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The procedure for calculating the trajectory of a baseball in air is described.

Trajectory of a Projectile with Air Resistance

A baseball flying through the air is acted upon by only two forces, namely the aerodynamic force and gravity. Because the ball is spherical and probably has little spin, the aerodynamic force is nearly all drag and acts opposite to the direction of velocity. We can write the equation of motion as

drag + gravity = mass × acceleration

This is a vector equation and each vector has a horizontal and a vertical component. Because the drag is a non-linear function of velocity, we will not be able to find an analytic solution and a numerical solution will be necessary. Hence, we have two second order non-linear differential equations. If you want to be a skilled engineer, you need to be able to solve equations of this sort with ease, because most of the real problems of the engineering world do not yield to simple linear analysis.

The usual technique is to replace the two second order differential equations with four first order equations. The four variables are the x and y components of position and the x and y components of velocity. Call them x, y, vx and vy. The four equations are

dx/dt = vx
dy/dt = vy
d(vx)/dt = (1/mass) × (dragX + gravityX)
d(vy)/dt = (1/mass) × (dragY + gravityY)

and, of course, gravityX is zero.

Aerodynamic Drag of a Sphere

Everything is quite simple except for the calculation of the drag force. The traditional way to express the drag of an object such as a sphere is through a drag coefficient defined as the drag force divided by the product of the frontal area and the dynamic pressure. The dynamic pressure is defined as one-half the density times the square of the velocity. This gives a model of the varying atmospheric force on the sphere as the velocity and altitude change.

The drag coefficient of a sphere travelling at speeds typical of baseballs is about 0.5 but I have included a procedure taken from Chow, An Introduction to Computational Fluid Mechanics that will let you use this program to study other spheres such as BBs or ping-pong balls or cannon balls or even a 20-ton Soyuz entry capsule and observe their behavoir as they fly through the atmosphere. I recommend Chow's book for some examples for further study.

To use this routine, you need to know the Reynolds number of the flow which requires the density of the air as well as the viscosity of air. You can see the routines in the program that compute these quantities and eventually return the drag coefficient to the procedure that computes the acceleration. You have to know velocity, elevation, air density, drag coefficient, gravity and it all comes together in the calculation of acceleration.

Solution of the Differential Equations

Each of the four variables is a function of time. To perform the numerical integration, we take time steps of finite size and advance the solution. The best way to do this is to use a variable step-size differential equation solver that computes the step size as it develops the solution. However, for this simple problem, I felt it would be confusing to introduce variable step size. So the calculations are done with a simple fixed step size algorithm. Long fly balls stay in the air about five or six seconds, so a step size of 0.1 seconds gives something like 50 or 60 finite steps for the numerical equation solver. The most straightforward way to proceed is to use a fourth-order Runge-Kutta algorithm.

Correction of the final point

At each step along the trajectory, we check to see that the altitude is greater than the initial altitude. Eventually, of course, we will arrive at a point where this is no longer true. This is our clue to terminate the calculation. The fixed step size in time will most likely step over the point where the altitude is exactly equal to the initial altitude and give a point with negative altitude as the final point. To make the solution look good, it is best to do an interpolation in the final interval and land right on zero altitude.

Source code for the computer program

Finally, you can look at the source code of the computer program. This program is written without input data. To run different cases, you update the source code and recompile and rerun. You will need a Fortran compiler to run your revised source code.

Assumption of Zero Spin

The analysis above assumes that there is negligible spin on the ball and the total aerodynamic force is exactly opposite in direction to the velocity. This is not the case for a thrown ball. The craft of pitching is largely a study in putting the proper amount of spin on the ball. Calculating the path of a pitched ball is more complex. Actually, a batted ball can have quite a bit of spin but this is always associated with a glancing hit and the initial velocity will be low. The only way to get a long ball is to hit it squarely.

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