Question

A good baseball pitcher can throw a baseball toward home plate at \(85 \mathrm{mi} / \mathrm{h}\) with a spin of 1800 rev/min. How many revolutions does the baseball make on its way to home plate? For simplicity, assume that the 60 -ft trajectory is a straight line.

Short answer

The baseball makes approximately 14 revolutions on its way to the home plate.

Step-by-step solution

Convert Speed to Feet per Minute

First we need to convert the speed of the pitch from miles per hour to feet per minute, as the distance is given in feet and the spin in revolutions per minute. We know that 1 mile is equal to 5280 feet and 1 hour is equal to 60 minutes, so \(85 \, \text{mi/hr} = 85 \, \text{mi/hr} \times \frac{5280 \, \text{ft}}{1 \, \text{mi}} \times \frac{1 \, \text{hr}}{60 \, \text{min}} = 7480 \, \text{ft/min}\)

Calculate Time

Next, we will find out how long it takes for the baseball to complete the trajectory to the home plate, using the formula for time, which is time = distance / speed. So the time it takes for the baseball to travel 60 feet is \(time = \frac{distance}{speed} = \frac{60 \, \text{ft}}{7480 \, \text{ft/min}} = 0.00802 \, \text{minutes}.\)

Calculate Number of Revolutions

Finally, we use the time it takes for the pitch to reach home plate and the rate of spin to calculate how many revolutions the ball makes. We know the baseball is spinning at a rate of 1800 revolutions per minute, so we can use the formula \( \text{revolutions} = \text{rate of spin} \times \text{time} = 1800 \, \text{rev/min} \times 0.00802 \, \text{minutes} = 14.4 \, \text{revolutions}\) thus the baseball makes approximately 14 revolutions on its way to home plate.

Key concepts

Angular Velocity

Angular velocity is the rate at which an object rotates or spins around a central point. It's expressed in units like revolutions per minute (RPM) or radians per second. In our baseball example, angular velocity describes how fast the ball spins as it travels towards home plate. Understanding angular velocity helps to calculate how many spins or turns an object makes over a specific period.

When we say the baseball spins at 1800 revolutions per minute, it means that every minute, the ball completes 1800 full spins around its axis. To find out how many revolutions it makes over the short journey to home plate, we need to link this angular velocity with time. This concept comes in handy across various applications, be it in sports, engineering, or even physics experiments.

Linear Velocity

Linear velocity refers to how fast an object moves along a straight path. Unlike angular velocity, which involves rotation, linear velocity is about straightforward motion. In this baseball scenario, the pitcher throws the ball at 85 miles per hour, which is its linear velocity.

To smoothly transition into calculations, it's often converted into other units, such as feet per minute. This is useful because it allows us to work with the given distance more easily. Linear velocity is essential for determining the time it takes an object to cover a specified distance. Imagine you're driving a car; knowing your linear velocity helps you estimate how long a journey will take.

Conversion of Units

Converting units is a vital skill in solving physics problems. It helps ensure consistency across measurements and enables smoother calculations. In the baseball problem, we deal with different units: the speed is in miles per hour, the distance in feet, and the spin in revolutions per minute.

Converting Miles per Hour to Feet per Minute

To convert 85 miles per hour to feet per minute, remember:

• 1 mile equals 5280 feet
• 1 hour equals 60 minutes

Using these conversions is simple as:\[85 \, \text{mi/hr} \times \frac{5280 \, \text{ft}}{1 \, \text{mi}} \times \frac{1 \, \text{hr}}{60 \, \text{min}} = 7480 \, \text{ft/min}\]Conversion of units aligns various measurements, making complex problems more manageable.

Distance Calculation

Distance calculation involves finding out how far an object travels over a period. It's straightforward but crucial, especially when solving problems involving motion. In our baseball scenario, the distance the ball travels is given as 60 feet.

Using the formula for time, \(\text{time} = \frac{\text{distance}}{\text{speed}}\), we can determine how long the travel takes. For our example:\[\text{time} = \frac{60 \, \text{ft}}{7480 \, \text{ft/min}} = 0.00802 \, \text{minutes}\]This brief time span allows us to calculate how many complete rotations or spins the ball makes using its angular velocity. Understanding distance calculation helps you connect the dots in any physics problem involving motion, from simple everyday activities to complicated scientific inquiries.