertically. The only force acting on it is gravity, which accelerates the stone downward at 9.8 m/s².

• For vertical motion, the stone's initial vertical speed is affected by the balloon's descent, resulting from the initial speed at the moment of release.
• We calculate the change in vertical velocity over time, knowing that the stone falls for 6 seconds before it hits the ground.

This analysis allows us to determine the vertical velocity just before impact and the vertical trajectory of the stone during its descent.

Relative Velocity

Relative velocity helps us understand how the motion of one object appears from another moving object. Here, the stone's motion is analyzed from both the basket's and the ground observer's perspectives.

• From the basket's perspective, the stone is thrown horizontally at 15.0 m/s, while the basket and balloon move downwards at a constant speed.
• For an observer on the ground, the downward velocity of the balloon influences the initial conditions of the stone's release.

By considering these different perspectives, we can calculate the actual path and velocity components of the stone relative to all observers involved.

Equations of Motion

The equations of motion are essential for predicting the future position and velocities of objects. They are a key part of kinematics and are used extensively in this exercise.

• Equations like \( h = v_i t + \frac{1}{2} a t^2 \) help us find the initial height of the balloon at stone release and at impact.
• The horizontal distance covered and the final velocities in both directions are calculated using the basic constant velocity and acceleration formulas.

These equations help break down the motion into understandable parts, providing insights into how each component of the stone's motion contributes to its final position and velocity.