Wednesday, June 22, 2022
Statcast Lab: Angular v Linear Speed
Suppose we have this bat moving from one point to the other (90 degrees apart, overhead view) at a constant speed around the knob of the bat. The travel time is 0.03 seconds (30 milliseconds). What is the speed of the bat?
The key question is "which point"?
(click image to embiggen)
If you think in terms of "angles", then the speed of the bat is the same regardless of which point of the bat you select. This is called angular speed. In this example, the bat moves 90 degrees in 0.03 seconds, or 3000 degrees per second. I agree with you: I have (or had anyway) no idea what that represents. Is that very fast or very slow or normal?
Mathematicians prefer to use radians, not degrees. Since a circle is 360 degrees, or 2PI radians, that means we have 57.296 degrees per radian. 3000 degrees per second is 52 radians per second. Since a radian is unitless, we can say 52/s is the angular speed. And this 52/s speed is true whether you select the red point or the blue point or any other point on the bat. The entire bat will go from 0 degrees to 90 degrees in 0.03 seconds at the same (angular) speed: That's either 3000 degrees per second or 52 radians per second or 52/s.
We are of course more used to linear speed, which is the displacement of the distance divided by the elapsed time. We've already established that the time is constant for any part of the bat, which in our illustration is 0.03 seconds. Now, what about the distance? It's obvious in the image that the blue point of the bat will need to travel much less distance than the red point on the bat.
The blue point is 17 inches from the rotational point (in this illustration, the bat is rotating around the knob), while the red point is 34 inches. The distance it travels can be determined from the circumference of a circle: 2PI times radius. Since this bat travels 90 degrees, or one-fourth of a circle, then the distance it travels is 2PI times radius / 4, which works out to radius * 1.57. The blue point therefore travels 17 x 1.57 = 26.7 inches. And the red point, with twice the radius, will cover twice the distance or 53.4 inches.
The (linear) speed is distance / time. Therefore, the blue point travels at a speed of 26.7/0.03 = 890 inches / second, or 74 feet / second, or 50.5 mph. Naturally, the red point being twice as far away, travelling twice as much distance (over the same amount of time) will be moving at twice the speed of the red point: 101 mph.
Now, we don't really care about the middle of the bat, or the head of the bat. What we care about is when we have a perfectly swung bat is the speed at the ideal collision point, which batters would term "the sweetspot". The sweetspot is about 6-7 inches from the head (top) of a 33.5-34 inch bat. Or about 27 inches from the knob (bottom) of the bat. For convenience, I'll term the location that is 27 inches from the rotational point as R27.
Since the distance travelled for the bat is proportional to its distance from the knob, we can measure the (linear) speed of a bat at any point on the bat. And so, a point that is 27 inches from the knob will travel at a speed of 27/34 of the speed of the head of the bat. In this illustration, where the head is travelling at 101mph, the sweetspot is traveling at 27/34 x 101, or 80mph.
In other words, an angular speed of 52/s is the same thing as a linear speed at R27 of 80mph. Here is how the speed of the bat looks like, along every point along the bat. So in order to know the "bat speed", you need to pick a point along the bat. The Sweetspot point works. Except when it doesn't. When is that? In this illustration, I've noted that the knob is the rotational point, which makes the calculations fairly straightforward. Next time, we'll talk about the speed of the bat when the bat is rotating at a point other than the knob. And that's where the complications kick in. That's when using the Sweetspot point might not work so well.
(click image to embiggen)
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