A football is kicked off with an initial speed of 63 ft/s at a projection angle of 45 degrees. A receiver on the goal line 67 yd away in the direction of the kick starts running to meet the ball at that instant. What must be his minimum speed (in feet/second) if he is to catch the ball before it hits the ground?
find the vertical and horizontal components of the initial velocity.
from the vertical velocity, find the time in air:
hf=ho+viv*t-1/2 g t^2 solve for t time in air.
then, using the horizontal component find how far it goes.
distance=vih*timeinair.
Next, find out speed (distance he has to run in the given time)
To find the receiver's minimum speed required to catch the ball before it hits the ground, we need to split the initial velocity of the football into horizontal and vertical components.
First, let's find the horizontal component. Since the kick is made at an angle of 45 degrees, the initial velocity can be split equally between the horizontal and vertical components. So, the horizontal component (Vx) is given by:
Vx = V * cos(θ)
where V is the initial speed of the football and θ is the angle of projection.
Vx = 63 ft/s * cos(45°)
Vx = 63 ft/s * √(2)/2
Vx = 63 ft/s * 0.707
Vx ≈ 44.541 ft/s
Now, we need to determine the time it takes for the ball to reach the receiver. We can use the horizontal distance and the horizontal velocity to find the time as follows:
distance = velocity * time
67 yd = 44.541 ft/s * time
Converting 67 yards to feet:
67 yd = 67 yd * 3 ft/yd
67 yd = 201 ft
201 ft = 44.541 ft/s * time
Solving for time:
time = 201 ft / 44.541 ft/s
time ≈ 4.514 s
Now, considering the vertical motion, we can find the maximum height the ball reaches using the formula:
h_max = (V * sin(θ))^2 / (2 * g)
where g is the acceleration due to gravity.
h_max = (63 ft/s * sin(45°))^2 / (2 * 32.2 ft/s^2)
h_max ≈ 39.054 ft
Since the receiver starts at the goal line and wants to catch the ball before it hits the ground, the vertical distance he needs to run is equal to the maximum height of the ball.
Therefore, the receiver must run a distance of at least 39.054 ft to catch the ball.
Now, we can calculate the minimum speed required for the receiver to catch the ball. The total distance the receiver needs to cover is the sum of the horizontal distance and the vertical distance:
Total distance = horizontal distance + vertical distance
Total distance = 201 ft + 39.054 ft
Total distance ≈ 240.054 ft
To find the minimum speed (Vy) required for the receiver to catch the ball, we divide the total distance by the time:
Vy = Total distance / time
Vy = 240.054 ft / 4.514 s
Vy ≈ 53.238 ft/s
Therefore, the receiver must have a minimum speed of approximately 53.238 ft/s to catch the ball before it hits the ground.