ME 3456: Dynamics Fall 2024
Laboratory Experiment #1:
Projectile Trajectories with Aerodynamic Drag
Abstract
In introductory physics courses, you learned the mathematical theory behind 1- dimensional and 2-dimensional projectile motion under ideal conditions. However, in real world applications, conditions are rarely ideal, and much more complex measurements and calculations are required to accurately predict projectile motion. In this experiment, you will learn numerical methods for predicting and calculating motion behavior in the presence of air resistance. To do this, measurements of the motion of a foam ball are taken and compared with different mathematical models with and without air resistance. This allows students to experimentally determine drag force.
Learning Objectives
• Dynamics of Ideal Motion
• Theory of Aerodynamic Drag force
• Numerical Evaluation of Data
• Comparison of Analytical, Numerical, and Experimental Data
I. Problem Statement
The purpose of this laboratory is to study the effect of air resistance on the motion of a small foam ball. We will use air resistance information extracted from a vertical fall of the ball to predict the trajectory of the ball when undergoing projectile motion. We will also quantify the air resistance effect by comparing with the ideal motion predicted when there is no resistance. The findings will also be used to predict the flight of a baseball in Fenway Park with and without the effect of air resistance.
We start by looking at the pure vertical fall case. The motion of the ball can be predicted analytically by considering that the air resistance introduces a negative acceleration component (upwards, opposed to the direction of motion) that is proportional to the square of the falling speed.
II. Introduction
a. Aerodynamic Drag Force:
In introductory physics, equations of motion for projectile motion are derived assuming that aerodynamic drag is negligible. In reality, there will be a drag force acting on the ball, in a direction exactly opposite to its velocity vector. This force can be estimated via the following results from fluid mechanics.
When a body moves “sufficiently quickly” through a fluid, the drag is not caused by the fluid viscosity, but rather by its mass, i.e., a force is needed to accelerate the fluid "intersected" by the path of the ball. This force is proportional to the square of the speed. In that case, the drag force (FD) may be approximated by:
(1)
where ρf is the air density, is the area of the projection of the ball on a plane perpendicular to the direction of motion, v is the velocity vector of the ball (with respect to the air), and CD is the drag coefficient, an experimentally determined quantity. The negative sign in Eq. (1) signifies that the drag force acts in the direction opposite to that of the velocity vector. For a smooth sphere, CD ≈ 0.47. However, a foam ball does not have a smooth surface, and roughness (like the dimples on a golf ball) can change the drag coefficient.
For a spherical ball of diameter Dd, If the foam ball has a density ρb, its mass So the drag force from the air will introduce an acceleration given by:
(2)
b. Terminal Velocity
Terminal velocity refers to the maximum velocity that an object in free-fall can reach when falling in the presence of aerodynamic drag. It occurs when the drag force acting opposite the motion becomes equal to the gravitational force (weight) acting in the direction of the motion, as shown in Figure 1 for the case of a vertically falling ball.
Figure 1: Free body diagram of an object in free fall.
Because the motion is rectilinear, the downward acceleration of the ball is given by
(3)
Where is an acceleration-based drag coefficient. Notice that the value of D decreases (i.e., the drag effect decreases) when the density of the ball material and/or the ball diameter increase.
We can use (3) to solve for the ball velocity as a function of time. We have learned in class that for rectilinear motion, when the acceleration is given as a function of velocity, we can use the following approach:
Assuming the ball starts from rest, we have:
(4)
It can be shown that the analytical solution of this integral is:
(5)
Notice that for large t, the value of v approaches When v = vf the acceleration becomes zero (0), and the velocity remains constant. vf is called the terminal velocity of the falling object. A typical plot of v vs. time starting from rest is shown in Figure 2.
Figure 2: Plot of velocity vs. time for an object in free-fall in the presence of aerodynamic drag.
The vertical position of the ball (measured downwards from the initial resting position), can be obtained by integrating (5). The solution is:
(6)
c. General trajectory in the presence of aerodynamic drag:
When the motion is not purely vertical, that is, when the ball starts with an initial velocity component on the horizontal direction, it is not possible to analytically solve for the ball velocity from equation (2). In this case numerical integration techniques, such as using finite differences, need to be used to approximately compute the horizontal and vertical components of the velocity as well as to generate the ball trajectory.
The document “Projectile Motion Including Drag Effect” included in the Canvas/Modules section for Lab 1 explains how the finite-differences method can be used to compute the velocity and the position of the ball as time advances. The modules page also includes a MATLAB script that uses finite differences to do just that. You will be using that script. as you work on the report for this Lab.
The main objective of this lab is to extract the drag coefficient, CD from the motion of a foam ball during free vertical fall, and use it to predict the motion expected when balls of the same material are launched with different initial speeds and orientations. Additionally, by comparing with the expected motion in the absence of drag, you will be able to quantify the drag effect.