Flying Balls of Wood and Metal

Classical Mechanics Level 5

A solid spherical ball of radius $R$ is launched from the origin of the $xy$ plane with speed $v_0$ at an angle $\theta$ with respect to the positive $x$ axis. The ambient gravitational acceleration $g$ is in the negative $y$ direction.

While in flight, the ball experiences an air drag force directed opposite to its velocity. The air drag force magnitude is:

$F_D = \frac{1}{2} \, \rho_{air} \, v^2 C_D A$

In the above equation, $\rho_{air}$ is the density of the air, $v$ is the scalar speed of the ball, $C_D$ is the drag coefficient of the ball, and $A$ is the cross-sectional area of the ball.

The experiment is performed twice: the first time with a ball made out of oak wood, and the second time with a ball made out of aluminum. Let the range of the ball be the $x$ coordinate when the ball lands at $y = 0$ .

How much more range, in meters, does the aluminum ball have than the oak wood ball?

Details and Assumptions:
1) $R = 0.05 \, \text{m}$
2) $v_0 = 30 \, \text{m/s}$
3) $\theta = \pi/4 \, \text{rad}$
4) $g = 10 \, \text{m/s}^2$
5) $\rho_{air} = 1.205 \, \text{kg/m}^3$
6) $C_D = 0.47$ (unitless)
7) $\rho_{oak} = 510 \, \rho_{air}$
8) $\rho_{aluminum} = 2117 \, \rho_{air}$

The answer is 18.65.

1 solution

Karan Chatrath
May 1, 2021

This problem uses a completely numerical approach. I used Octave as my programming tool.

When aerodynamic forces are neglected, one knows that the motion of a projectile is independent of its mass. By accounting for drag force, it is seen that the size of the object and its mass have an effect on its motion.

In this case, I conclude that given two balls of the same size, the one made of Aluminium is more aerodynamic than the one made of wood. The metal ball is about four times heavier than the wooden ball. So if they are projected with the same speed and at the same angle, by virtue of its inertia, the drag force has less of a retarding effect on the metal ball than on the wooden ball. Hence the metal ball travels further.

The simulation code is attached below:

clear all
clc

% Parameters:
R                   = 0.05;
vo                  = 30;
theta               = pi/4;
g                   = 10;
rho_air             = 1.205;
rho_AL              = 2117*rho_air;
rho_oak             = 510*rho_air;
Cd                  = 0.47;

% Time step:
dt                  = 1e-4;

% Motion of the oak and aluminium ball respectively - Two function calls:
[t_oak,r_oak,v_oak] = Motion(R,rho_oak,rho_air,g,Cd,vo,theta,dt);
[t_AL,r_AL,v_AL]    = Motion(R,rho_AL,rho_air,g,Cd,vo,theta,dt);

% Required Answer:
ANSWER     = r_AL(1,end) - r_oak(1,end)
% ANSWER   ~ 18.649

__________________________________________________________________________________________________________

function [t,r,v] = Motion(R,rho,rho_air,g,Cd,vo,theta,dt)

  % This function accepts system and numerical parameters, and initial conditions,
  % and solves for the motion of the system.

  % Initial conditions for position and velocity vectors:
  r(:,1)    = [0;0];
  v(:,1)    = vo*[cos(theta);sin(theta)];

  % Mass and cross section area of the ball:
  M         = (4/3)*pi*R^3*rho;
  A         = pi*R^2;

  % Time and time index initialisation:
  t(1)      = 0;
  k         = 1;

  % Simulation run until the ball reaches the ground level 
  while r(2,k) >= 0

    Fg           = [0;-M*g];                                    % Gravity force

    vx           = v(1,k);                                      % X component velocity
    vy           = v(2,k);                                      % Y component velocity

    Fd           = -0.5*rho_air*A*Cd*sqrt(vx^2 + vy^2)*v(:,k);  % Drag force

    a            = (Fd + Fg)/M;                                 % Newton's 2nd Law

    % Semi implicit Euler:
    v(:,k+1)     = v(:,k) + dt*a;                               % Numerical Integration
    r(:,k+1)     = r(:,k) + dt*v(:,k+1);                        % Numerical Integration
    t(k+1)       = t(k) + dt;                                   % Time advancing
    k            = k + 1;                                       % Index advancing

  end


end

Nice solution! My take is that the aluminium ball travels further since there's more energy put into it to launch it at the same initial velocity and angle than the less heavy wooden ball, and the drag force depends only on the shape of the ball, which is the same.

Gediminas Sadzius - 1 month, 1 week ago

Flying Balls of Wood and Metal

The answer is 18.65.

1 solution

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