Barn-Pole Paradox Calculator
Explore the famous relativity paradox demonstrating length contraction and the relativity of simultaneity
The length of the pole in its rest frame
The length of the barn in its rest frame
Velocity of the pole relative to the barn (must be less than 299,792,458 m/s)
How to Use This Calculator
Enter the Rest Length of the Pole
Input the length of the pole as measured when it's at rest (in meters). This is typically longer than the barn to create the paradox.
Enter the Length of the Barn
Input the length of the barn in its rest frame (in meters). For the paradox, this should be shorter than the pole's rest length.
Enter the Velocity
Input the velocity of the pole relative to the barn in meters per second. Must be less than the speed of light (299,792,458 m/s).
Calculate and Interpret Results
Click "Calculate Paradox" to see how length contraction affects the measurements from both reference frames. The results will show whether the pole fits from each perspective.
Formula
Length Contraction: L = L₀ / γ
Lorentz Factor: γ = 1 / √(1 - v²/c²)
where L₀ is the rest length, v is velocity, and c is the speed of light
Example 1: Moderate Speed
Given: Pole rest length = 10 m, Barn length = 8 m, Velocity = 200,000,000 m/s (66.7% c)
γ = 1 / √(1 - 0.667²) = 1.342
Contracted pole length (from barn's frame) = 10 / 1.342 = 7.45 m
From barn's frame: 7.45 m < 8 m, so pole fits!
Contracted barn length (from pole's frame) = 8 / 1.342 = 5.96 m
From pole's frame: 10 m > 5.96 m, so pole does not fit!
Example 2: High Speed
Given: Pole rest length = 20 m, Barn length = 10 m, Velocity = 260,000,000 m/s (86.7% c)
γ = 1 / √(1 - 0.867²) = 2.028
Contracted pole length = 20 / 2.028 = 9.86 m
From barn's frame: 9.86 m < 10 m, so pole fits!
Contracted barn length = 10 / 2.028 = 4.93 m
From pole's frame: 20 m > 4.93 m, so pole does not fit!
About the Barn-Pole Paradox
The Barn-Pole Paradox is a classic thought experiment in special relativity that demonstrates the counterintuitive nature of length contraction and the relativity of simultaneity. It shows how observers in different inertial frames can reach different conclusions about whether a pole fits inside a barn.
The Paradox Scenario
Imagine a pole with rest length L₀ that is longer than a barn with length L. When the pole moves at relativistic speeds toward the barn, length contraction causes it to appear shorter from the barn's reference frame. From the barn's perspective, the pole might fit entirely inside. However, from the pole's reference frame, it's the barn that appears contracted, making it seem impossible for the pole to fit.
When to Use This Calculator
- Physics Education: Teaching students about special relativity and length contraction
- Understanding Relativity: Visualizing how measurements depend on the observer's frame
- Paradox Resolution: Exploring how simultaneity resolves apparent contradictions
- Relativistic Calculations: Calculating Lorentz factors and contracted lengths
- Research: Analyzing relativistic scenarios in physics research
Why Use Our Calculator?
- ✅ Instant Results: Calculate length contraction and Lorentz factors immediately
- ✅ Dual Perspectives: See measurements from both reference frames simultaneously
- ✅ Educational: Clear explanations of the physics behind the paradox
- ✅ 100% Free: No registration or payment required
- ✅ Accurate: Uses precise relativistic formulas
- ✅ Interactive: Experiment with different velocities and lengths
The Resolution
The paradox is resolved by understanding that simultaneity is relative. Events that occur simultaneously in one reference frame (like both barn doors closing at the same time) do not occur simultaneously in another frame. From the barn's frame, the doors can close simultaneously when the pole is inside. From the pole's frame, the doors close at different times, allowing the pole to pass through without contradiction.
Common Applications
Teaching Special Relativity: The Barn-Pole Paradox is a standard example in physics courses to illustrate how length contraction works and why it doesn't lead to logical contradictions when properly understood.
Understanding Relativistic Effects: This calculator helps visualize how objects appear shorter when moving at relativistic speeds, which is crucial for understanding particle physics, astrophysics, and GPS systems.
Research Applications: Scientists use similar calculations when studying relativistic particles, cosmic rays, and high-energy physics experiments.
Tips for Best Results
- Use velocities close to but less than the speed of light to see significant effects
- Make the pole's rest length longer than the barn to create the paradox
- Try different velocities to see how the Lorentz factor changes
- Compare results from both frames to understand the relativity of measurements
- Remember that velocities must always be less than 299,792,458 m/s
Frequently Asked Questions
What is the Barn-Pole Paradox?
The Barn-Pole Paradox is a thought experiment in special relativity that demonstrates how a pole can appear to fit inside a barn from one reference frame while appearing too long from another frame. It illustrates length contraction and the relativity of simultaneity.
How is the paradox resolved?
The paradox is resolved by recognizing that simultaneity is relative. Events that are simultaneous in one frame (like both barn doors closing) are not simultaneous in another frame. This allows both perspectives to be correct without contradiction.
What is length contraction?
Length contraction is a relativistic effect where objects appear shorter in the direction of motion when observed from a frame in which they are moving. The formula is L = L₀/γ, where L₀ is the rest length and γ is the Lorentz factor.
Why does the pole fit from one frame but not another?
From the barn's frame, the pole is contracted and appears shorter, so it can fit. From the pole's frame, the barn is contracted and appears shorter, so the pole cannot fit. Both perspectives are correct because simultaneity is relative—the timing of when the doors close is different in each frame.
What is the Lorentz factor?
The Lorentz factor (γ) is a quantity that appears in special relativity and increases as velocity approaches the speed of light. It's calculated as γ = 1/√(1 - v²/c²) and determines how much time dilation and length contraction occur.
Can I use this for velocities at or above the speed of light?
No. According to special relativity, nothing with mass can travel at or above the speed of light. The calculator will reject velocities ≥ c because they would result in undefined or imaginary values for the Lorentz factor.