The initial velocity of the kicked football was approximately 19.7 m/s .
Assumptions:
Ignore air resistance.
Consider the Earth as a flat plane with constant gravity (9.8 m/s²).
Given:
Launch angle (θ) = 51.0° ,Horizontal distance (d) = 38.7 m ,Initial velocity (v₀) = Unknown
Steps to Solve:
Break down the initial velocity:
The initial velocity can be decomposed into its horizontal and vertical components:
v₀x = v₀ * cos(θ)
v₀y = v₀ * sin(θ)
where:
v₀x is the horizontal component of the initial velocity.
v₀y is the vertical component of the initial velocity.
Analyze the horizontal motion:
The horizontal motion of the football is independent of gravity and can be described by the following equation:
d = v₀x * t
where:
t is the time taken for the football to reach the ground.
Analyze the vertical motion:
The vertical motion of the football is influenced by gravity and can be described by the following equation:
h = v₀y * t - 0.5 * g * t²
where:
h is the vertical distance traveled by the football (which is 0 at the ground level).
g is the acceleration due to gravity (9.8 m/s²).
Relate horizontal and vertical distances:
Since the football hits the ground, the vertical distance traveled is 0. This allows us to relate the horizontal and vertical components of the initial velocity:
0 = v₀ * sin(θ) * t - 0.5 * g * t²
Solve for the horizontal component of the initial velocity:
From the horizontal motion equation, we can express the time as:
t = d / v₀x
Substitute this expression for t in the vertical motion equation:
0 = v₀ * sin(θ) * (d / v₀x) - 0.5 * g * (d / v₀x)²
Simplify the equation:
0 = d * sin(θ) - 0.5 * g * d / v₀x
Solve for v₀x:
v₀x = √( 2 * g * d / sin(θ) )
Calculate the initial velocity:
Now, we can use the equation for v₀x to calculate the initial velocity:
v₀ = v₀x / cos(θ)
v₀ = √( 2 * g * d / sin(θ) ) / cos(θ)
Substitute the given values:
v₀ = √( 2 * 9.8 m/s² * 38.7 m / sin(51.0°) ) / cos(51.0°)
v₀ ≈ 19.7 m/s
Therefore, the initial velocity of the kicked football was approximately 19.7 m/s.