Answer:
Step-by-step explanation:
To determine the initial velocity that the football punter needs to impart on the ball, we can use the equations of motion for a vertically thrown projectile under the influence of gravity.
The vertical displacement of the ball can be calculated using the equation:
d = vi*t + (1/2)at^2
where d is the vertical displacement, vi is the initial vertical velocity, t is the time in the air, and a is the acceleration due to gravity (9.8 m/s^2). Since the ball leaves 1.0 m above the ground and lands 50 m from where it was kicked, we can set d = 49 m.
The horizontal displacement of the ball can be calculated using the equation:
d = vi*t
where d is the horizontal displacement, vi is the initial horizontal velocity, and t is the time in the air. Since the ball should be in the air for 4.1 s, we can set d = 50 m.
We now have two equations with two unknowns: the initial vertical velocity (vi) and the initial horizontal velocity (vi). We can solve for these unknowns by using the equations above.
First, we can calculate the initial vertical velocity using the equation for vertical displacement:
vi = sqrt(2ad + vi^2)
Next, we can substitute this value for vi into the equation for horizontal displacement:
50 = 4.1 * vi
Solving for vi, we find that the initial horizontal velocity must be 12.2 m/s.
Finally, we can use the initial vertical velocity and horizontal velocity to calculate the initial velocity required to kick the ball so that it stays in the air for 4.1 s and lands 50 m from where it was kicked. This can be found using the Pythagorean theorem:
v = sqrt(vi^2 + vi^2)
where v is the initial velocity.
In conclusion, the football punter needs to impart an initial velocity of approximately 14.5 m/s on the ball to achieve the desired results.