Understanding the Motion of the Stone

Oct 13, 2025 by talynt.org 20 views

Introduction

In this problem, we are tasked with determining the distance a stone fell in the 3rd second. To solve this problem, we need to consider the motion of the stone and the runner. The stone is thrown from a 20 m high building, while the runner is moving at a constant speed of $1.5 \, \text{m} \cdot \text{s}^{-1}$ at ground level.

Understanding the Motion of the Stone

The stone is thrown from a height of 20 m, and it will fall under the influence of gravity. We can use the equation of motion to describe the position of the stone as a function of time. The equation of motion for an object under constant acceleration is given by:

s(t) = s_0 + v_0t + \frac{1}{2}at^2

where $s(t)$ is the position of the stone at time $t$ , $s_0$ is the initial position, $v_0$ is the initial velocity, $a$ is the acceleration, and $t$ is time.

Calculating the Distance Fallen in the 3rd Second

To determine the distance the stone fell in the 3rd second, we need to calculate the position of the stone at time $t=3$ seconds. We can use the equation of motion to do this. The acceleration due to gravity is $g = 9.8 \, \text{m} \cdot \text{s}^{-2}$ .

import math
s0 = 20  # initial position (m)
v0 = 0   # initial velocity (m/s)
a = 9.8  # acceleration due to gravity (m/s^2)
t = 3    # time (s)
s = s0 + v0t + 0.5a*t**2
print("The distance the stone fell in the 3rd second is:", s, "m")

Understanding the Motion of the Runner

The runner is moving at a constant speed of $1.5 \, \text{m} \cdot \text{s}^{-1}$ at ground level. We can use the equation of motion to describe the position of the runner as a function of time. The equation of motion for an object moving at a constant velocity is given by:

s(t) = s_0 + v_0t

where $s(t)$ is the position of the runner at time $t$ , $s_0$ is the initial position, $v_0$ is the initial velocity, and $t$ is time.

Calculating the Distance Traveled by the Runner in the 3rd Second

To determine the distance traveled by the runner in the 3rd second, we need to calculate the position of the runner at time $t=3$ seconds. We can use the equation of motion to do this.

import math
s0 = 0  # initial position (m)
v0 = 1.5  # initial velocity (m/s)
t = 3    # time (s)
s = s0 + v0*t
print("The distance traveled by the runner in the 3rd second is:", s, "m")

Combining the Motions of the Stone and the Runner

To determine the distance between the stone and the runner in the 3rd second, we need to combine the motions of the stone and the runner. We can do this by calculating the position of the stone and the runner at time $t=3$ seconds and then finding the difference between their positions.

import math
s0_stone = 20  # initial position of the stone (m)
v0_stone = 0   # initial velocity of the stone (m/s)
a_stone = 9.8  # acceleration due to gravity (m/s^2)
t = 3    # time (s)
s0_runner = 0  # initial position of the runner (m)
v0_runner = 1.5  # initial velocity of the runner (m/s)
s_stone = s0_stone + v0_stonet + 0.5a_stone*t**2
s_runner = s0_runner + v0_runner*t
distance = abs(s_stone - s_runner)
print("The distance between the stone and the runner in the 3rd second is:", distance, "m")

Conclusion

Q: What is the equation of motion for an object under constant acceleration?

A: The equation of motion for an object under constant acceleration is given by:

s(t) = s_0 + v_0t + \frac{1}{2}at^2

where $s(t)$ is the position of the object at time $t$ , $s_0$ is the initial position, $v_0$ is the initial velocity, $a$ is the acceleration, and $t$ is time.

Q: What is the acceleration due to gravity?

A: The acceleration due to gravity is $g = 9.8 \, \text{m} \cdot \text{s}^{-2}$ .

Q: How do you calculate the position of the stone at time $t=3$ seconds?

A: To calculate the position of the stone at time $t=3$ seconds, we can use the equation of motion:

s = s_0 + v_0t + \frac{1}{2}at^2

where $s_0$ is the initial position, $v_0$ is the initial velocity, $a$ is the acceleration, and $t$ is time.

Q: What is the distance the stone fell in the 3rd second?

A: The distance the stone fell in the 3rd second is given by:

s = 20 + 0 \cdot 3 + \frac{1}{2} \cdot 9.8 \cdot 3^2 = 44.1 \, \text{m}

Q: How do you calculate the distance traveled by the runner in the 3rd second?

A: To calculate the distance traveled by the runner in the 3rd second, we can use the equation of motion:

s = s_0 + v_0t

where $s_0$ is the initial position, $v_0$ is the initial velocity, and $t$ is time.

Q: What is the distance traveled by the runner in the 3rd second?

A: The distance traveled by the runner in the 3rd second is given by:

s = 0 + 1.5 \cdot 3 = 4.5 \, \text{m}

Q: How do you calculate the distance between the stone and the runner in the 3rd second?

A: To calculate the distance between the stone and the runner in the 3rd second, we can use the equation of motion for the stone and the runner:

s_{stone} = s_0 + v_0t + \frac{1}{2}at^2

s_{runner} = s_0 + v_0t

where $s_0$ is the initial position, $v_0$ is the initial velocity, $a$ is the acceleration, and $t$ is time.

Q: What is the distance between the stone and the runner in the 3rd second?

A: The distance between the stone and the runner in the 3rd second is given by:

distance = |s_{stone} - s_{runner}| = |44.1 - 4.5| = 39.6 \, \text{m}

Q: What is the significance of the distance between the stone and the runner in the 3rd second?

A: The distance between the stone and the runner in the 3rd second is significant because it represents the distance between the two objects at a specific time. This distance can be used to determine the relative motion of the two objects and can be used in various applications such as physics, engineering, and computer science.

Q: How can the equation of motion be used in real-world applications?

A: The equation of motion can be used in various real-world applications such as:

Calculating the trajectory of a projectile
Determining the motion of a vehicle
Modeling the behavior of a system
Predicting the outcome of a physical event

Q: What are some common mistakes to avoid when using the equation of motion?

A: Some common mistakes to avoid when using the equation of motion include:

Not considering the initial conditions
Not accounting for external forces
Not using the correct units
Not checking the validity of the solution

Q: How can the equation of motion be simplified or modified for specific applications?

A: The equation of motion can be simplified or modified for specific applications by:

Using approximations or simplifications
Considering specific initial conditions
Accounting for external forces
Using numerical methods or approximations

Q: What are some advanced topics related to the equation of motion?

A: Some advanced topics related to the equation of motion include:

Relativistic mechanics
Quantum mechanics
Chaos theory
Nonlinear dynamics

Q: How can the equation of motion be used to model complex systems?

A: The equation of motion can be used to model complex systems by:

Using numerical methods or approximations
Considering multiple variables and interactions
Accounting for external forces and constraints
Using advanced mathematical techniques such as differential equations and linear algebra.