Curve Balls and Vectors Does a baseball pitch really curve or is it some sort of
ID: 2863937 • Letter: C
Question
Curve Balls and Vectors Does a baseball pitch really curve or is it some sort of optical illusion? Assume the pitcher's mound is at the point (60, 0, 0) and that home plate is at the origin of our coordinate system. Suppose the pitcher throws the ball toward home plate and gives it a spin of s revolutions per second about a vertical axis through the center of the ball. This spin is described by the spin vector s where s points along the axis of revolution with length s.
From aerodynamics we learn that this spin causes a difference in air pressure on the sides of the ball and results in a spin acceleration on the ball given by , where c is some empirical constant and is the velocity vector. The total acceleration of the ball is then , where g is the usual acceleration due to gravity.
If the ball is thrown so that the spin vector is pointing upward, we find that
For a ball pitched along the x-axis, v1 is much larger than v2, and the approximation is sufficiently accurate for our purposes. The acceleration vector of the ball may then be taken as .
Now suppose that the pitcher throws the ball from the initial position of (60, 0, 6) in feet with an initial velocity vector of <-120, 3, 4> in ft/sec and with a spin of s = 40 rps. A reasonable value of c is c = 0.005 ft/sec2 per ft/sec of velocity and rev/sec of spin, though a precise value depends on whether the pitcher has scuffed the ball or contaminated it with some foreign substance (aka a spitball). ?
1. Determine the vector function that represents the position of the baseball from the pitcher to the plate.
2. Complete the table below finding the location of the baseball for increments of 1/8 second. t (in sec.) x y z 0 1/8 2/8 3/8 4/8
3. Find a different initial velocity (assuming the same s and c as above) that would cause the ball to cross the plate 9'' left of center and 24'' above the ground. Make the initial velocity have an x-component of -120. Find the vector function that represents the position of the baseball. Also, determine when the pitch gets to the plate.
4. Find another vector functions for the position of the baseball, one with a spin of 30 rps, assuming the same c, and with the initial velocity vector <-120, 3, 4>
Curve Balls and Vectors Does a baseball pitch really curve or is it some sort of optical illusion? Assume the pitcher's mound is at the point (60, 0,0) and that home plate is at the origin of our coordinate system. Suppose the pitcher throws the ball toward home plate and gives it a spin ofs revolutions per second about a vertical axis through the center of the ball. This spin is described by the spin vector s where s points along the axis of revolution with length s From aerodynamics welearm that this spin causes a difference in air pressure on the sides of the ball and results in a spin acceleration on the ball given by where c is some empirical constant and v = (v-v, v) is the velocity vector. The total acceleration of the ball is then where g is the usual acceleration due to gravity If the ball is thrown so that the spin vector s sk is pointing upward, we find that For a ball pitched along the x-axis. v1 is much larger than V2, and the approximation s × v ~ sv/ is sufficiently accurate for our purposes. The acceleration vector of the ball may then be taken as Now suppose that the pitcher throws the ball from the initial position of (60, 0, 6) in feet with an initial velocity vector of in ft/sec and with a spin ofs-40 rps. A reasonable value of cis c = 0.005 ft/sec2 per ft/sec of velocity and rev/sec of spin, though a precise value depends on whether the pitcher has scuffed the ball or contaminated it with some foreign substance (aka a spitball)Explanation / Answer
Find the vector function for the position of the baseball, one with a spin of 30rps, assuming the same C2 and with the initial velocity vector <-120,3,4>
y = f(x) that we studied in the first part of this book is of course that the “output” values are now three-dimensional vectors instead of simply numbers. It is natural to wonder if there is a corresponding notion of derivative for vector functions. In the simpler case of a function y = s(t), in which t represents time and s(t) is position on a line, we have seen that the derivative s (t) represents velocity; we might hope that in a similar way the derivative of a vector function would tell us something about the velocity of an object moving in three dimensions. One way to approach the question of the derivative for vector functions is to write down an expression that is analogous to the derivative we already understand, and see if we can make sense of it. This gives us r (t) = lim t0 r(t + t) r(t) t = lim t0 hf(t + t) f(t), g(t + t) g(t), h(t + t) h(t)i t = lim t0 h f(t + t) f(t) t , g(t + t) g(t) t , h(t + t) h(t) t i = hf (t), g (t), h (t)i, if we say that what we mean by the limit of a vector is the vector of the individual coordinate limits. So starting with a familiar expression for what appears to be a derivative, we find that we can make good computational sense out of it—but what does it actually mean? We know how to interpret r(t + t) and r(t)—they are vectors that point to locations in space; if t is time, we can think of these points as positions of a moving object at times that are t apart. We also know what r = r(t + t) r(t) means—it is a vector that points from the head of r(t) to the head of r(t + t), assuming both have their tails at the origin. So when t is small, r is a tiny vector pointing from one point on the path of the object to a nearby point. As t gets close to 0, this vector points in a direction that is closer and closer to the direction in which the object is moving; geometrically, it approaches a vector tangent to the path of the object at a particular point. . ...................................................................................................................................................................... ..... ... ............. ....................................................................................................................................... ....... .... r(t) r r(t + t)
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