Chapter 2: Problem 26

Consider the flight of a golf ball with mass \(46 \mathrm{~g}\) and a diameter of \(42.7\) \(\mathrm{mm} .\) Assume it is projected at \(30^{\circ}\) with a speed of \(36 \mathrm{~m} / \mathrm{s}\) and no spin. a. Ignoring air resistance, analytically find the path of the ball and determine the range, maximum height, and time of flight for it to land at the height that the ball had started. b. Now consider a drag force \(f_{D}=\frac{1}{2} C_{D} \rho \pi r^{2} v^{2}\), with \(C_{D}=0.42\) and \(\rho=1.21 \mathrm{~kg} / \mathrm{m}^{3}\). Determine the range, maximum height, and time of flight for the ball to land at the height that it had started. c. Plot the Reynolds number as a function of time. [Take the kinematic viscosity of air, \(v=1.47 \times 10^{-5}\).] d. Based on the plot in part \(c\), create a model to incorporate the change in Reynolds number and repeat part b. Compare the results from parts a, b, and d.

Short Answer

Expert verified

In the absence of air resistance, concepts of basic projectile motion help calculate the path, range, maximum height, and flight time. When considering air resistance and changes in the Reynolds number, more complex equations, involving concepts from fluid mechanics, are used to compute these parameters. The results using air resistance altered by the Reynolds number show how drag force significantly affects the trajectory, range, and flight time of the golf ball.

Step by step solution

Solve for the Projectile Motion With No Air Resistance

Firstly, resolve the initial velocity into its horizontal and vertical components using the projection angle: \( v_{x0} = v_{0} \cos(\theta) \), \(v_{y0} = v_{0} \sin(\theta)\) where \( v_{x0} \) and \( v_{y0} \) are the initial horizontal and vertical velocities respectively, \( v_{0} \) is the initial speed and \( \theta \) is the projection angle. \nIn this case, we can substitute \( v_{0} = 36 m/s \) and \( \theta = 30^{\circ} \) to calculate the initial velocities. \nThe equations of motion for horizontal and vertical directions are \( x = v_{x0} t \) and \( y = v_{y0} t - \frac{1}{2} g t^{2} \) respectively, where x and y are the horizontal and vertical displacements, t is the time and g is the acceleration due to gravity (9.81 m/s\(^2\)). Solving these equations will result in the path, range, maximum height and time of flight of the ball.

Calculate the Range With Air Resistance

Now introduce air resistance, which is defined as \(f_{D}=\frac{1}{2} C_{D} \rho \pi r^{2} v^{2}\), where \(C_{D}\) is the drag coefficient, \(\rho\) is the air density, r is the radius of the ball, and v is the speed. The equations of projectile motion with air resistance become more complex, taking the form of differential equations. Solve these for the range, maximum height, and flight time.

Plotting the Reynolds number

The Reynolds number can be computed as \( Re = \frac{vd}{\nu}\) where v is the velocity, d is the diameter of the golf ball, and \(\nu\) is the kinematic viscosity. Plot this number as a function of time.

Incorporate Reynolds Number Into The Model And Comparison

We can use the computed Reynolds number to adjust the drag coefficient in our model and recalculate the range, maximum height, and flight time. This involves updating the drag force equation. Compare these results to those in parts a and b.