10 pts A football kicker can give the ball an initial speed of vo 29.7 m/s. In o

Question

10 pts A football kicker can give the ball an initial speed of vo 29.7 m/s. In order to score field goal, the ball must go over the cross bar that is at a height of h-3.44 m above the ground. If the kicker kicks the ball from a point d-33.4 m from the goalposts, find the minimum and maximum angles at which he must kick the ball. Give the difference between the minimum and maximum. (Hint: The minimum angle is where the ball just clears the cross bar on its way up and the maximum angle is that where the ball just clears the cross bar on its way down.) a Ve

Explanation / Answer

Given,

v0 = 29.7 m/s ; h = 3.44 m ; d = 33.4 m

We know that

d = vo cos(theta) t

t = d/(v0 cos(theta))

also, Y = vy t + 1/2 ay t^2

y = v0 sin(theta) x d/v0 cos(theta) - 1/2 g d^2/v0^2 cos^2(theta)

y = tan(theta) - gd^2/2 v0^2 cos^2(theta)

from trigonometric identity, 1/cos^2(theta) = 1 + tan^2(theta)

Y = x tan(theta) - gd^2/2 v0^2 - g d^2 tan^(theta)/2 v0^2

g d^2/2 v0^2 tan^2(theta) - d tan(theta) + h + gd^2/v0^2

1/2 gd^2/v0^2 = 0.5 x 9.8 x 33.4^2/29.7^2 = 6.2

6.2 tan^2(theta) - 33.4 tan(theta) + 9.64 = 0

tan(theta) = 5.081 ; 0.306

theta = 78.87 deg ; 17.01 deg

Hence, (theta-min) = 17.01 deg ; theta-max = 78.87 deg

difference = 78.87 - 17.01 = 61.86 deg

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