Question

A typical Major League fastball is thrown at approximately \(88 \mathrm{mph}\) and with a spin rate of \(110 \mathrm{rpm} .\) If the distance between the pitcher's point of release and the catcher's glove is exactly \(60.5 \mathrm{ft},\) how many full turns does the ball make between release and catch? Neglect any effect of gravity or air resistance on the ball's flight.

Short answer

Answer: 0 full turns

Step-by-step solution

Convert mph to ft/s

Convert the velocity of the fastball from 88 mph to feet per second (ft/s). 1 mile = 5280 feet and 1 hour = 3600 seconds. So, we have the following conversion formula: \(88\frac{\text{miles}}{\text{hour}} \times \frac{5280\text{ feet}}{1\text{ mile}} \times \frac{1\text{ hour}}{3600\text{ seconds}}.\)

Calculate the fastball's velocity in ft/s

Now, calculate the fastball's velocity in ft/s. \(88 \frac{\text{miles}}{\text{hour}} \times \frac{5280\text{ feet}}{1\text{ mile}} \times \frac{1\text{ hour}}{3600\text{ seconds}}\approx 128.96\text{ ft/s}.\)

Convert rpm to turns/s

Convert the spin rate of the fastball from 110 rpm to turns per second (turns/s). 1 minute = 60 seconds. So, we have the following conversion formula: \(110\frac{\text{turns}}{\text{minute}} \times \frac{1\text{ minute}}{60\text{ seconds}}.\)

Calculate the spin rate in turns/s

Now, calculate the spin rate of the fastball in turns/s. \(110\frac{\text{turns}}{\text{minute}} \times \frac{1\text{ minute}}{60\text{ seconds}}\approx 1.83\text{ turns/s}\)

Find the travel time

Calculate the time it takes for the fastball to travel the given distance of 60.5 ft. Speed = distance / time Re-arrange for time: time = distance / speed \(\text{time} = \frac{60.5\text{ ft}}{128.96\text{ ft/s}}\approx 0.47\text{ seconds}\)

Calculate the total turns

Multiply the time it takes for the ball to travel the distance by the spin rate in turns/s to find out how many total turns the ball makes between release and catch. Total turns = turns/s × time \(\text{Total turns} = 1.83\text{ turns/s} \times 0.47\text{ seconds} \approx 0.86\text{ turns}\). Since only a whole number of full turns can be made, the number of full turns that the ball makes between release and catch is 0.

Key concepts

Projectile Motion

Projectile motion is a type of motion that occurs when an object is thrown or projected into the air, subject to only the acceleration of gravity. In the context of our baseball fastball problem, factors often include gravity and air resistance. In this specific exercise, however, any effects related to gravity or air resistance are ignored for simplification.

Despite its complexity in many real-world scenarios, basic projectile motion can be understood through simple two-dimensional physics. Understanding the components:

• Horizontal motion: Constant velocity since no forces are acting horizontally (ignoring air resistance).
• Vertical motion: Gravity acts downhill, influencing the vertical velocity and position over time.

While this exercise avoids gravity's influence, understanding how projectiles move helps connect physics to real-world applications, from sports to engineering.

Angular Velocity

Angular velocity is the rate of rotation of an object and measures how fast something spins around an axis. In terms of a baseball, Angular velocity is reflected in its spin rate. For example:

• "Revolutions Per Minute (rpm)" describes how many complete turns an object makes in one minute.
• To find the turns per second, we divide the rpm by 60 since there are 60 seconds in a minute.

In our exercise, the given spin rate is 110 rpm. Dividing by 60, we get approximately 1.83 turns per second. This simplistic approach gives insight into how fast the baseball is spinning over the short distance of its journey, crucial for athletes' technique improvement in sports.

Unit Conversion

Unit conversion is a fundamental skill in physics to translate different measurements into a common unit system. This simplifies calculations and ensures consistency:

In the baseball fastball problem, velocity and spin rate are given in different units than what might be numerically convenient.

• Velocity is provided in miles per hour (mph) and needs conversion to feet per second (ft/s) for calculating travel time.
• The conversion uses the facts that 1 mile = 5280 feet and 1 hour = 3600 seconds. This converts 88 mph to approximately 128.96 ft/s.
• Spin rate is initially in revolutions per minute (rpm), needing conversion to turns per second (turns/s) by dividing by 60.

Converting to consistent units is essential for accurately solving physics problems and avoiding errors.