This simulation plots the trajectory of a non-rotating sphere when launched; the surface roughness determines the sphere's drag profile. First, select the relative surface roughness using the drop-down menu,where \( \large{ \varepsilon } \) is the depth of imperfections and \( \small{ D } \) is the sphere's diameter. Then, choose a launch angle and launch velocity using the sliders:then press . After the sphere has landed, you may press "Launch" again with different settings to compare trajectories on the same plot. Otherwise, press to clear the previous trajectories.The drag coefficient is displayed as a white circle on the "Reynold's Number versus Drag Coefficient" plot. This plot also displays the shape of the drag coefficient curve, which depends on the sphere's roughness.

Three forces act on an object in motion: the gravitational force \( F_{g} \), the drag force \( F_{D} \), and the lift force \( F_{L} \).The gravitational force is \( F_{g} = gm \), where \( g \) is gravitational acceleration and \( m \) is the object's mass. The lift force \( F_{L} \) is zero in this simulation because the spheres are not rotating. A golf ball's geometry creates lift when the ball rotates. The drag force equation is $$ F_{D} = \frac{ C_{D} \rho A_{c} V^{2} }{ 2 } $$ where \( V \) is the velocity and \( A_{c} \) is the cross-sectional area of a sphere with diameter \( D \). The drag coefficient \( C_{D} \) is a function of the angular velocity, the sphere's surface roughness, and the Reynold's number \( Re = \frac{ \rho V D }{ \mu } \) where \( \mu \) is the dynamic viscosity of air. Surface roughness has a large effect on the drag coefficient, and the dimples on golf balls are designed to minimize \( C_{D} \) at low velocities. Values for drag coefficients obtained from Munson et al. [1]

1. Munson, B.R., Okiishi, T.H., Huebsch, W.W., Rothmayer, A.P. (2013). Fundamentals of Fluid Mechanics (7th edition). John Wiley & Sons.