To determine the initial speed of the ball, we can use the principles of projectile motion and the given information.
The vertical motion of the ball can be analyzed separately from the horizontal motion since there is no horizontal acceleration (assuming no air resistance).
First, let's focus on the vertical motion. The ball is kicked into the air at an angle of 45°, which can be split into its vertical and horizontal components. The vertical component determines the height reached by the ball.
We can use the equation for vertical displacement (Δy) in projectile motion:
Δy = v₀y * t - (1/2) * g * t²
Where: v₀y is the initial vertical component of velocity (which we want to find). t is the time of flight. g is the acceleration due to gravity (approximately 9.8 m/s²).
At the highest point, the vertical velocity becomes zero, so we can rearrange the equation:
0 = v₀y - g * t
From this equation, we can solve for the time of flight (t):
t = v₀y / g
Now, let's consider the horizontal motion. The horizontal distance traveled by the ball can be determined using the equation:
Δx = v₀x * t
Where: v₀x is the initial horizontal component of velocity. t is the time of flight.
In projectile motion, the vertical and horizontal components of velocity are related as follows:
v₀x = v₀y = v₀ * sin(45°)
Since we're looking for the initial speed (v₀), we need to find either v₀x or v₀y.
Now, let's substitute the expression for t from the vertical motion equation into the horizontal motion equation:
Δx = v₀ * sin(45°) * (v₀ * sin(45°) / g)
Given that the height reached by the ball (Δy) is 12 m, we can use it to find the time of flight (t):
Δy = v₀y * t - (1/2) * g * t² 12 = v₀ * sin(45°) * t - (1/2) * g * t²
Solving this equation will give us the time of flight (t).
Once we have the time of flight, we can substitute it back into the equation for horizontal distance to find Δx.
Finally, we can rearrange the equation for horizontal distance to solve for v₀:
v₀ = Δx / (t * sin(45°))
By plugging in the appropriate values into this equation, you can calculate the initial speed (v₀) of the ball.