A football is kicked 58.1 meters. If the ball is in the air 4.60 s, with what initial velocity was it kicked? magnitude:

direction: above the horizontal

To find the initial velocity of the football, we can use the equation of motion:

Δy = v₀y * t - (1/2) * g * t²

Where:
- Δy is the vertical displacement (which is the height the football reaches above the initial point)
- v₀y is the initial vertical velocity
- t is the time in the air
- g is the acceleration due to gravity (9.8 m/s²)

Since the question mentions that the ball is in the air for 4.60 seconds, and we want to find the initial velocity (v₀), we need to determine the initial vertical velocity (v₀y).

Since the ball is kicked above the horizontal, it means there is an initial vertical component to the velocity. The magnitude of the initial velocity can be found using the horizontal distance kicked (58.1 meters) and the time in the air (4.60 seconds).

We can use the horizontal distance and time to find the horizontal component of the velocity (v₀x) using the equation:

Δx = v₀x * t

Solving for v₀x, we get:

v₀x = Δx / t

Substituting the given values, we get:

v₀x = 58.1 meters / 4.60 seconds

Now we need to find the vertical component of the initial velocity (v₀y). Since we know the initial velocity is above the horizontal, we can assume that the angle is not 0 degrees but rather some angle above the horizontal.

To find the magnitude of the initial velocity (v₀), we can use the Pythagorean theorem:

v₀ = sqrt(v₀x² + v₀y²)

Since we have v₀x from the previous calculation, we can solve for v₀y:

v₀y = v₀ * sin(θ)

where θ is the angle above the horizontal.

Now we can solve for v₀ using the given information:

v₀ = sqrt(v₀x² + v₀y²)

We have found v₀x and v₀y, so we can substitute these values into the equation:

v₀ = sqrt((58.1 / 4.60)² + v₀y²)

Now let's solve for v₀y:

Δy = v₀y * t - (1/2) * g * t²

Since we want Δy to be the maximum height the ball reaches, we can set Δy to be equal to half of the horizontal distance (58.1 meters) i.e., Δy = 29.05 meters.

29.05 = v₀y * 4.60 - (1/2) * 9.8 * 4.60²

Now we need to solve for v₀y:

29.05 = v₀y * 4.60 - 1/2 * 9.8 * 4.60²
v₀y = (29.05 + 1/2 * 9.8 * 4.60²) / 4.60

Now we can substitute the value of v₀y back into the equation for v₀:

v₀ = sqrt((58.1 / 4.60)² + v₀y²)

Using the calculated values of v₀x (horizontal component of the velocity) and v₀y (vertical component of the velocity), we can find the magnitude of the initial velocity (v₀).