Problem 83 - Pearson Physics

Question

A radar gun is often used to measure the speed of a Major League pitch by reflecting a beam of radio waves off the moving ball. Describe how the Doppler effect can give the speed of the ball from a measurement of the frequency of the reflected beam.

Step-by-Step Solution

Answer

The Doppler effect causes a frequency shift in reflected waves, allowing velocity computation from the radar's formula.

Step 1: Understanding the Doppler Effect

The Doppler effect describes the change in observed frequency of a wave when the source or observer is moving relative to the medium. For a moving source and stationary observer, like the radar gun measuring a moving ball, this effect can alter the frequency of reflected waves.

Step 2: Initial Frequency Emission

The radar gun emits a radio wave with a certain frequency \( f_0 \). This initial frequency is transmitted to the moving baseball.

Step 3: Frequency Shift on Reflection

As the radio wave hits the moving baseball, the motion of the ball causes the frequency of the wave to change. The frequency of the wave reflected back to the radar gun becomes \( f' \), which differs from \( f_0 \).

Step 4: Relating Frequency Change to Velocity

Using the formula for the Doppler effect, the change in frequency (\( \Delta f = f' - f_0 \)) can be related to the speed \( v \) of the baseball. The formula is approximately \( \Delta f = \dfrac{2v f_0}{c} \), where \( c \) is the speed of light.

Step 5: Solving for the Ball's Speed

Rearranging the formula, the speed \( v \) of the baseball can be measured as \( v = \dfrac{c \Delta f}{2f_0} \). By measuring the frequency shift \( \Delta f \), the radar gun calculates the speed of the baseball.

Key Concepts

Radar Gun Frequency

A radar gun is used to measure speed by utilizing radio waves, specifically the frequency at which these waves are emitted and received. When the radar gun emits these waves, they travel until they hit a moving object, like a baseball. The initial frequency of the radio wave, denoted as \( f_0 \), is a crucial part of the process. This frequency acts as a reference point for determining any changes.

Radar guns are designed to emit these frequencies steadily and precisely, allowing them to accurately track how the waves change once they bounce back. This initial frequency is always known, which is key when it comes to understanding the Doppler effect and the resulting changes in frequency. The whole setup is like sending a beacon of specific frequency, then waiting to hear how it sounds when it returns.

Wave Frequency Change

The true magic happens when the emitted radio wave hits a moving object. Here, the Doppler Effect comes into play, causing a shift or change in the frequency of the wave when it reflects back. This occurs because the object, such as a speeding baseball, affects the way the wave behaves. The frequency of the returning waves, noted as \( f' \), is altered compared to the original emitted frequency \( f_0 \).

The relative motion between the radar gun and the moving ball causes this change. If the object is moving closer, the waves compress, increasing the frequency. If moving away, the waves stretch, lowering the frequency. This change, or \( \Delta f \), is a critical part of measuring speed: it's the difference between the radar gun's emitted frequency and what it detects after reflection. This frequency shift is what allows us to calculate how fast the ball is moving.

Measuring Speed with Doppler Effect

To determine the object's speed, we use the relationship created by the Doppler Effect between the change in frequency and the velocity of the object. The formula used is \( \Delta f = \frac{2v f_0}{c} \), where \( \Delta f \) is the change in frequency, \( v \) is the velocity of the object, and \( c \) is the speed of light.

This formula can be rearranged to solve for the speed \( v \) of the baseball. By measuring \( \Delta f \), the frequency change, we can plug this value back into the formula to get \( v = \frac{c \Delta f}{2f_0} \). The radar gun uses this calculation to understand how fast the ball is traveling. This process is cleverly designed to ensure precision, as the radar gun must accurately detect even slight changes in frequency to measure the speed effectively. By capturing even minute shifts in frequency, the device can calculate the speed of objects with remarkable accuracy.