a place kicker must kick a football from a
Lint 36 meters from the goal. As a result of the kick, the ball must clear the crossbar, which is 3.04 meters high. When kicked, the ball leaves the ground with a speed of 20 meters per second at an angle of 50 degrees to the horizontal.
By how much does the ball clear or fall short of clearing the crossbar?
Does the ball approach the crossbar while still rising or falling?
Vo = 20m/s[50o]
Xo = 20*cos50 = 12.9 m/s
Yo = 20*sin50 = 15.3 m/s.
V = Vo + g*Tr = 0 @ max Ht.
Tr = -Yo/g = -15.3/-9.8 = 1.56 s. = Rise
time.
h = Yo*Tr + 0.5g*Tr^2
h max = 15.3*1.56 - 4.9*1.56^2=11.9 m.
h = 0.5g*t^2 = 11.9-3.04 = 8.90 m.
4.9t^2 = 8.90
t^2 = 1.82
Tf = 1.35 s. = Fall time.
a. Dx = Xo*(Tr+Tf) = 12.9 * (1.56+1.35) = 37.5 m.
D = 37.5-36 = 1.5 m to spare
b. Falling.
To determine by how much the ball clears or falls short of clearing the crossbar, we need to analyze the vertical motion of the ball.
1. First, let's find the initial vertical velocity (Viy) of the ball. Given the initial velocity (V0) of 20 m/s and the launch angle (θ) of 50 degrees, we can use trigonometry to find the vertical component of the initial velocity:
Viy = V0 * sin(θ)
= 20 * sin(50°)
2. Next, let's find the time it takes for the ball to reach its maximum height (t_max), using the equation:
Vf = Viy - g * t_max
Since at the maximum height the vertical velocity (Vf) becomes zero, we can solve for t_max:
0 = Viy - g * t_max
t_max = Viy / g
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
3. Now, let's find the maximum height (H_max) reached by the ball:
H_max = Viy^2 / (2 * g)
4. Finally, let's find the total time of flight (T) for the ball, which is the time it takes for the ball to reach the crossbar:
T = 2 * t_max
Now, we can calculate the answers to the questions:
To determine if the ball clears or falls short of the crossbar, we subtract the height of the crossbar (3.04 meters) from the maximum height (H_max).
Clearance = H_max - 3.04 meters
If the clearance is positive, the ball clears the crossbar. If the clearance is negative, it falls short of clearing the crossbar.
To determine if the ball approaches the crossbar while still rising or falling, we compare the clearance with zero:
If the clearance is positive (greater than 0), the ball is still rising when it reaches the crossbar.
If the clearance is negative (less than 0), the ball is falling when it reaches the crossbar.