Final answer:
The speed of the ball when the goal keeper catches it is 16.18 m/s.
Step-by-step explanation:
To find the speed of the ball when the goal keeper catches it, we need to break down the initial velocity of the ball into its horizontal and vertical components. The horizontal component remains constant throughout the flight and is given by:
horizontal component = initial speed * cos(angle)
The vertical component of the velocity changes due to gravity. The time of flight can be calculated using the vertical component of velocity and the acceleration due to gravity:
time of flight = 2 * (vertical component of velocity) / acceleration due to gravity
Once we have the time of flight, we can find the vertical component of velocity when the ball lands using:
vertical component of velocity when the ball lands = initial vertical component of velocity - (acceleration due to gravity * time of flight)
Finally, we can find the speed of the ball when the goal keeper catches it using the vertical and horizontal components of velocity:
speed of the ball when the goal keeper catches it = sqrt((horizontal component)^2 + (vertical component of velocity when the ball lands)^2)
Plugging in the given values:
initial speed = 19.0 m/s
angle = 32.0 degrees
distance to the goal keeper = 29.0 m
acceleration due to gravity = 9.8 m/s^2
We can calculate the horizontal and vertical components of velocity using the given values:
horizontal component = 19.0 m/s * cos(32.0 degrees) = 16.12 m/s
vertical component of velocity when the ball lands = 19.0 m/s * sin(32.0 degrees) - (9.8 m/s^2)(2 * (19.0 m/s * sin(32.0 degrees)) / 9.8 m/s^2) = -1.70 m/s
Lastly, we can calculate the speed of the ball when the goal keeper catches it using the horizontal and vertical components of velocity:
speed of the ball when the goal keeper catches it = sqrt((16.12 m/s)^2 + (-1.70 m/s)^2) = 16.18 m/s