Question
Three metallic objects were released in a vacuum chamber from a height of 4.0 meters. The following data illustrates the mass of each object and the force exerted by each object upon colliding with the chamber floor.
Object | Force (N) | Mass (kg)
1 | 19.6 | 2.0
2 | 78.4 | 8.0
3 | 156.8 | 16.0
Which of the provided statements accurately characterizes the correlation between the mass of the descending objects (measured in kilograms) and the force exerted by these objects (measured in newtons)?
Object | Force (N) | Mass (kg)
1 | 19.6 | 2.0
2 | 78.4 | 8.0
3 | 156.8 | 16.0
Which of the provided statements accurately characterizes the correlation between the mass of the descending objects (measured in kilograms) and the force exerted by these objects (measured in newtons)?
Answers
GPT-4o mini
The data provided in the table indicates a clear correlation between the mass of each object and the force they exert upon colliding with the floor.
To analyze this correlation, we can look at the relationship between force \( F \), mass \( m \), and acceleration \( a \) using Newton's second law of motion:
\[
F = m \cdot a
\]
In this scenario, since all objects are falling under the influence of gravity alone, the acceleration \( a \) would be equal to the acceleration due to gravity, which is approximately \( 9.8 \, \text{m/s}^2 \).
If we calculate the force exerted by each object based on their mass, we can check the values:
1. For Object 1:
\[
F_1 = m_1 \cdot g = 2.0 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 19.6 \, \text{N}
\]
2. For Object 2:
\[
F_2 = m_2 \cdot g = 8.0 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 78.4 \, \text{N}
\]
3. For Object 3:
\[
F_3 = m_3 \cdot g = 16.0 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 156.8 \, \text{N}
\]
The calculated forces match the provided force values for each object.
From this analysis, we can conclude that the relationship between mass and force is directly proportional. This means as the mass of the object increases, the force exerted upon impact also increases proportionally. The specific ratio of force to mass in this case remains constant and equal to the acceleration due to gravity.
Thus, the accurate characterization would be: **The force exerted by each object when colliding with the floor is directly proportional to its mass.**
To analyze this correlation, we can look at the relationship between force \( F \), mass \( m \), and acceleration \( a \) using Newton's second law of motion:
\[
F = m \cdot a
\]
In this scenario, since all objects are falling under the influence of gravity alone, the acceleration \( a \) would be equal to the acceleration due to gravity, which is approximately \( 9.8 \, \text{m/s}^2 \).
If we calculate the force exerted by each object based on their mass, we can check the values:
1. For Object 1:
\[
F_1 = m_1 \cdot g = 2.0 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 19.6 \, \text{N}
\]
2. For Object 2:
\[
F_2 = m_2 \cdot g = 8.0 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 78.4 \, \text{N}
\]
3. For Object 3:
\[
F_3 = m_3 \cdot g = 16.0 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 156.8 \, \text{N}
\]
The calculated forces match the provided force values for each object.
From this analysis, we can conclude that the relationship between mass and force is directly proportional. This means as the mass of the object increases, the force exerted upon impact also increases proportionally. The specific ratio of force to mass in this case remains constant and equal to the acceleration due to gravity.
Thus, the accurate characterization would be: **The force exerted by each object when colliding with the floor is directly proportional to its mass.**