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The data below was recorded during an investigation of the relationship between potential energy and the mass and height of an object.

Height Data

When a 2 kg weight was lifted 3 meters above the floor, it had 58.8 Joules of potential energy.When a 2 kg weight was lifted 6 meters above the floor, its potential energy doubled.
Mass Data

When a 3 kg weight was lifted 2 meters above the floor, it had 58.8 Joules of potential energy.When a 6 kg weight was lifted 2 meters above the floor, its potential energy doubled.
Which of the following graphs BEST represents the relationship between potential energy and the mass and height of an object?

3 answers

To determine the relationship between potential energy (PE), mass (m), and height (h), we can use the formula for gravitational potential energy:

\[ PE = m \cdot g \cdot h \]

where:

  • \(PE\) is the potential energy,
  • \(m\) is the mass of the object,
  • \(g\) is the acceleration due to gravity (approximately \(9.81 , \text{m/s}^2\)),
  • \(h\) is the height above the ground.

Analyzing the Given Data

  1. Height Data

    • For a 2 kg weight lifted to 3 meters: \[ PE = 2 , \text{kg} \cdot 9.81 , \text{m/s}^2 \cdot 3 , \text{m} \approx 58.86 , \text{J} \]
    • For a 2 kg weight raised to 6 meters, the potential energy doubles: \[ PE_{6m} = 2 , \text{kg} \cdot 9.81 , \text{m/s}^2 \cdot 6 , \text{m} \approx 117.72 , \text{J} \]

  2. Mass Data

    • For a 3 kg weight lifted to 2 meters: \[ PE = 3 , \text{kg} \cdot 9.81 , \text{m/s}^2 \cdot 2 , \text{m} \approx 58.86 , \text{J} \]
    • For a 6 kg weight lifted to 2 meters, the potential energy doubles: \[ PE_{6kg} = 6 , \text{kg} \cdot 9.81 , \text{m/s}^2 \cdot 2 , \text{m} \approx 117.72 , \text{J} \]

Observations

  • The potential energy is directly proportional to both mass and height. Thus, increasing either mass or height results in a linear increase in potential energy.
  • When mass is doubled at a constant height (2 m), the potential energy also doubles.
  • When height is doubled at a constant mass (2 kg), the potential energy doubles.

Graph Representation

The best representation of this relationship in a graph would show a linear increase in potential energy with both mass and height. Specifically:

  • A 3D graph with mass on one axis, height on another, and potential energy on the third axis would clearly demonstrate this relationship.
  • Alternatively, a 2D graph with potential energy on the y-axis and either mass or height on the x-axis would also illustrate that potential energy increases linearly with increases in mass or height.

In conclusion, the BEST graph would be one that indicates a linear relationship between potential energy and both mass and height, ideally with two separate lines or surfaces indicating how potential energy changes when either mass or height is varied independently.

what would the slope look like

In the context of a graph representing the relationship between potential energy (PE), mass (m), and height (h), let's discuss the slope specifically in two common scenarios:

Scenario 1: Graph of PE vs. Mass (at constant height)

  • Graph: Potential Energy (PE) on the y-axis and Mass (m) on the x-axis.
  • Constant Height (h): For a specific height, the equation simplifies to \( PE = m \cdot g \cdot h \).
  • Slope: The slope of the line on this graph would be given by the formula: \[ \text{slope} = g \cdot h \] This slope is constant for a specific height. If \( g = 9.81 , \text{m/s}^2 \), the slope will be \( g \cdot h \) for that height. As mass increases, potential energy increases linearly, resulting in a straight line with a positive slope.

Scenario 2: Graph of PE vs. Height (at constant mass)

  • Graph: Potential Energy (PE) on the y-axis and Height (h) on the x-axis.
  • Constant Mass (m): In this case, the equation becomes \( PE = m \cdot g \cdot h \).
  • Slope: The slope of the line on this graph would be: \[ \text{slope} = m \cdot g \] This slope is constant for a specific mass. Here, with a fixed mass, as height increases, the potential energy also increases linearly, depicted as a straight line with a positive slope.

Summary of Slope Characteristics

  • Positive Slope: In both scenarios, the slope will be positive, indicating that as either mass or height increases, the potential energy also increases.
  • Steeper Slope: The slope will be steeper for larger values of mass (when plotting PE vs. height) or for larger values of height (when plotting PE vs. mass).

Overall Visualizing the Graph

  1. PE vs. Mass: A straight line starting from the origin, with the slope dependent on the height.
  2. PE vs. Height: Also a straight line originating at the origin, with the slope depending on the mass.

In both cases, the straight-line nature indicates a direct, linear relationship between mass, height, and potential energy, consistent with the equation for potential energy.