Problem 103 - Fundamentals of Physics

Question

If a baseball pitcher throws a fastball at a horizontal speed of \(160 \mathrm{~km} / \mathrm{h},\) how long does the ball take to reach home plate \(18.4 \mathrm{~m}\) away?

Step-by-Step Solution

Answer

The ball takes approximately 0.414 seconds to reach home plate.

Step 1: Convert Speed to Meters per Second

First, we need to convert the speed from kilometers per hour to meters per second, since we are dealing with meters for distance. The conversion factor is 1 km/h = 0.27778 m/s.Calculate the speed: \[ 160 \times 0.27778 = 44.4448 \text{ m/s} \]

Step 2: Calculate Time Using the formula

Use the formula for time, which is:\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]Here, the distance is 18.4 meters and the speed is 44.4448 m/s.\[ \text{Time} = \frac{18.4}{44.4448} = 0.414 \text{ seconds} \]

Key Concepts

Speed Conversion

When tackling problems involving different units, speed conversion is crucial. Imagine being in a situation where you're given speed in kilometers per hour, but you need to work in meters per second. To make this conversion, you use a simple unit rate: 1 km/h is equivalent to approximately 0.27778 meters per second. So, what does this mean practically?
For instance, if you're measuring the speed of a baseball at 160 km/h, you multiply by 0.27778 to convert it into meters per second. This gives you:
\[ 160 \times 0.27778 = 44.4448 \text{ m/s} \] This conversion is vital because it aligns the units with the distance, simplifying further calculations. It ensures that every value you're working with is harmonious and makes calculating time or determining any other outcomes straightforward.

Distance Calculation

Although the original exercise gives the distance directly, understanding distance calculation is vital for mastering projectile motion problems. Distance, in most basic terms, can be thought of as the total stretch or expanse an object covers during its motion. The given distance, like the 18.4 meters in our exercise, is how far the baseball travels. In scenarios where this is not provided, you typically calculate it using other known values and applying appropriate kinematic formulas.
Distance calculations might be used to determine how different variables such as velocity or time affect how far an object travels. In multi-part projects or real-life scenarios, these calculations help predict outcomes and plan more effectively.

Time Calculation

Calculating time is a critical component in many physics problems, especially in projectile motion. Time tells us how long an object, like a baseball, will take to reach a specific point. The key formula here is:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \] Divide the distance by the converted speed to get the time in seconds. From our exercise, with a distance of 18.4 meters and speed of 44.4448 meters per second, this comes to:
\[ \text{Time} = \frac{18.4}{44.4448} = 0.414 \text{ seconds} \] Use this calculation method in various scenarios from sports to transport dynamics, enabling accurate prediction of how different elements interact in motion over time.