Problem 68 - University Physics with Modern Physics

Question

A baseball coach uses a radar device to measure the speed of an approaching pitched baseball. This device sends out electromagnetic waves with frequency \(f_{0}\) and then measures the shift in frequency \(\Delta f\) of the waves reflected from the moving baseball. If the fractional frequency shift produced by a baseball is \(\Delta f / f_{0}=2.86 \times 10^{-7}\) , what is the baseball's speed in \(\mathrm{km} / \mathrm{h} ?\) (Hint: Are the waves Doppler- shifted a second time when reflected off the ball?)

Step-by-Step Solution

Answer

The baseball's speed is 154.44 km/h.

Step 1: Understanding the Doppler Effect

We need to consider that the radar measures a Doppler shift in frequency upon detection, meaning the frequency changes due to the motion of the baseball. This shift occurs twice: first when the radar waves hit the baseball and again when the waves reflect back to the detector.

Step 2: Determine the Formula for Speed Calculation

The formula for Doppler shift in frequency due to relative motion is given by:\[ \Delta f = \frac{2v}{c} f_0 \]where \(v\) is the speed of the baseball, \(c\) is the speed of light (approximately \(3 \times 10^8\) m/s), and \(\Delta f / f_{0} = 2.86 \times 10^{-7}\) is the given fractional frequency shift. The factor of 2 accounts for the double shift.

Step 3: Rearrange the Formula to Solve for the Baseball's Speed

Rearranging the formula \( \Delta f = \frac{2v}{c} f_0 \) to solve for \(v\):\[ v = \frac{\Delta f}{f_{0}} \times \frac{c}{2} \]

Step 4: Calculate the Baseball's Speed

Substitute the given values:\[ v = 2.86 \times 10^{-7} \times \frac{3 \times 10^{8}}{2} \]Simplifying this gives:\[ v = 2.86 \times 10^{-7} \times 1.5 \times 10^{8} \]\[ v = 4.29 \times 10^{1}\]\[ v = 42.9 \text{ m/s} \]

Step 5: Convert Speed to Kilometers Per Hour

To convert the speed from meters per second to kilometers per hour, use the conversion:\[ 1 \text{ m/s} = 3.6 \text{ km/h} \]Therefore, \(v = 42.9 \text{ m/s} \times 3.6 = 154.44 \text{ km/h} \).

Key Concepts

Electromagnetic Waves

Electromagnetic waves are vital tools for various technologies in our everyday lives. They are waves of electric and magnetic fields that travel through space at the speed of light, approximately \(3 \times 10^8\) meters per second.
Unlike sound or water waves, electromagnetic waves do not need a medium to travel, making them essential for communication, such as in radar systems.
A radar device specifically uses radio waves, which are a type of electromagnetic wave. These waves are emitted, bounce off a moving object such as a baseball, and return as echoes. The change in the frequency of these returning waves can reveal the speed of the object. This is because the motion of the object affects the waves, a phenomenon described by the Doppler Effect.

Frequency Shift

The Doppler Effect is responsible for the frequency shift of electromagnetic waves. When an object moves toward or away from an observer, the frequency of the waves it reflects changes. This change is known as the frequency shift.
For the Doppler Effect, there are two primary shifts to consider. First, when waves emitted by the radar hit the moving baseball, and then when waves are reflected back. This reflects a double shift rather than just a single alteration.

• The formula for the frequency shift is: \[ \Delta f = \frac{2v}{c} f_0 \]
• \(\Delta f\) is the difference in frequency.
• \(v\) is the speed of the baseball.
• \(c\) is the speed of light.
• \(f_0\) is the original frequency emitted by the radar.

Given the fractional shift \(\Delta f / f_{0} = 2.86 \times 10^{-7}\), one can rearrange the formula to find the speed of the baseball.

Speed Calculation

Once the relationship between the frequency shift and velocity is established, you can calculate the speed of an approaching baseball. The aim is to find \(v\), which represents the speed of the baseball. Using the formula:
\[ v = \frac{\Delta f}{f_{0}} \times \frac{c}{2} \]
Given the fractional frequency shift \(\Delta f / f_{0} = 2.86 \times 10^{-7}\), the Doppler shift formula can be directly applied. Substituting the known values:\
\[ v = 2.86 \times 10^{-7} \times \frac{3 \times 10^{8}}{2} \]
Yields
\[ v = 42.9 \text{ m/s} \]
Converting from meters per second to kilometers per hour involves multiplying by 3.6 (this converts velocity to a more common unit for speed, as \(1 \text{ m/s} = 3.6 \text{ km/h}\)):
\[ v = 42.9 \text{ m/s} \times 3.6 = 154.44 \text{ km/h} \]
Therefore, the baseball is traveling at approximately 154.44 km/h.