4.7
حركة المقذوف: مثال
إن نظرية حركة المقذوفات مفيدة جدًا للاعبي العديد من الألعاب الرياضية لتحسين أدائهم. فمثلاً، يحتاج رامي الرمح إلى رمي الرمح بطريقة تجعله يقطع أبعد مساف…
ضع في اعتبارك رامي السهام الذي يطلق سهما بحيث يتبع مسارا مقذوفا. تذكر أن نطاق المقذوف يعتمد على مربع السرعة الابتدائية و sin2θ.
الآن ، sin2θ له قيمة قصوى تساوي 1 ، عندما تكون ثيتا تساوي 45 درجة. في هذه الحالة ، سيكون مدى المقذوف هو الحد الأقصى للسرعة الابتدائية المحددة.
ضع في اعتبارك سهمين تم إطلاقهما بزاويتين 30 و 60 درجة ، وهما نفس السرعة الابتدائية البالغة 50 مترا في الثانية. التسارع الناتج عن الجاذبية لكلا السهمين هو 9.8 متر لكل ثانية مربعة.
لذلك ، بالتعويض عن قيم السرعة والزاوية ، فإن النطاق الذي يغطيه كلا السهمين يساوي 220.9 مترا.
هنا ، بما أن sin (180 − sin2θ) يساوي 2θ ، فإن نطاق المقذوف لزوايا الإطلاق التكميلية هو نفسه.
ومع ذلك ، فإن الحد الأقصى للارتفاع الذي يصل إليه كل سهم يتناسب مع مربع السرعة الابتدائية و sin2θ. لذلك ، يختلف الحد الأقصى للارتفاع لكلتا الحالتين.
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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.