4.7
Lançamento de projéteis: exemplo
A teoria do lançamento de projéteis é muito útil para jogadores de diversos esportes melhorarem seu desempenho. Por exemplo, um lançador de dardos pre…
Considere um arqueiro atirando uma flecha de forma que siga uma trajetória de projétil. Lembre-se de que o alcance de um projétil depende do quadrado da velocidade inicial e do seno2θ.
Agora, sin2θ tem um valor máximo de 1, quando é igual a 45°. Nesse caso, o alcance do projétil seria máximo para a velocidade inicial dada.
Considere duas flechas lançadas em ângulos de 30 e 60°, com a mesma velocidade inicial de 50 metros por segundo. A aceleração da gravidade para ambas as setas é de 9,8 metros por segundo ao quadrado.
Portanto, substituindo os valores de velocidade e ângulo, o alcance percorrido por ambas as setas é igual a 220,9 metros.
Aqui, como sin(180 − sin2θ) é igual a 2θ, o alcance do projétil para ângulos de lançamento complementares é o mesmo.
No entanto, a altura máxima que cada seta atinge é proporcional ao quadrado da velocidade inicial e sen2θ. Portanto, a altura máxima é diferente para ambos os casos.
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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.