📚 Study Notes: Gravitational Potential Energy and Kinetic Energy (9702 Syllabus 5.2)

Welcome, future physicists! This chapter is all about understanding the two most fundamental forms of mechanical energy: the energy of movement (Kinetic Energy) and the energy of position (Gravitational Potential Energy).


Understanding these concepts is crucial because they allow us to use the powerful principle of the Conservation of Energy to solve complex motion problems without needing detailed acceleration calculations. Think of it as a shortcut!

1. Kinetic Energy (E\(_k\)): The Energy of Motion

1.1 Defining Kinetic Energy

Kinetic Energy (E\(_k\)) is the energy an object possesses due to its motion. Anything that is moving has kinetic energy. The faster an object moves, and the more massive it is, the more kinetic energy it has.


  • Key Terms:
    • Unit: The unit of energy is the Joule (J).
    • Prerequisite: Remember that energy is a scalar quantity (it has magnitude but no direction).

1.2 The Kinetic Energy Formula

The formula you need to recall and use for kinetic energy is:

$$E_k = \frac{1}{2}mv^2$$

Where:

  • \(E_k\) is the Kinetic Energy (in J)
  • \(m\) is the mass of the object (in kg)
  • \(v\) is the speed (or magnitude of velocity) of the object (in m s\({^{-1}}\))

Analogy: Notice the square on velocity (\(v^2\)). This tells us that speed is much more important than mass in determining the energy of a moving object. If you double the mass, you double the energy. If you double the speed, you quadruple the energy! (Imagine the difference between a slow car and a fast car hitting a wall.)


⚠ Common Mistake: Always ensure the mass \(m\) is in kilograms (kg) and speed \(v\) is in metres per second (m s\({^{-1}}\)) before using this formula. If units are not standard, your answer will not be in Joules!

1.3 Derivation of \(E_k = \frac{1}{2}mv^2\) (Required by Syllabus)

This derivation shows how the energy gained by an object is equal to the work done on it. We use the definition of work done and one of the equations of motion for uniform acceleration.

Step 1: Start with Work Done and Newton's Second Law

Work Done (\(W\)) = Force (\(F\)) \(\times\) distance (\(s\)).
Using Newton's Second Law (\(F = ma\)), we substitute \(F\):

$$W = (ma)s$$


Step 2: Use an Equation of Motion (SUVAT)

We assume the object starts from rest (\(u = 0\)) and accelerates over distance \(s\) to a final speed \(v\). We use:

$$v^2 = u^2 + 2as$$

Since \(u = 0\):

$$v^2 = 2as$$


Step 3: Rearrange SUVAT to find \(as\)

We rearrange the equation above to isolate the term \(as\):

$$as = \frac{1}{2}v^2$$


Step 4: Substitute back into the Work Done Equation

Now substitute \(as = \frac{1}{2}v^2\) back into the \(W = m(as)\) equation from Step 1:

$$W = m\left(\frac{1}{2}v^2\right)$$

Since the work done on the object is equal to the kinetic energy gained:

$$E_k = \frac{1}{2}mv^2$$

➤ Quick Review: Kinetic Energy
  • Formula: \(E_k = \frac{1}{2}mv^2\)
  • Dependent on: Mass (linear) and Speed (square)
  • Concept: Energy of movement

2. Gravitational Potential Energy (GPE)

2.1 GPE in a Uniform Gravitational Field

Gravitational Potential Energy (\(\Delta E_p\)) is the energy stored in an object because of its position within a gravitational field. When you lift a book, you do work against gravity, and that work is stored as GPE.


⚠ Crucial Syllabus Point: For AS Level, we only deal with GPE changes near the Earth's surface where the gravitational field is considered uniform (i.e., gravitational acceleration \(g\) is constant).

2.2 The GPE Change Formula

When an object is moved through a height difference \(\Delta h\) in a uniform field, the change in GPE is:

$$\Delta E_p = mg\Delta h$$

Where:

  • \(\Delta E_p\) is the change in GPE (in J)
  • \(m\) is the mass of the object (in kg)
  • \(g\) is the acceleration of free fall (gravitational field strength, typically \(9.81\text{ m s}^{-2}\))
  • \(\Delta h\) is the change in vertical height (in m)

Did you know? Since \(W = mg\) (Weight), the formula simply means Change in GPE = Weight \(\times\) Change in height.

2.3 Derivation of \(\Delta E_p = mg\Delta h\) (Required by Syllabus)

The change in GPE is defined as the work done against the gravitational force (weight) to lift an object.


Step 1: Define Work Done

Work done, \(W\), is the force multiplied by the distance moved in the direction of the force.

$$W = Fs$$


Step 2: Identify Force and Displacement

When lifting an object of mass \(m\):

  • The minimum upward Force (\(F\)) needed is equal to the object's Weight, \(W\): \(F = mg\).
  • The Displacement (\(s\)) is the vertical height change, \(\Delta h\).

Step 3: Substitute to find Change in GPE

Since the work done against gravity is stored as GPE:

$$\text{Work Done} = \text{Force} \times \text{Distance}$$ $$\Delta E_p = (mg) \times (\Delta h)$$ $$\Delta E_p = mg\Delta h$$

Remember: This formula gives the change in potential energy. Absolute GPE depends on choosing a zero reference point (like the ground). For calculation purposes, we only care about the height difference, \(\Delta h\).

➤ Quick Review: Gravitational Potential Energy
  • Formula: \(\Delta E_p = mg\Delta h\)
  • Context: Only valid in a uniform field (near Earth's surface)
  • Concept: Energy of position or height

3. Applying Energy Conservation

In many physics problems, an object gains kinetic energy by losing potential energy, or vice versa. This is where the Principle of Conservation of Energy (from Syllabus 5.1) becomes your best friend.

3.1 The Energy Exchange

In an ideal system (where we assume negligible air resistance or friction), the total mechanical energy remains constant. Any loss of GPE results in an equal gain in \(E_k\), and vice versa.


$$E_{\text{Total}} = E_k + E_p = \text{Constant}$$

Therefore, for a falling or rolling object (ignoring air resistance):

$$\text{Loss in GPE} = \text{Gain in } E_k$$ $$\Delta E_p = \Delta E_k$$

Or, using the formulas:

$$mg\Delta h = \frac{1}{2}m(\Delta v^2)$$


Example: The Rollercoaster

At the top of the hill (maximum height, \(h\)), the car has maximum GPE and (usually) minimum \(E_k\). As it rushes down the hill, GPE is converted directly into \(E_k\), making it go faster. At the bottom (minimum \(h\)), it has maximum \(E_k\) and minimum GPE.

3.2 Accounting for Energy Losses (Friction)

If resistive forces (like air resistance or friction) are present, they do work which converts mechanical energy into thermal (heat) energy. The total energy is still conserved, but the total mechanical energy is not.

$$\text{Initial } E_{\text{Total}} = \text{Final } E_{\text{Total}}$$

$$\text{Initial } E_k + \text{Initial } E_p = \text{Final } E_k + \text{Final } E_p + \text{Work Done Against Friction}$$

Use the following relationship to solve problems involving losses:

$$\text{Loss in GPE} = \text{Gain in } E_k + \text{Energy converted to heat (Work Done by Resistance)}$$

3.3 Common Errors to Avoid

1. Mass Cancellation: When equating \(mg\Delta h = \frac{1}{2}mv^2\), the mass \(m\) cancels out! This is a great shortcut, as it means the speed reached by a falling object (in a vacuum) is independent of its mass.

2. The Root of the Problem: When calculating speed from \(E_k\), students often forget the final step: taking the square root. Remember that \(v^2\) is calculated first!

3. Direction vs. Magnitude: \(E_k\) uses speed (\(v\), a scalar). If a question asks for velocity (a vector), you might need to combine your speed calculation with knowledge of the direction of motion.

3.4 Step-by-Step Problem Solving Strategy

When tackling problems involving GPE and \(E_k\), follow these steps:

Step 1: Choose Reference Points
Define the initial and final points of the motion. Set the zero reference height (\(h=0\))—usually the lowest point the object reaches.

Step 2: List Initial Energies
Calculate \(E_{k\text{ initial}}\) and \(E_{p\text{ initial}}\) at the starting point.

Step 3: List Final Energies
Calculate \(E_{k\text{ final}}\) and \(E_{p\text{ final}}\) at the ending point.

Step 4: Apply Conservation

If there are no losses (ideal case):
$$\text{Total Initial Energy} = \text{Total Final Energy}$$
$$(E_{k} + E_{p})_{\text{initial}} = (E_{k} + E_{p})_{\text{final}}$$

If work is done against resistance (\(W_f\)):
$$(E_{k} + E_{p})_{\text{initial}} = (E_{k} + E_{p})_{\text{final}} + W_f$$

Step 5: Solve for the Unknown
Substitute your known values and solve the resulting algebraic equation.

➤ Key Takeaway: Synthesis

Kinetic energy and gravitational potential energy are interchangeable forms of mechanical energy. For AS Level, the GPE formula \(\Delta E_p = mg\Delta h\) is only valid when dealing with small height changes near a planetary surface where \(g\) is assumed constant.