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This tutorial will discuss how to model the flight of a golf ball using the Newton Tool. The tutorial will use the Cartesian coordinates x, y, and z. The model that will be used is described in an American Journal of Physics article by MacDonald and Hanzely.

Acceleration Equations

The three acceleration equations in the MacDonald and Hanzely model are:

The z direction is vertical. The y direction is down range from the tee; thus, the ball's initial velocity lies in the y-z plane. The x direction is thus perpendicular to both the y and z directions. A right-hander's slice would thus travel in the positive x direction. The input to the equations consists of B, a constant that gives the conditions of the air, C_D, the coefficient of drag, C_L, the coefficient of lift, a, the spin angle, and the components of the wind velocity.

The spin is taken to lie in the x-z plane; thus, the unit vector parallel to the spin ω is e_ω = cos a e_x + sin a e_z. Thus, for example, if a=0, the ball has only backspin, and if a= π/2 the ball has only spin to the side (remember the angle a is in radians).

The coefficient of drag measures the strength of the air resistance on the flight of the ball. MacDonald and Hanzely suggest a coefficient given by

where v is the speed relative to the ground.

The coefficient of lift measures the strength of the lift on the ball due to its spin. MacDonald and Hanzely suggest a coefficient given by

where again v is the speed relative to the ground. Notice the angle dependence of the lift term in the z acceleration. If a > π/2, the golfer is "topping" the ball, which means there is a negative contribution to the z acceleration.

The wind enters the accelerations through the vector u, which is given by

Since v is the velocity relative to the ground and v_w is the wind velocity, u is the velocity relative to the air.

With the equations above, there is enough information to use the Newton Tool. To simulate the golf shot we need to specify the value for a, B, and the 3 components of the wind velocity. Initially, assume there is no wind. An appropriate value for B is 0.00512. Also, we'll set a=0.1 radians, which means the spin axis is about 0.1 × 180^o / π = 5.72^o off from pure back spin. Since the units used here are feet and seconds, the magnitude of the acceleration due to gravity is 32.16 ft/s². For simplicity, use 32 ft/s². With all of this information, the equations of motion are:

x acceleration

-0.00512*(vx^2+vy^2+vz^2)^(1/2)*((46.0/(vx^2+vy^2+vz^2)^(1/2))*(vx)+(33.4/(vx^2+vy^2+vz^2)^(1/2))*(vy)*sin(0.1))

y acceleration

-0.00512*(vx^2+vy^2+vz^2)^(1/2)*((46.0/(vx^2+vy^2+vz^2)^(1/2))*(vy)-(33.4/(vx^2+vy^2+vz^2)^(1/2))*((vx)*sin(0.1)-(vz)*cos(0.1)))

z acceleration

-32-0.00512*(vx^2+vy^2+vz^2)^(1/2)*((46.0/(vx^2+vy^2+vz^2)^(1/2))*(vz)-(33.4/(vx^2+vy^2+vz^2)^(1/2))*(vy)*cos(0.1))

If you have not already done so, launch the Newton Tool. Choose 3 dimensions. Copy and paste the above equations into the fields on the Acceleration Panel and then click next.

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Initial Conditions

The initial conditions for this problem require a bit of thought too. All three of the initial positions should just be 0.0. If we want to fire the golf ball at an angle of 11 degrees and with an initial velocity of 200ft/s, enter 38.16 into the initial vz, 0.0 into the initial vx, and 196.32 into the initial vy. These numbers were calculated by multiplying the velocity of 200ft/s by the sin and cos functions. 200*sin(11^o) and 200*cos(11^o) for vz and vy, respectively. Choose 5.6 seconds for the Final Time since that will turn out to be the travel time of the ball. Click Calculate.

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Plots

Using the Plot panel, several different meaningful plots can be generated. For example, plotting y position on the y-axis, and z position on the x-axis. Since negative z positions are non-physical, change the minimum y-axis value to 0. This yields the following plot.

Make a plot of z position versus time to confirm that 5.6 seconds is indeed the travel time of the ball before it hits the ground. Now check the motion in the x direction. Choose to plot x position versus Time, change the maximum and minimum values of the plots to 'Default', and click 'View Plot'.

Our initial conditions have led to a 7 yard hook for a righthanded golfer. Try many other combinations to generate other interesting plots.

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Challenges

Using the equations given in the Acceleration section can you add wind to the problem? Can you add an initial x velocity to make the ball land near x=0? Can you vary the coefficients of drag and lift to understand how the dimples on the golf ball help carry it farther?

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