Question
Two balls are hurled at a constant rate and a consistent angle from the top of a 35-foot-tall
building. The preceding scatterplot represents the number of seconds it took for each ball to reach
the ground from a certain height. Each point represents a piece of data collected from each throw.
For example, \((4,9)\) represents that after 4 seconds a ball will be 9 feet in the air. Which model is
the best fit for this data?
Option #1: linear
Option #2: quadratic
Question 4
Option #
Option #3: exponential
building. The preceding scatterplot represents the number of seconds it took for each ball to reach
the ground from a certain height. Each point represents a piece of data collected from each throw.
For example, \((4,9)\) represents that after 4 seconds a ball will be 9 feet in the air. Which model is
the best fit for this data?
Option #1: linear
Option #2: quadratic
Question 4
Option #
Option #3: exponential
Answers
GPT-4o mini
To determine the best model for the data described, we need to consider the physics of the scenario. When balls are thrown vertically from a height, their motion can be modeled using the equations of projectile motion, which typically involves a quadratic relationship.
1. **Linear Model**: A linear model would imply that the height of the balls decreases at a constant rate over time. This is not applicable here because the height of an object in free fall is affected by acceleration due to gravity, which does not yield a linear relationship.
2. **Quadratic Model**: The motion of the balls as they fall from the building follows a parabolic trajectory, which can be described by a quadratic function. The height of the balls with respect to time can be expressed as a quadratic equation (in the standard form \( h(t) = -gt^2 + vt + h_0 \)), where \( g \) is the acceleration due to gravity, \( v \) is the initial velocity, and \( h_0 \) is the initial height.
3. **Exponential Model**: An exponential model would suggest that the rate of change of height is proportional to the height itself, which is not the case for free-falling objects under the influence of gravity.
Given that we are considering the trajectory of balls thrown from a height, the appropriate model for the height vs. time relationship in this scenario is **Option #2: quadratic**.
1. **Linear Model**: A linear model would imply that the height of the balls decreases at a constant rate over time. This is not applicable here because the height of an object in free fall is affected by acceleration due to gravity, which does not yield a linear relationship.
2. **Quadratic Model**: The motion of the balls as they fall from the building follows a parabolic trajectory, which can be described by a quadratic function. The height of the balls with respect to time can be expressed as a quadratic equation (in the standard form \( h(t) = -gt^2 + vt + h_0 \)), where \( g \) is the acceleration due to gravity, \( v \) is the initial velocity, and \( h_0 \) is the initial height.
3. **Exponential Model**: An exponential model would suggest that the rate of change of height is proportional to the height itself, which is not the case for free-falling objects under the influence of gravity.
Given that we are considering the trajectory of balls thrown from a height, the appropriate model for the height vs. time relationship in this scenario is **Option #2: quadratic**.