NASA Logo - Web Link to Vertical Line

+ Text Only Site
+ Non-Flash Version
+ Contact Glenn

Computer drawing of ballistic flight with the
 equations that describe the motion including drag.

The study of the flight of a baseball and the "bend" of a soccer kick are excellent ways for students to learn the basics of forces and the response of an object to external forces. A ball in flight has no engine to produce thrust, so the resulting flight is similar to the flight of shell from a cannon, or a bullet from a gun. This type of flight is called ballistic flight and assumes that weight is the only force acting on the ball. In reality, a baseball or a soccer ball in flight generates a moderate amount of aerodynamic drag and is not strictly ballistic. On this page we develop the equations which describe the motion of a flying ball including the effects of drag.

At launch the ball is inclined at some angle to the vertical, so we resolve the initial velocity into a vertical and horizontal component. Unlike the ballistic flight equations, the horizontal equation includes the action of aerodynamic drag on the ball. We will first consider the vertical component and then develop the equations for the horizontal component.

In the vertical plane, the only forces acting on the ball are the forces of weight and drag. There is a characteristic velocity which appears in many of the equations that is called the terminal velocity because it is the constant velocity that the object sustains during a coasting descent. Terminal velocity is noted by the symbol Vt.

During the vertical descent, for a light object, the weight and drag of an object are equal and opposite. There is no net force acting on the ball and the vertical acceleration is zero.

a = 0

W = D


W = m * g

where m is the mass of the object and g is the gravitational acceleration equal to 32.2 ft/sec^2 or 9.8 m/sec^2 on the surface of the Earth. (The gravitational acceleration has different values on the Moon and on Mars.) The drag is given by the drag equation:


where r is the gas density, Cd is the drag coefficient which characterizes the effects of shape of the ball, A is the cross-sectional area of the ball, and Vt is the terminal velocity.





Now, turning to the ascent trajectory, the ball is traveling at an initial vertical velocity Vo. With the positive vertical coordinate denoted by y, the net vertical force Fnet acting on the ball is given by:

Fnet = -W -D


Fnet = m a = -W - D



Notice that the acceleration changes with time. Multiply the last term by g/g and use the definition of the terminal velocity to obtain:


The acceleration is the time rate of change of velocity :


Integrating this differential equation:








on the right hand side to obtain:



This is the equation for the velocity at any time during the vertical ascent. At the top of the trajectory, the velocity is zero. We can solve the velocity equation to determine the time when this occurs:




dv/dt = dv/dy * dy/dt

dv/dt = v * dv/dy

We previously determined that




Integrating both sides:







Here's a Java calculator which solves the vertical equations presented on this page:




Button to Download a Copy of the Program


The horizontal equations are a little easier, since the only net force acting on the ball is the drag:

Fnet = m a = - D

m a = - (.5 * Cd * r * A * u^2)

a = - (Cd * r * A * u^2) / (2 * m)

where u is the horizontal velocity. We can use the terminal velocity to simplify this equation:

a = du / dt = - g * u^2 / Vt^2

(1 / u^2) du = - (g / Vt^2) dt

Integrating the equations, with the limits on the velocity from the intial velocity Uo to U, we obtain:

u = dx/dt = Vt^2 * Uo / (Vt^2 + g * Uo * t)

The horizontal velocity is inversely dependent on the time. We can similarly solve for the location x at any time by integrating the velocity equation:

x = (Vt^2 / g) * ln( (Vt^2 + g * Uo * t) / Vt^2 )

You can study the aerodynamic effects on a falling object with the DropSim simulation available at this web site. You can also study the flight characteristics of a ball with drag by using the on-line HitModeler simulation program for a hit baseball, or the SoccerNASA simulation program for a kicked soccer ball.

Have Fun!

Guided Tours


Navigation ..

Button to Display Aerodynamics Index Button to Display Model Rocket Index
NASA Baseball Home Page
NASA Soccer Home Page
Beginner's Guide Home Page


     First Gov Image

+ Inspector General Hotline
+ Equal Employment Opportunity Data Posted Pursuant to the No Fear Act
+ Budgets, Strategic Plans and Accountability Reports
+ Freedom of Information Act
+ The President's Management Agenda
+ NASA Privacy Statement, Disclaimer,
and Accessibility Certification


NASA Logo   
Editor: Nancy Hall
NASA Official: Nancy Hall
Last Updated: May 05 2015

+ Contact Glenn