Understanding Distance OP in Projectile Motion

Introduction

In the realm of physics, understanding the dynamics of projectiles is essential for a variety of applications ranging from sports to engineering. One classic example involves an arrow being shot horizontally towards a point O, which is 20.0 meters away from the shooter, and hitting point P directly below O after 0.10 seconds. This scenario can be analyzed through basic projectile motion principles to determine the distance OP. This article delves into the mathematical and physical aspects of this problem, presenting a detailed step-by-step analysis and offering insights into the underlying concepts.

Problem Statement

An arrow is shot horizontally towards point O which is 20.0 meters away from the shooter. The arrow hits point P, which is directly below O, after 0.10 seconds. The challenge is to determine the distance OP.

Analysis and Solution

To solve this problem, we need to analyze the motion of the arrow in both the horizontal and vertical directions separately. The horizontal direction doesn't experience acceleration (assuming no air resistance), while the vertical motion is affected by gravity.

Vertical Motion

The vertical displacement s can be calculated using the formula for free-fall motion:

s 1/2 g t2

Where:

s is the vertical displacement (distance OP) g is the acceleration due to gravity (9.8 m/s2) t is the time of flight (0.10 seconds)

Substituting the given values:

s 1/2 * 9.8 * (0.10)2

s 0.049 meters

s 4.9 cm

Horizontal Motion

The horizontal motion is determined by the initial horizontal velocity vo. However, the problem does not ask for the horizontal velocity; instead, it focuses on the vertical displacement.

The horizontal velocity can be calculated as:

vo distance / time

vo 20 m / 0.10 s

vo 200 m/s or approximately 720 km/h

However, this speed is unrealistically high for a traditional bow and arrow. Bows and crossbows typically shoot arrows at speeds around 140 m/s or 500 km/h. Thus, the given horizontal speed of 200 m/s is not physically possible in the context of these weapons.

Conclusion

The distance OP is determined primarily by the vertical displacement due to gravity. Given the time of flight and constant gravity, the arrow will fall approximately 4.9 cm during the horizontal flight path of 20 meters. The horizontal speed is unnecessary for determining distance OP but is crucial for understanding the overall motion of the arrow.

Key Takeaways

Vertical Displacement: The primary factor in determining the distance OP is the vertical displacement due to gravity. Physical Context: The given horizontal speed of 200 m/s is unrealistic for a traditional bow and arrow, emphasizing the importance of physical context in problem-solving. Application: Understanding projectile motion is crucial for various applications, including archery, aerospace engineering, and sports science.