The game of baseball has fascinated fans of the game, and also physicists (someof us are both), for many years. Our goal is to understand how far a real baseball should be expected to travel when hit by a typical power hitter. Model the dffect of backspin on the range of a batted ball. Assume an angular velocity of 2000 rpm by using the Euler method.
The basic equations of motion for a baseball are the same as those of the cannon shell,(1)
and we will again use the Euler method in our simulations.
It is interesting to estimate the distance a baseball would travel in a vacuum, that is, without atmospheric drag.This range can be calculated analytically, or numerically, by using (1) with B_2=0. A good power hitter can give the ball an initial speed of about 110 mph, and if it is hit at an initial angle of starting from a height of 1 m, the range in a vacuum would be 248 m, or approximately 815 ft. So far as we know, no one has ever hit a baseball that far. So we need to consider the effect of drag coeffecient,which is illustrate below.
We need to solve the Euler equations (1) in order to calculate the trajectory of a baseball, including the velocity dependence of the drag force. It turns out that the drag factor for a normal baseball is described reasonably well by the function (2)
with v_d=35m/s and . Using this parameterization of the drag we can construct a program to calculate the trajectory of our batted baseball.
To treat the problem of a thrown or pitched ball, we must deal with two different effects. One effect is the spin of the ball; this will turn out to dominate the motion of a curve ball. The other effect concerns the difference in the drag codfficients for rough and smooth balls, which we encountered in the above part.
For a ball spinning about an axis perpendicular to the direction of travel, this speed will be different on opposite edges of the ball, as illustrated below.
The Magnus force is believed to make an important contribution for objects such as baseballs.
Thus the net spin-dependent force have the general form (3)
The numerical codfficient then takes care of averaging the drag force over the face of the ball, as just discussed; we will determine its value from experimental measurements.For simplicity we will assume that
is a constant in the following. Over the velocity range that is usually of interest to a pitcher, 50-110 mph, we estimate from the data given that
, where m=149 g is the mass of the ball. For a typical curve ball, the magnitude of the Magnus force (3) is about one-third the weight of the ball.
The equations of motion for a sidearm curve ball are then
Here and below we assume that is a constant.
The effect of backspin on the range of a batted ball can be considered according to this. We just need to change the initial angle from 0 to 35.
If we need to anaylze the effect of backspin, we first give trajectory of a batted baseball with the efects of atmospheric drag and without the effect of backspin. The curve is given below:
We pick different initial velocity 50,80,110 mph and assume initial angle is . We can find that the range will increase with veclocity speeding up.
We can see that when speed is 50 mph, the range is about 36 m; when speed is 80 mph, the range is about 73 m; when speed is 110 mph, the range is about 119 m.
After learning the effects of spin on a pitched ball, we can easily imagine that the situation of a batted ball is almost the same, expect that the initial angle will be much larger. So we also assume the initial angle is and change the initial speed of 50,80,110 mph. And we can have a 3D curve below:
And then we can analyze this picture by giving x-y and x-z chart below:
We can see that with the effect of backspin, the batted baseball will greatly move from 0 to 50 m towards the back direction of the Magus force. In contrast to the first circumstance (without the effect of backspin), we can get the range of the batted baseball turns to 39 m (initial speed is 50 mph), 80 m (initial speed is 80 mph), 130 m (inital speed is 110 mph), which are surpassing the former one.
The effect of backspin is that increases the range of the batted baseball greatly. So when sportsmen play baseball, they can use skills to spin the baseball properly in order to increase the range of the batted baseball.
Computational Physics by Nicholas J. Giordano, Hisao Nakanishi