Question

Can the expression for gravitational potential energy \(U_{\mathrm{g}}(y)=m g y\) be used to analyze high-altitude motion? Why or why not?

Short answer

Explain your answer. Answer: No, the expression \(U_{\mathrm{g}}(y) = mgy\) is not suitable for analyzing high-altitude motion as it relies on the assumption of a constant gravitational field, which breaks down at high altitudes. Instead, the general potential energy formula, \(U(r)=-\frac{GMm}{r}\), should be used for high-altitude motion analysis, as it accounts for the varying gravitational field with distance from the Earth's center.

Step-by-step solution

Understand the expression for gravitational potential energy

The gravitational potential energy of an object is the energy that it possesses due to its position in the gravitational field. The formula for gravitational potential energy is given by \(U_{\mathrm{g}}(y) = mgy\), where \(m\) is the mass of the object, \(g\) is the acceleration due to gravity, and \(y\) is the height of the object from the reference point, often Earth's surface.

Assumptions in the expression for gravitational potential energy

The formula \(U_{\mathrm{g}}(y) = mgy\) is derived under certain assumptions. The primary assumption is that the Earth's gravitational field is uniform, i.e., the value of \(g\) is constant throughout the region of interest. This assumption holds true for motion near the Earth's surface where the changes in the gravitational field are negligible. However, at very high altitudes, the Earth's gravitational field is no longer uniform as gravity weakens with distance from the Earth's center.

Considering high-altitude motion

High-altitude motion refers to motion at great distances from the Earth's surface, where the gravitational field is not uniform. The expression \(U_{\mathrm{g}}(y) = mgy\) is based on the assumption of a constant gravitational field. At high altitudes, it becomes crucial to account for the varying gravitational field by using the general potential energy formula, \(U(r)=-\frac{GMm}{r}\), where \(M\) is the Earth's mass, \(r\) is the distance from the object to the center of the Earth, and \(G\) is the gravitational constant.

Conclusion

Therefore, the expression \(U_{\mathrm{g}}(y) = mgy\) cannot be used to analyze high-altitude motion, as it relies on the assumption of a constant gravitational field. This assumption breaks down at high altitudes where changes in the gravitational field become significant. Instead, the general potential energy formula that considers the changes in the gravitational field should be used for high-altitude motion analysis.

Key concepts

High-Altitude Motion

High-altitude motion involves movements or positioning of objects significantly far from the Earth's surface. At these altitudes, we observe certain differences that can affect how we calculate energy or forces acting on objects, especially when dealing with gravitational potential energy.
One critical issue with high-altitude motion is that the gravitational field is no longer uniform. At such distances, the gravitational pull is weaker than it is near the surface. This variability can influence the motion and energy calculations for any objects in high-altitude environments, including satellites orbiting Earth.

• Normal gravitational assumptions near the Earth may no longer apply.
• Energy properties, such as potential energy, require more accurate formulas to reflect how gravity changes with altitude.

Consequently, for any analysis of high-altitude motion, it becomes essential to employ formulas that appropriately account for the differing conditions due to the weakening of gravity with distance.

Gravitational Field Uniformity

The notion of gravitational field uniformity is central to many calculations involving motion and energy near the Earth's surface. In a uniform gravitational field, the gravitational force is constant, meaning that the acceleration due to gravity (\( g \)) does not vary, which simplifies computations.
For everyday physics applications on or near the Earth's surface, this assumption holds well. However, as one moves away from the Earth, this assumption of constant gravity begins to fail.

• If you're miles above the surface, the value of \( g \) decreases.
• This change means that simple equations, like the typical potential energy formula \( U_g = mgy \), are less accurate.

To accurately describe potential energy at different altitudes, formulas must adjust to the changes in gravitational force or field. This often comes into play in space missions or when studying environmental changes at different atmospheric layers.

Potential Energy Formula

Potential energy describes the energy an object holds due to its position relative to other systems, particularly concerning gravitational fields. The traditional formula for gravitational potential energy, \( U_{\mathrm{g}}(y) = mgy \), is effective for situations where the gravitational field is constant.
This formula works well near the Earth's surface because the gravitational field is fairly uniform. However, its immediate assumption of uniformity disqualifies it in scenarios involving significant heights, such as those encountered in high-altitude motion.

• At high altitudes, relying on \( mgy \) would lead to inaccuracies.
• Instead, one should use the general formula:\( U(r)=-\frac{GMm}{r} \).

This formula accurately considers the changing gravitational field as distance from the Earth's center increases and is essential for precise calculations in space science and engineering applications.