Aplacekicker must kick a football from a point 36.0 m (about 40 yards) from the goal. half the crowd hopes the ball will clear the crossbar, which is 3.05 m high. when kicked, the ball leaves the ground with a speed of 22.8 m/s at an angle of 51.0° to the horizontal. (a) by how much does the ball clear or fall short of clearing the crossbar? (enter a negative answer if it falls short.)
(b) does the ball approach the crossbar while still rising or while falling?

a) The ball clears the crossbar by 10.6 m

b) The ball approaches the crossbar while falling

Explanation:

The position of the ball is described by the vector position r (see attached figure):

r = (x0 + v0 t cos α ; y0 + v0 t sin α + 1/2 g t²)

where:

x0 = initial horizontal position

v0 = magnitude of the initial velocity

t = time

α = launching angle  

y0 = initial vertical position

g = acceleration due to gravity (-9.8 m/s²)

The vector r is composed by rx and ry (see figure):

r = (rx ; ry)

a) Let´s find the time at which the ball flies a distance of  36.0 m. If at that time the vertical component of the vector r, ry, is equal or greater than 3.05 m, then the ball will clear the crossbar.

rx = x0 + v0 t cos α = 36.0 m

Since the origin of the system of reference is located where the kicker is, x0 = 0.

36.0 m = v0 t cos  α

36.0 m /(v0 cos  α) = t

36.0 m / (22.8 m/s * cos 51.0°) = t

t = 2.51 s

Now let´s calculate the height of the ball at that time:

ry = y0 + v0 t sin α + 1/2 g t²

Since the kicker is on the ground, y0 = 0

ry = 22.8 m/s * 2.51 s * sin 51.0° - 1/2 * 9.8 m/s² * (2.51 s)² = 13.6 m

Since the crossbar is 3.05 m high, the ball clears it by (13.6 m - 3.05 m) 10.6 m

b) Please see the figure to figure this out ;)

If the ball approaches the crossbar while still rising, the vertical component (vy) of the velocity vector will be positive. In change, if the ball approaches the crossbar while falling the vertical component of the velocity will be negative. See the figure.

The velocity vector is given by this equation:

v = (vx ; vy)

v = ( v0 cos α ; v0 sin α + g t)

Let´s see the vertical component at time t = 2.51

vy = v0 sin α + g t

vy = 22.8 m/s * sin 51.0° - 9.8 m/s² * 2.51 s

vy = -6.88 m/s

Then, the ball approaches the crossbar while falling.

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