Question

New York Yankees pitcher Roger Clemens threw a fastball at a horizontal speed of \(160 \mathrm{~km} / \mathrm{h}\), as verified by a radar gun. How long did it take for the ball to reach home plate, which is \(18.4 \mathrm{~m}\) away?

Short answer

The ball took approximately \(0.414 \mathrm{~s}\) to reach the home plate.

Step-by-step solution

Convert Speed into Correct Units

First, the speed is given in km/h but the distance is presented in meters. In order to use the formula, speed and distance should be in the same unit. Therefore, convert the speed from km/h to m/s by using the conversion factor \(1 \mathrm{~km} / \mathrm{h} = 1000 \mathrm{~m} / 3600 \mathrm{~s}\). So the speed in m/s becomes \(160 \times 1000 / 3600 = 44.44 \mathrm{~m/s}\)

Calculate Time

Now using the formula \(time = distance / speed\), we can calculate the time it took for the ball to reach the home plate. Substituting the values we have, \(time = 18.4 \mathrm{~m} / 44.44 \mathrm{~m/s} = 0.414 \mathrm{~s}\)

Interpret the Result

The calculated time refers to how long it took for the ball to travel from the pitcher's hand to the home plate. This calculation assumes that the speed of the ball didn't change during the flight, which is an idealization.

Key concepts

Horizontal speed calculation

Calculating horizontal speed is a fundamental aspect of understanding projectile motion. Horizontal speed measures how fast an object is moving along a straight line parallel to the ground. When a baseball is thrown, like in Roger Clemens' pitch, it travels with this type of speed.
To find horizontal speed when it’s initially not given in the same unit as distance, we use conversion techniques. For example, speed often needs converting from kilometers per hour (km/h) to meters per second (m/s). This conversion allows the speed to be in the same units as the distance (meters).
The conversion factor for km/h to m/s is \[1 ext{ km/h} = \frac{1000 ext{ meters}}{3600 ext{ seconds}} = \frac{5}{18} ext{ m/s} \]Hence, by multiplying the speed in km/h by this conversion factor, we obtain the speed in m/s as shown in the exercise. Keen insight into these calculations builds a strong foundation for tackling physics problems involving motion along the horizontal axis.

Unit conversion

Unit conversion is a critical skill in physics that ensures all measurements are in compatible units for calculations. When calculating time, speed, or distance, it is essential to have all quantities in the same unit system.
The common practice for converting units involves using conversion factors, which are ratios that express the same quantity in different units. For instance, converting a speed from kilometers per hour to meters per second helps in maintaining consistency across unit systems. This consistency allows complex physics equations, like those in projectile motion, to be solved accurately.
In the pitching example, the speed conversion from 160 km/h to meters per second, which gives 44.44 m/s, is done to match the distance unit (meters). Understanding unit conversion allows a seamless transition between different forms of measurements, increasing precision and avoiding errors in calculations.

Time calculation in physics

Timing is crucial in physics, especially when assessing how long it takes an object to travel from one point to another. Here, time was calculated using the equation \[time = \frac{distance}{speed}\]This formula is derived from the basic speed formula, where speed equals distance divided by time.
To figure out how long Clemens' fastball took to reach home plate, we first needed the speed in the same unit as the distance, which requires preliminary unit conversion. Once the speed was converted to 44.44 m/s, finding the time was straightforward.
At its core, time calculation in physics provides insights into the dynamics of motion. By calculating time, we can predict when an object will arrive at a point or understand behavior during its travel. Simplified as 0.414 seconds in this context, such calculations not only apply to baseballs but to any real-world application involving motion.