Instructions
Welcome to the world of special relativity! You are going to practice calculating the immense energy contained within ordinary objects using Albert Einstein's famous equation: $E=mc^2$.
The Formula Breakdown:
- E stands for Energy (the total energy released if the mass is converted).
- m stands for Mass (how much stuff is in the object).
- c² stands for the Speed of Light, squared. This is a gigantic fixed number that is the core of the equation.
To make the calculations manageable for practice, we will use a simplified constant for $c^2$, which we will call the Relativity Multiplier (RM).
PRACTICE CONSTANT: For all problems below, use RM = 10,000.
Your Task: Calculate the Energy (E) by multiplying the Mass (m) by 10,000.
Section 1: Calculating Potential Energy
Complete the table below by finding the Energy (E) for each mass, using the formula: $E = M \ imes 10,000$. Use a calculator or mental math to solve.
| Object | Mass (M) in Units | Calculation (M x 10,000) | Potential Energy (E) in Units |
|---|---|---|---|
| Example: Small Pebble | 5 | $5 \ imes 10,000$ | 50,000 |
| Pencil | 10 | ||
| Apple | 150 | ||
| Baseball | 145 | ||
| Textbook | 800 | ||
| Water Bottle | 550 |
Section 2: Conceptual Understanding
Einstein’s equation shows that a tiny amount of mass holds a vast amount of energy. Answer the following short questions based on the relationship $E=mc^2$.
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If you double the mass ($m$) of an object, what happens to the resulting potential energy ($E$)?
(Circle one) A. It stays the same. B. It is cut in half. C. It also doubles. D. It triples.
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Imagine two objects. Object A has 25 Units of Mass. Object B has 200 Units of Mass.
Without doing the full calculation, how much greater is Object B's potential energy compared to Object A's potential energy?
Hint: Think about how many times 25 fits into 200.
Answer: Object B has ___________ times the potential energy of Object A.
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Why is the constant $c^2$ (the speed of light squared) so important in the equation? (Choose the best answer.)
A. It makes the mass measurement more accurate. B. It shows that even small masses contain extremely large amounts of energy. C. It only matters for objects traveling at the speed of light.
Section 3: The Power of Scale (Challenge Questions)
In the real world, the actual value of $c^2$ is approximately 90,000,000,000,000,000 (90 quadrillion!).
If you were calculating the energy of a small grain of sand with a mass of only 0.001 kg using the real $c^2$ value:
Real Formula: $E = 0.001 \ imes 90,000,000,000,000,000$
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When you multiply $0.001$ (which is $1/1000$) by a large number, you effectively move the decimal place three spots to the left (or remove three zeros).
What is the total energy (E) released by the grain of sand in Joules?
E = ______________________ Joules
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Real-World Context: A single megawatt-hour (MWh) of electricity—enough to power about 300 homes for an hour—is roughly 3,600,000,000 Joules.
Based on your answer to Question 1, is the energy contained in a single grain of sand enough to power 300 homes for an hour?
Answer and Explanation:
Answer Key
Section 1: Calculating Potential Energy
| Object | Mass (M) in Units | Calculation (M x 10,000) | Potential Energy (E) in Units |
|---|---|---|---|
| Example: Small Pebble | 5 | $5 \ imes 10,000$ | 50,000 |
| Pencil | 10 | $10 \ imes 10,000$ | 100,000 |
| Apple | 150 | $150 \ imes 10,000$ | 1,500,000 |
| Baseball | 145 | $145 \ imes 10,000$ | 1,450,000 |
| Textbook | 800 | $800 \ imes 10,000$ | 8,000,000 |
| Water Bottle | 550 | $550 \ imes 10,000$ | 5,500,000 |
Section 2: Conceptual Understanding
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If you double the mass ($m$) of an object, what happens to the resulting potential energy ($E$)? C. It also doubles. (E and M are directly proportional)
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Without doing the full calculation, how much greater is Object B's potential energy compared to Object A's potential energy? Calculation: 200 / 25 = 8 Answer: Object B has 8 times the potential energy of Object A.
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Why is the constant $c^2$ (the speed of light squared) so important in the equation? B. It shows that even small masses contain extremely large amounts of energy.
Section 3: The Power of Scale (Challenge Questions)
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What is the total energy (E) released by the grain of sand in Joules? $90,000,000,000,000,000 \ imes 0.001 = 90,000,000,000,000$ E = 90,000,000,000,000 Joules (90 trillion Joules)
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Based on your answer to Question 1, is the energy contained in a single grain of sand enough to power 300 homes for an hour? MWh equivalent: 3,600,000,000 Joules Grain of Sand: 90,000,000,000,000 Joules
Answer and Explanation: Yes. The energy in the grain of sand (90 trillion Joules) is much larger than the required energy (3.6 billion Joules). It could power those homes for many hours.