You have two small, identical boxes that generate 440 Hz notes. While holding one, you drop the other from a 20-m-high balcony. How many beats will you hear before the falling box hits the ground? You can ignore air resistance.

Verified step by step guidance

1. Determine the time it takes for the falling box to hit the ground using the kinematic equation for free fall: $t=\sqrt{\frac{2h}{g}}$, where $h$ is the height (20 m) and $g$ is the acceleration due to gravity (9.8 m/s²).
2. Calculate the velocity of the falling box just before it hits the ground using the equation $v=\sqrt{2gh}$. This velocity will cause a Doppler shift in the frequency of the sound emitted by the falling box.
3. Determine the observed frequency of the falling box using the Doppler effect formula: $f$_obs = f_sourcev_soundv_sound - v_falling, where $v$_sound is the speed of sound in air (approximately 343 m/s), and $v$_falling is the velocity of the falling box.
4. Calculate the beat frequency using the formula: $f$_beat = |f_obs - f_source|, where $f$_source is the original frequency of the stationary box (440 Hz).
5. Multiply the beat frequency by the time of fall to determine the total number of beats heard before the falling box hits the ground: $n=f$_beat × t.

Key Concepts

Frequency and Beats

Frequency refers to the number of cycles of a wave that occur in a unit of time, measured in Hertz (Hz). When two sound waves of slightly different frequencies interact, they produce a phenomenon known as beats, which is the periodic variation in sound intensity. The beat frequency is equal to the absolute difference between the two frequencies, allowing us to calculate how many beats will be heard over a given time period.

Free Fall and Time of Flight

Free fall describes the motion of an object under the influence of gravity alone, without any air resistance. The time it takes for an object to fall from a certain height can be calculated using the equation t = √(2h/g), where h is the height and g is the acceleration due to gravity (approximately 9.81 m/s²). This concept is crucial for determining how long the second box will be in the air before it hits the ground.

Sound Wave Propagation

Sound waves are mechanical waves that propagate through a medium, such as air, and their speed is influenced by factors like temperature and pressure. In this scenario, the sound generated by the boxes travels at a constant speed, allowing us to consider the time it takes for the sound of the falling box to reach the observer. Understanding sound wave propagation is essential for calculating the total number of beats heard during the fall.