A place kicker must kick a football from a point 37.1 m from a goal. As a result of the kick, the ball must clear the crossbar, which is 3.05 m high. When kicked the ball leaves the ground with a speed of 19.8 m/s at an angle of 53° to the horizontal.
(a) By how much does the ball clear or fall short of clearing the crossbar?
(b) Does the ball approach the crossbar while still rising or while falling?
Vo = 19.8m/s@53Deg.
Xo = hor = 19.8cos53 = 11.92m/s.
Yo = ver = 19.8sin53 = 15.81m/s.
a. t = (Yf - Yo) / g,
t(up) = (0 - 15.81) / -9.8 = 1.613s.
d=Vo*t + 4.9*t^2 = 12.753 - 3.05=9.703m
0 + 4.9t^2 = 9.703,
t^2 = 1.98,
t(dn) = 1.41s. to fall to 3.05m.
Dh = Xo(t(up)+t(dn)),
Dh = 11.92(1.41+1.613) = 36m=hor dist.
Falls short by: 37.1 - 36 = 1.1m.
b. While falling.
a. Falls short by 3 m
To solve this problem, we need to break it down into different components. Let's tackle each part step by step:
(a) By how much does the ball clear or fall short of clearing the crossbar?
To find out how much the ball clears or falls short of clearing the crossbar, we need to determine the maximum height the ball reaches during its flight.
We can use the kinematic equation to find the maximum height (H) reached by the ball:
H = (V^2 * sin^2θ) / (2 * g),
where V is the initial velocity (19.8 m/s) and θ is the angle of launch (53°), and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting the given values into the equation:
H = (19.8^2 * sin^2(53°)) / (2 * 9.8)
Now, evaluate this expression to find the maximum height.
Next, we need to calculate the horizontal distance covered by the ball. We can use the following equation:
R = (V^2 * sin(2θ)) / g,
where R is the range (horizontal distance).
Substituting the values:
R = (19.8^2 * sin(2 * 53°)) / 9.8
Now, evaluate this expression to find the horizontal distance.
Finally, subtract the height of the crossbar (3.05 m) from the maximum height (H) to determine how much the ball clears or falls short of the crossbar.
If the result is positive, the ball clears the crossbar. If it's negative, the ball falls short of clearing the crossbar.
(b) Does the ball approach the crossbar while still rising or while falling?
By analyzing the angle of projection (53°) and considering that the ball reaches a maximum height, we can conclude that the ball approaches the crossbar while falling.