Welcome to the World of Gravitational Potential Energy!

Hello there! In this chapter, we are going to explore Gravitational Potential Energy (GPE) within the context of Projectile Motion. If you have ever wondered why it takes more effort to climb a flight of stairs than to walk on flat ground, or why a ball dropped from a higher floor hits the ground harder, you are already thinking about GPE!

We will focus specifically on uniform gravitational fields (like the one we experience near the surface of the Earth). By the end of these notes, you’ll be able to derive the GPE formula yourself and use it to solve physics problems like a pro. Don't worry if things seem a bit "heavy" at first—we'll break it down step-by-step!


1. Understanding Weight and the Gravitational Field

Before we talk about energy, we need to remember what Weight is. In Physics, weight isn't just a number on a scale; it is a force.

What is a Gravitational Field?

Imagine a "region of influence" around the Earth. Any object with mass placed in this region feels a pull toward the center of the Earth. This region is called a gravitational field.

The Concept of Weight

Weight is the name we give to the gravitational force acting on an object's mass. We calculate it using the formula:
\(W = mg\)

  • \(W\) is the Weight (measured in Newtons, \(N\)).
  • \(m\) is the Mass of the object (measured in kilograms, \(kg\)).
  • \(g\) is the Gravitational Field Strength. Near the Earth's surface, we treat this as a constant value of approximately \(9.81 \text{ N kg}^{-1}\) (or \(9.81 \text{ m s}^{-2}\)).

Quick Tip: In the "Projectile Motion" section of your syllabus, we always assume the gravitational field is uniform. This means that no matter how high or low the projectile goes (within a reasonable distance near Earth), the value of \(g\) stays the same and always points straight down!

Key Takeaway: Weight is the force of gravity pulling on a mass within a field.


2. What is Gravitational Potential Energy (GPE)?

Energy is the capacity to do work. Gravitational Potential Energy (\(E_p\)) is a type of "stored" energy an object has because of its position in a gravitational field.

Analogy: Think of GPE like a "savings account" for energy. When you lift a ball up, you are "depositing" energy into it. When you let go and it falls, the ball "withdraws" that energy to move faster!

Did you know? We usually talk about the change in potential energy (\(\Delta E_p\)) rather than an absolute value. We choose a "reference level" (like the ground) where we decide the energy is zero, and measure everything from there.


3. Deriving the Formula: \(\Delta E_p = mg\Delta h\)

One of your syllabus requirements is to be able to derive this formula from the definition of Work Done. It's actually quite simple if you follow these three steps!

Step 1: Start with Work Done

Recall that Work Done (\(W\)) by a force is the product of the force and the displacement in the direction of that force:
\(Work = Force \times displacement\)

Step 2: Apply it to Lifting an Object

To lift an object of mass \(m\) at a constant speed, you must apply a force equal to its Weight (\(mg\)). If you lift it through a vertical height change of \(\Delta h\), then:

  • The Force applied = \(mg\)
  • The Displacement = \(\Delta h\)
Step 3: Combine them

The work you do to lift the object is transferred into the object's gravitational potential energy store. Therefore:
\(\Delta E_p = (mg) \times (\Delta h)\)

The Final Equation:
\(\Delta E_p = mg\Delta h\)

Key Takeaway: The change in GPE is simply the weight of the object multiplied by how far it moved vertically.


4. Using the Formula to Solve Problems

When you see a projectile motion problem involving height, you will likely need this equation. Here is how to use it correctly:

The Variables:

  • \(\Delta E_p\): Change in GPE (measured in Joules, \(J\)).
  • \(m\): Mass (must be in kilograms, \(kg\)).
  • \(g\): Acceleration of free fall (\(9.81 \text{ m s}^{-2}\)).
  • \(\Delta h\): Change in vertical height (measured in meters, \(m\)).

Common Mistakes to Avoid:

1. Units: Students often forget to convert grams (\(g\)) to kilograms (\(kg\)). Always check your units first!
2. Horizontal Distance: Remember, GPE only cares about vertical height. If a ball moves 10 meters horizontally but stays at the same height, its GPE does not change.
3. The "Delta" (\(\Delta\)): This symbol means "change in." If a ball falls from 5m to 2m, \(\Delta h\) is 3m. The GPE decreases because the height decreased.


5. Connecting GPE to Projectile Motion

In projectile motion, we often look at how energy converts from one form to another. Because we assume there is no air resistance (unless the question says otherwise), the total mechanical energy is conserved.

  • At the launch point: The projectile has maximum Kinetic Energy (KE) but low GPE.
  • At the highest point: The projectile has reached its maximum height (\(h_{max}\)). Here, its GPE is at its maximum, and its vertical kinetic energy is zero.
  • As it falls: GPE is converted back into Kinetic Energy, and the projectile speeds up.

Don't worry if this seems tricky! Just remember: Higher up = More GPE; Lower down = Less GPE.


Quick Review Box

The Equation: \(\Delta E_p = mg\Delta h\)
Uniform Field: Means \(g\) is constant (\(9.81 \text{ m s}^{-2}\)).
Direction: Only vertical displacement matters for GPE.
Conservation: In a vacuum, loss in GPE = gain in KE.


Summary Takeaways

1. Weight is a force exerted on a mass by a gravitational field (\(W=mg\)).
2. GPE is energy stored due to an object's vertical position.
3. You can derive \(\Delta E_p = mg\Delta h\) by recognizing that the work done to lift an object is its weight times the height gain.
4. Always use SI units (kg, m, s) to ensure your energy comes out in Joules (J).
5. In projectile motion, GPE increases as the object rises and decreases as it falls.