4.7
발사체 모션: 예
투사체 운동 이론은 여러 스포츠 선수들이 경기력을 향상시키는 데 매우 유용합니다. 예를 들어, 창던지는 사람은 창을 최대한 멀리 던져야 합니다. 창던지기 선수는 창던지기의 초기 속도를 높이기 위해 짧은 준비 시간을 가집니다. 발사체의 범위는 최대 45° 그래서 창던지는…
궁수가 발사체 궤적을 따라 화살을 쏘는 것을 생각해 보십시오. 발사체의 범위는 초기 속도의 제곱과 sin2θ에 따라 달라진다는 것을 기억하십시오.
이제 sin2θ는 theta가 45°일 때 최대값이 1입니다. 이 경우 발사체의 범위는 주어진 초기 속도에 대해 최대가 됩니다.
30°와 60° 각도로 발사된 두 개의 화살표가 초당 50미터의 동일한 초기 속도를 갖는다고 가정해 보겠습니다. 두 화살표의 중력으로 인한 가속도는 초당 9.8미터입니다.
따라서 속도와 각도 값을 대체하면 두 화살표가 덮는 범위는 220.9m가 됩니다.
여기서 sin(180 - sin2θ)은 2θ와 같기 때문에 상보적 발사 각도에 대한 발사체의 범위는 동일합니다.
그러나 각 화살표가 도달하는 최대 높이는 초기 속도와 sin2θ의 제곱에 비례합니다. 따라서 최대 높이는 두 경우 모두 다릅니다.
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Q1: Why does a projectile launched at 45 degrees travel farther than one launched at other angles?
The range of a projectile depends on the square of initial speed and sin(2θ). Since sin(2θ) reaches its maximum value of 1 when θ equals 45°, the range is maximized at this angle for any given initial velocity. This is why javelin throwers aim for launch angles as close to 45° as possible to achieve maximum distance.
Q2: Do arrows launched at 30 and 60 degrees with the same initial speed travel the same distance?
Yes, complementary launch angles produce identical ranges. For arrows launched at 30° and 60° with an initial velocity of 50 m/s, both travel 220.9 meters. This occurs because sin(2θ) equals sin(180° − 2θ), making the range equation yield the same result for complementary angle pairs.
Q3: How does launch angle affect the maximum height a projectile reaches?
Maximum height is proportional to the square of initial velocity and sin²θ. Since sin²θ varies with launch angle, projectiles launched at different angles reach different maximum heights even when initial speed is identical. A 60° launch angle produces greater maximum height than a 30° angle, though both cover the same range.
Q4: What happens to projectile motion when the range becomes very large?
When range is large compared to Earth's circumference, the Earth's curvature becomes significant. The surface drops away from the projectile, changing gravity's direction along the path. This allows the projectile to travel farther than predicted by standard range equations because it has greater distance to fall before reaching the ground.
Q5: At what horizontal velocity does an object enter orbit around Earth?
An object enters orbit when its horizontal velocity matches Earth's surface drop rate. Since Earth's surface drops approximately 5 meters every 8000 meters, and an object falls 5 meters in 1 second without air resistance, a horizontal velocity of 8000 m/s achieves orbit. This approximates the speed of space shuttles and satellites in low Earth orbit.
Q6: How does initial speed affect projectile range and maximum height?
Both range and maximum height depend on the square of initial speed. Doubling initial velocity quadruples both the range and maximum height achieved. This quadratic relationship means that small increases in launch speed produce substantial improvements in projectile performance, which is why athletes like javelin throwers use run-ups to increase initial velocity.
Q7: Why do sports athletes use the physics of projectile motion to improve performance?
Understanding projectile motion helps athletes optimize launch angles and initial speeds for maximum distance or height. Javelin throwers apply this knowledge by increasing initial speed through run-ups and aiming for angles near 45° to maximize range. This physics-based approach significantly improves athletic performance across multiple sports.