Questions: For questions 1-6 use the image to the right (assume air resistance is not present): 1. In order to throw the ball the shortest, the football player should throw the ball at a) 15° b) 45° c) 75° d) Any angle, as long as the speed is the same, they land in the same place 2. The ball will stay in the air the shortest (for the least time) if it is thrown at a) 15° b) 45° c) 75° d) Any angle, as long as the speed is the same, they spend the same time in the air 3. The ball has the greatest vertical velocity when launched at a) 15° b) 45° c) 75° d) Horizontal velocity is the same at each angle

For questions 1-6 use the image to the right (assume air resistance is not present): 1. In order to throw the ball the shortest, the football player should throw the ball at a) 15° b) 45° c) 75° d) Any angle, as long as the speed is the same, they land in the same place 2. The ball will stay in the air the shortest (for the least time) if it is thrown at a) 15° b) 45° c) 75° d) Any angle, as long as the speed is the same, they spend the same time in the air 3. The ball has the greatest vertical velocity when launched at a) 15° b) 45° c) 75° d) Horizontal velocity is the same at each angle
Transcript text: For questions 1-6 use the image to the right (assume air resistance is not present): 1. In order to throw the ball the shortest, the football player should throw the ball at a) $15^{*}$ b) $45^{\circ}$ c) $75^{\circ}$ d) Any angle, as long as the speed is the same, they land in the same place 2. The ball will stay in the air the shortest (for the least time) if it is thrown at a) $15^{\circ}$ b) $45^{\circ}$ c) $75^{\circ}$ d) Any angle, as long as the speed is the same, they spend the same time in the air 3. The ball has the greatest vertical velocity when launched at a) $15^{\circ}$ b) $45^{\circ}$ c) $75^{\circ}$ d) Horizontal velocity is the same at each angle
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Solution

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Paso 1: Determinar el ángulo para la distancia más corta
  • La distancia horizontal de un proyectil es máxima a \(45^{\circ}\).
  • Para la distancia más corta, se debe elegir el ángulo más pequeño.
  • Por lo tanto, el ángulo de \(15^{\circ}\) resultará en la distancia más corta.
Paso 2: Determinar el ángulo para el menor tiempo en el aire
  • El tiempo de vuelo de un proyectil depende de su componente vertical de velocidad.
  • Un ángulo más pequeño significa una menor componente vertical.
  • Por lo tanto, el ángulo de \(15^{\circ}\) resultará en el menor tiempo en el aire.
Paso 3: Determinar el ángulo para la mayor velocidad vertical
  • La componente vertical de la velocidad inicial es \(v_0 \sin(\theta)\).
  • Para maximizar esta componente, se debe maximizar \(\sin(\theta)\).
  • \(\sin(75^{\circ})\) es mayor que \(\sin(45^{\circ})\) y \(\sin(15^{\circ})\).
  • Por lo tanto, el ángulo de \(75^{\circ}\) dará la mayor velocidad vertical.
Respuesta Final
  1. La respuesta correcta es A.
  2. La respuesta correcta es A.
  3. La respuesta correcta es C.
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