Why a longer arm makes a catapult launch faster
In short: Imagine two catapults spinning at the exact same rate. If one has a longer arm than the other, which projectile travels faster? The answer lies in the fundamental relationship between circular motion and linear distance, a principle that dictates everything from orbital mechanics to playground swings.
The relationship between linear velocity (v) and angular velocity (ω) is defined by the radius (r) of the circular path. Specifically, the formula is v = r × ω. This means that even if two objects rotate at the same angular speed—meaning they complete one full turn in the same amount of time—the object further from the center will always possess a higher linear velocity.
To visualize this, consider a stone tied to a string rotating at a constant speed. If the stone moves at 3 m s-1 on a string that is 0.75 m long, its angular velocity is calculated as 4 rad s-1. Because the stone is at a specific distance from the center, its speed is directed along a tangent to the circle at every point. While the magnitude of that speed remains constant in a uniform rotation, its direction is in a state of constant change.
This distinction is crucial in physics: angular velocity measures the rate of change of the angle (measured in radians per second), while linear velocity measures the actual distance covered over time (measured in meters per second). In practical applications, such as the pumpkin catapult example, increasing the arm length directly scales the output speed without requiring the motor or mechanism to spin any faster. It is a simple multiplication of geometry and motion.
Understanding these units is also a matter of choosing the right tool for the job. While degrees are common in everyday speech, radians are considered a more natural unit for physics. One radian is defined as the angle subtended at the center of a circle by an arc length equal to the radius. By using radians, the relationship between linear and angular motion becomes a direct, elegant product.
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Sources
- Summary by Intellect · source: Linear velocity comparison from radius and angular velocity: Worked example | Khan Academy
- Also: Episode 225: Quantitative circular motion
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