Question

Jake attempts a 36-yard field goal in a football game. For his attempt to be a success, the football needs to pass through the uprights and over the crossbar that is 10 feet above the ground. Jake kicks the ball from the ground with an initial velocity of 68 feet per second, at an angle of 34° with the horizontal. What is true of Jake's attempt?

The kick is good! The ball passes about 10 feet above the crossbar.

The kick is good! The ball clears the crossbar by approximately 14 feet.

The kick is good! The ball clears the crossbar by about 4 feet.

The kick is not successful. The ball is 2 feet too low.

Answer 1

Question:

1) The kick is good! The ball passes about 10 feet above the crossbar.

2) The kick is good! The ball clears the crossbar by approximately 14 feet.

3) The kick is good! The ball clears the crossbar by about 4 feet.

4) The kick is not successful. The ball is 2 feet too low.

Answer:

The correct option is;

3) The kick is good! The ball clears the crossbar by about 4 feet.

Explanation:

Here we have

Horizontal distance from goal, x = 36 yards = 108 ft

Vertical distance of goal, y = 10 ft

Angle of elevation of ball = 34°

Initial velocity of ball, v₀ = 68 ft/s

∴ Vertical component of velocity =
`v_y` = v₀×sin(θ₀)×t - g×t

Horizontal component of velocity = vₓ = 68×cos(34) = 56.375 ft/s

The equation for projectile motion is as follows

x = x₀ + v₀×cos(θ₀)×t

y = y₀ + v₀×sin(θ₀)×t - 1/2×g×t²

Where:

x₀ = Initial horizontal displacement of the ball = 0

y₀ = Initial vertical displacement of the ball = 0

t = Time of flight of the ball

Therefore;

108 = 0 + 68×cos(34)×t which gives;

t = 108/56.375 = 1.916 seconds

Hence, y = y₀ + v₀×sin(θ₀)×t - 1/2×g×t² gives;

y = 0 + 68×sin(34)×1.916 - 1/2×32.2×1.916²

y = 72.85 - 59.09 = 13.76 ft, hence the vertical location of the ball at 36 yard is 13.76 ft where the crossbar is at 10 ft hence the ball clears the crossbar by 13.76 ft - 10 ft = 3.76 ft ≈ 4 ft

Therefore, the kick is good! The ball clears the crossbar by about 4 feet.