To represent the distance a major-league outfielder can throw a baseball in terms of speed, you can use the equation of motion for a projectile (assuming the throw occurs at an angle to the horizontal). However, if you just want a basic linear relationship between speed and distance, we can simplify it.
Since you've given the speed in both miles per hour (mph) and feet per second (ft/s), let's convert both speeds for consistency:
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Speed in mph: \[ 105 \text{ mph} = 105 \times \frac{5280 \text{ feet}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3600 \text{ seconds}} \approx 154 \text{ ft/s} \] (Note : Using 105 mph directly, you can see the given speed of 150 ft/s is approximately within the range.)
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If \( v \) is the speed in ft/s and \( d \) is the distance thrown, a simple linear relationship can be described as: \[ d = k \cdot v \] where \( k \) is a constant that depends on factors such as the angle of the throw, height, initial velocity, and so forth.
Given your example of a maximum throw speed of 105 mph (or approximately 154 ft/s), the equation becomes: \[ d = k \cdot v \]
You could even state: \[ d \leq 150 \text{ ft/s} \cdot k \] where \( k \) would need to be estimated or determined based on the exact mechanics of the throw.
If there is a specific height or angle of release you're considering, we could use equations of projectile motion to get a more precise formula.
Let me know if you need further help or clarification!
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