To calculate the initial speed of the ball, we can break down the motion into horizontal and vertical components.
First, let's consider the horizontal motion. The ball travels a distance of 4000 meters horizontally in 10 seconds. Therefore, the horizontal component of velocity (Vx) can be calculated using the formula:
Vx = distance / time
Vx = 4000 m / 10 s Vx = 400 m/s
Now, let's consider the vertical motion. The ball follows a projectile motion, and we know that the time of flight is 10 seconds. The vertical displacement (Vy) can be determined using the formula:
Vy = (1/2) * g * t^2
where g is the acceleration due to gravity (approximately 9.8 m/s^2) and t is the time of flight.
Vy = (1/2) * 9.8 m/s^2 * (10 s)^2 Vy = 490 m
Since the ball was kicked at an angle of 45 degrees to the horizontal, the vertical component of the initial velocity (Vyi) is equal to the horizontal component of the initial velocity (Vxi) because the angles are symmetrical.
Now, we can calculate the initial speed (Vi) using the Pythagorean theorem:
Vi = sqrt(Vx^2 + Vy^2)
Vi = sqrt((400 m/s)^2 + (490 m/s)^2) Vi ≈ sqrt(160000 m^2/s^2 + 240100 m^2/s^2) Vi ≈ sqrt(400100 m^2/s^2) Vi ≈ 632.46 m/s
Therefore, the initial speed of the ball is approximately 632.46 m/s.