Question

which of the following graphs illustrates the relationship between height and potential energy?

Explanation:

To determine the graph illustrating the relationship between height and gravitational potential energy, we recall the formula for gravitational potential energy (\(U\)):

$$ U = mgh $$

where \(m\) is mass, \(g\) is acceleration due to gravity, and \(h\) is height. Assuming \(m\) and \(g\) are constant, \(U\) is directly proportional to \(h\) (i.e., \(U \propto h\)). This means as height (\(h\)) increases, potential energy (\(U\)) increases linearly.

Analyzing the Graphs:

• A linear, upward - sloping graph (positive slope) represents a direct proportionality.
• Graphs with horizontal lines (constant \(U\) regardless of \(h\)) or downward - sloping lines (inverse relationship) do not match \(U = mgh\).

The graph with a straight line passing through the origin (or with a positive slope) and showing \(U\) increasing as \(h\) increases is the correct one. If the last graph (right - most) has a positive, linear slope (e.g., a line starting near the origin and rising as height increases), it illustrates \(U \propto h\).

\(\boldsymbol{\text{The graph with a positive linear relationship between height and potential energy (e.g., the right - most graph if it is upward - sloping and linear)}}\)

Answer:

To determine the graph illustrating the relationship between height and gravitational potential energy, we recall the formula for gravitational potential energy (\(U\)):

$$ U = mgh $$

where \(m\) is mass, \(g\) is acceleration due to gravity, and \(h\) is height. Assuming \(m\) and \(g\) are constant, \(U\) is directly proportional to \(h\) (i.e., \(U \propto h\)). This means as height (\(h\)) increases, potential energy (\(U\)) increases linearly.

Analyzing the Graphs:

• A linear, upward - sloping graph (positive slope) represents a direct proportionality.
• Graphs with horizontal lines (constant \(U\) regardless of \(h\)) or downward - sloping lines (inverse relationship) do not match \(U = mgh\).

The graph with a straight line passing through the origin (or with a positive slope) and showing \(U\) increasing as \(h\) increases is the correct one. If the last graph (right - most) has a positive, linear slope (e.g., a line starting near the origin and rising as height increases), it illustrates \(U \propto h\).

\(\boldsymbol{\text{The graph with a positive linear relationship between height and potential energy (e.g., the right - most graph if it is upward - sloping and linear)}}\)