# How high to reach terminal velocity

The forces on a rocket change dramatically during a typical flight. This figure shows the forces and resulting **velocity** of a rocket while **descending** through the atmosphere. The velocity at any time during the flight depends on the corresponding acceleration of the vehicle and the balance of forces acting on the vehicle. Forces, accelerations, and velocities are all vector quantities having both a magnitude and a direction. When describing the motion of an object caused by forces, one must account for both the magnitude and the direction.

During coasting flight, accelerations are produced in response to Newton's first law of motion. The forces present during coasting flight are the drag. and the weight. The weight is constant in magnitude and is always directed toward the center of the earth. The magnitude of the drag changes with the square of the velocity. The direction of the drag is along the rocket axis and opposed to the motion of the rocket. During ascent. the drag is in the same direction as the weight. But during descent, the drag opposes the weight.

For a descending rocket, the net vertical external force **F** is equal to the difference between the drag **D** and the weight **W**.

F = D - W

The magnitude of the drag is given by the drag equation. Drag **D** depends on a drag coefficient **Cd**. the atmospheric density **r**. the square of the air velocity **V**. and some reference area **A** of the object.

D = Cd * r * V ^2 * A / 2

Drag increases with the square of the speed. So as the rocket falls, we quickly reach conditions where the drag becomes equal to the weight, if the weight is small. When drag is equal to weight, there is no net external force on the object and the vertical acceleration goes to zero. With no acceleration, the object falls at a constant velocity as described by Newton's first law of motion. The constant vertical velocity is called the **terminal velocity** .

Using algebra, we can determine the value of the terminal velocity. At terminal velocity:

Solving for the vertical velocity **V**. we obtain the equation

V = sqrt ( (2 * W) / (Cd * r * A)

Typical values of the drag coefficient are given on a separate slide. For a model rocket, the value is near 0.75. Because model rockets do not fly very high, the air density is nearly constant and equal to the sea level value (.00237 slug/cu ft or 1.229 kg/cu m) on the surface of the Earth. For a rocket landing on Mars. the atmospheric density is quite low, and the terminal velocity correspondingly higher. But the weight on Mars is also lower than on Earth so the terminal velocity is not as high as predicted by the atmospheric effect alone.

Here's a Java calculator which will solve the equations presented

on this page:

This page shows an interactive Java calculator which solves the equations for the terminal velocity of a rocket in flight.

As mentioned above, the atmosphere and gravitational constant of a planet affects the terminal velocity. You select the planet using the choice button at the top left. You can perform the calculations in English (Imperial) or metric units. You must specify the weight or mass of your object. You can choose to input either the weight on Earth, or the local weight, or the mass of the object. Then you must specify the cross sectional area and the drag coefficient. Finally you must specify the atmospheric density. We have included models of the atmospheric density variation with altitude in the calculator. When you have the proper test conditions, press the red "Compute" button to calculate the terminal velocity.

You can download your own copy of this calculator for use off line. The program is provided as TermVel.zip. You must save this file on your hard drive and "Extract" the necessary files from TermVel.zip. Click on "Termvcalc.html" to launch your browser and load the program.

**Notice** In this calculator, you have to specify the drag coefficient. The value of the drag coefficient depends on the shape. of the object and on compressibility effects in the flow. For airflow near and faster than the speed of sound. there is a large increase in the drag coefficient because of the formation of shock waves on the object. So be very careful when interpreting results with large terminal velocities. If your drag coefficient includes compressibility effects, then your answer is correct. If your drag coefficient was determined at low speeds, and the terminal velocity is very high, you are getting the wrong answer because your drag coefficient does not include compressibility effects.

The terminal velocity equation tells us that an object with a small cross-sectional area, or a low drag coefficient, or a heavy weight will fall faster than an object with a large area, or high drag coefficient, or a light weight. A rocket with a small parachute will fall faster than with a large parachute because of these effects. Since a rocket will drift with the wind, you can make your rocket return closer to the pad on a windy day by using a smaller parachute.

If we have two objects with the same area and drag coefficient, like two identically sized spheres, the heavier object falls faster. This seems to contradict the findings of Galileo that all free falling objects fall at the same rate. But Galileo's principle only applies in a **vacuum**. where there is **NO** drag. Any two objects fall at the same rate on the Moon because there is a vacuum and no drag and only gravity acting on the objects. In general, the same two objects fall at different speeds on the Earth and on Mars because of aerodynamic drag.

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