Question

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

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.

Key concepts

Air Resistance

Air resistance plays a significant role in the study of projectile motion, particularly when dealing with objects traveling at high speeds, such as a golf ball in flight. This resistance, also known as drag, acts opposite to the direction of an object's velocity, ultimately slowing it down and altering its trajectory.

In the context of the golf ball problem, the drag force is modeled by the equation:

• \( f_{D} = \frac{1}{2} C_{D} \rho \pi r^{2} v^{2} \)

Here, \( C_{D} \) is the drag coefficient, which depends on the shape and surface roughness of the ball. \( \rho \) represents the air density, while \( r \) is the radius, and \( v \) is the ball's velocity.

With air resistance, the range, the maximum height, and the time of flight of a projectile are reduced compared to cases when air resistance is neglected. Solving the differential equations associated with the drag force requires numerical methods.

Reynolds Number

The Reynolds number is a dimensionless quantity that helps predict flow patterns in different fluid flow situations. It is crucial in determining whether the flow behaves in an orderly or turbulent manner. For the golf ball problem, the Reynolds number is computed using the formula:

• \( Re = \frac{vd}{u} \)

where \( v \) is the velocity of the ball, \( d \) is the diameter, and \( u \) is the kinematic viscosity of air.

As the golf ball moves through the air, its velocity changes, resulting in a varying Reynolds number over time. This variation affects the drag coefficient and, consequently, the drag force acting on the golf ball.

Plotting the Reynolds number against time helps visualize these changes and is essential in creating a more accurate model of the projectile's motion.

Differential Equations

Differential equations are fundamental in modeling the behavior of a projectile under the influence of air resistance. These equations take into account how the velocities in both the horizontal and vertical directions change over time due to drag forces.

The basic form of these equations with air resistance can be expressed as:

• In the horizontal direction: \( m \frac{dv_x}{dt} = -f_{D,x} \)
• In the vertical direction: \( m \frac{dv_y}{dt} = -mg - f_{D,y} \)

where \( m \) is the mass of the projectile, \( g \) stands for gravitational acceleration, and \( f_{D,x} \) and \( f_{D,y} \) are the components of the drag force in horizontal and vertical directions.

Analytical solutions for differential equations with variable drag force are typically complex or unavailable, requiring the use of numerical methods such as the Euler or Runge-Kutta methods to approximate the projectile's path. Understanding these equations is essential for predicting how drag affects projectile motion.