In a game of American football, a placekicker must kick a football from a point

Question

In a game of American football, a placekicker must kick a football from a point 36.0 m (about 40 yards) from the goal, and 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 24.0 m/s at an angle of 48.0° to the horizontal.

(a) By how much does the ball clear or fall short of clearing 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?

Explanation / Answer

Resolve the velocity of the ball into horizontal and vertical components respectively: 24cos48 and 24sin48 respectively
Horizontal: 16.06 m/s
Vertical: 17.84 m/s

(a) First ignore the goalpost and calculate the time taken of the projectile to touch down after being kicked:

Vertically, displacement is 0 since its final destination is on same level as its starting point.

s = ut + 0.5gt^2
0 = 17.84t - 0.5(9.8)t^2 Taking upwards as positive.
0 = t(17.84 - 0.5(9.8)t) To satisfy the LHS, t = 0 or t = 17.84/(0.5x9.8) = 3.64sec. This is correct because at t = 0, the ball has not been kicked and thus, it will not have any displacement. So, t = 3.64sec; the time taken for the entire journey. Now horizontally, let us calculate the distance travelled. s = vt s = 16.06 x 3.64 s = 58.5m Referring back to the goalpost, the goalpost is 36.0m away from the starting point. Since the ball will land 58.5m away from the starting point, the ball will clear the goalpost by (58.5-36) = 22.5m (b) Now let us find the peak of the projectile motion; if it is before the goalpost, it will seen as falling towards the goalpost and vice versa. time taken to reach peak is halve of the time taken for the entire journey = 3.64/2 = 1.82sec This, translated to horizontal distance; s = 16.06 x 1.82 = 29.2m Since this is before the goalpost, the ball at the peak of projectile will start to fall towards the ground. So the ball will approach the crossbar as it is falling.

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