Carlos is in an roof of a building of 20 meters tall, he throws a tennis ball vertically upwards at 15 m/s velocity after it reaches its highest height, the ball goes down until it reaches the ground (see the figure). Based on this information, answer the following.
a) Time taken to reach the highest height
b) The highest height in relation to building base
c) Total flight time since the ball is thrown until it reaches the ground
d) Its velocity when it crashes to the ground
QUIERO SABER LA RESPUESTA
Carlos está en una azotea de un edificio de 20 metros de altura, se lanza una pelota de tenis verticalmente hacia arriba a los 15 m / s de velocidad después de que alcance su altura máxima, la pelota va hacia abajo hasta que llega al suelo (ver la figura). Con base en esta información, responder a las siguientes.
A) Tiempo necesario para llegar a la mayor altura
b) La altura más alta en relación con la construcción de la base
c) El tiempo total de vuelo ya que la pelota es lanzada hasta que llega al suelo
d) Su velocidad cuando se bloquea al suelo
To answer these questions, we can use the equations of motion for an object in free fall. We'll need to consider the initial velocity, acceleration, and distance traveled.
a) Time taken to reach the highest height:
To find the time taken to reach the highest height, we can use the formula:
Time = Velocity / Acceleration.
In this case, as the ball is thrown vertically upwards, the initial velocity is positive (+15 m/s) and the acceleration due to gravity is negative (-9.8 m/s²), because it acts in the opposite direction to the motion of the ball.
So, plugging in the values:
Time = 15 m/s / (-9.8 m/s²).
Time = -1.53 seconds (rounded to 2 decimal places).
Note that we obtained a negative value for time, which means we are referring to the time it takes for the ball to reach its highest point, not the time since it was thrown.
b) The highest height in relation to the building base:
To find the highest height reached by the ball, we can use the formula:
Distance = (Velocity² - Initial Velocity²) / (2 * Acceleration).
Since we know the initial velocity is +15 m/s, the final velocity at the highest height is 0 m/s, and the acceleration is -9.8 m/s², plugging in the values:
Distance = (0² - 15²) / (2 * -9.8).
Distance = 11.99 meters (rounded to 2 decimal places).
So, the highest height reached by the ball is approximately 12 meters above the base of the building.
c) Total flight time since the ball is thrown until it reaches the ground:
To find the total flight time of the ball, we can double the time taken to reach the highest height and then add it to the time taken for the ball to fall back down.
Since the time taken to reach the highest height is -1.53 seconds (rounded to 2 decimal places), the time taken for the ball to fall back down is given by:
Time = √(2 * distance / acceleration),
where distance is twice the highest height, i.e., 2 * 12 = 24 meters.
Plugging in the values:
Time = √(2 * 24 / 9.8).
Time ≈ 1.99 seconds (rounded to 2 decimal places).
So, the total flight time of the ball is approximately 2 seconds.
d) Its velocity when it crashes to the ground:
To find the velocity with which the ball hits the ground, we can use the formula:
Final Velocity = Initial Velocity + (Acceleration * Time).
In this case, the initial velocity is +15 m/s, the acceleration is -9.8 m/s², and the time taken for the ball to reach the ground is 2 seconds.
Plugging in the values:
Final Velocity = 15 m/s + (-9.8 m/s² * 2 s).
Final Velocity = -4.6 m/s.
So, the velocity at which the ball crashes to the ground is approximately -4.6 m/s. The negative sign indicates that the velocity is directed downward.