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The speed of sound is a fundamental concept in physics that refers to how fast sound waves travel through a medium, typically air. It depends on factors such as air temperature and pressure. At an air temperature of \(20^{\circ} \mathrm{C}\), sound travels at approximately \(343 \mathrm{m/s}\). This speed is critical for various calculations, especially when dealing with situations where sound needs to be accurately timed, such as determining when an observer hears a distant event. In the given problem, using the speed of sound helps us compute how quickly the sound from the rifle shot reaches the observer. This value of \(343 \mathrm{m/s}\) is an average used in many physics problems to ensure consistency in calculations.

Projectile motion refers to an object moving in a curved path under the influence of gravity alone, after an initial force is applied. In this scenario, though the bullet is a projectile, we focus solely on its horizontal motion because gravity's effect on the bullet's vertical motion and air resistance are ignored for simplicity. In reality, projectile motion is quite complex and involves both horizontal and vertical components. The initial force from the rifle propels the bullet horizontally at a constant speed of \(840 \mathrm{m/s}\), which is crucial for determining how far the bullet travels before sound reaches the observer. Understanding projectile motion allows us to separate motions into independent components that can be analyzed and computed separately.

Horizontal motion is an essential part of the larger projectile motion, described previously. It describes the bullet's consistent movement in a straight line, unaffected by vertical forces, as assumed in this problem. Given that the bullet's horizontal speed is \(840 \mathrm{m/s}\), it highlights the bullet's swift journey across the firing range.

• Horizontal speed remains constant assuming no air resistance.
• Distance covered can be calculated simply using the formula \(d = \text{speed} \times \text{time}\).

In this problem, horizontal motion helps determine the precise distance a bullet travels during a specified period before the observer hears the sound. It simplifies the scenario, allowing a straightforward calculation of the bullet's path as a horizontal line.

Time calculation is at the heart of solving this physics problem. It involves determining how long it takes for sound to travel a certain distance to reach the observer. This calculation is performed using the formula \( t = \frac{\text{distance}}{\text{speed}} \), which helps us find the required time to travel a specific path.

• Calculate the time it takes for sound to travel \(25 \mathrm{m}\): \( t = \frac{25}{343} \approx 0.073 \mathrm{s}\).
• Determine the bullet's travel distance in the same period: \( d = 840 \times 0.073 \approx 61.32 \mathrm{m}\).

These time-based calculations are foundational in physics as they enable us to relate speed, distance, and overall motion dynamically. By understanding time calculation, it becomes clearer how dynamic processes unfold and interact over time.