Answer:
a)The initial speed of the ball was 24.8 m/s.
b)The ball passed through the goal posts 3.04 s after it was kicked.
Explanation:
The position vector of the ball at time "t" can be calculated using the following equation:
r = (x0 + v0 · t · cos α, y0 + v0 · t · sin α + 1/2 · g · t²)
Where:
r = position vector at time t.
x0 = initial horizontal position.
v0 = initial velocity.
t = time.
α = launching angle.
y0 = initial vertical position.
g = acceleration due to gravity (-9.8 m/s² considering the upward direction as positive).
Please, see the attached figure for a better understanding of the problem ( the following unit conversions were made 63.0 yards = 57.6 m and 10 ft = 3.05 m).
Notice that the position vector when the ball clears the bar is the following (see r 1 in the figure) :
r 1 = (57.6 m, 3.05 m)
Then, using the equation for the x and y-component of the position vector written above:
57.6 m = x0 + v0 · t · cos α
3.05 m = y0 + v0 · t · sin α + 1/2 · g · t²
Placing the origin of the frame of reference at the kicking point, x0 and y0 = 0. Then:
57.6 m = v0 · t · cos α
3.05 m = v0 · t · sin α + 1/2 · g · t²
We have two equations and two unknowns, so, we can solve the system:
Let´s solve the equation of the x-component for v0:
57.6 m = v0 · t · cos α
57.6 m / t · cos α = v0
Now, replacing v0 in the equation of the y-component of r
3.05 m = v0 · t · sin α + 1/2 · g · t²
3.05 m = (57.6 m / t · cos α) · t · sin α + 1/2 · g · t²
3.05 m = 57.6 m · tan 40.0° - 1/2 · 9.8 m/s² · t²
Solving for "t":
(3.05 m - 57.6 m · tan 40.0°) / - 1/2 · 9.8 m/s² = t²
t = 3.04 s
b) The ball passed through the goal posts 3.04 s after it was kicked.
a) The initial velocity can now be calculated:
57.6 m / t · cos α = v0
v0 = 57.6 m / 3.03 s · cos 40.0° = 24.8 m/s
The initial speed of the ball was 24.8 m/s
