Question

Sam attempts a 42-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.
Sam kicks the ball from the ground with an initial velocity of 70 feet per second, at an angle of 50° with the horizontal.


Which statement is true of Sam's attempt?



The kick is good! The football clears the crossbar by approximately 6 feet.


The kick is good! The ball clears the crossbar by nearly 2 feet.


The kick is good! The ball clears the crossbar by nearly 15 feet.


The kick is not successful. The ball hits the ground short of the crossbar.

Answer 1

Answer:

The correct option is;

The kick is good! The ball clears the crossbar by nearly 15 feet.

Step-by-step explanation:

Here we have;

Velocity of ball = 70 ft/s

Angle of motion = 50°

Therefore;

Vertical component of velocity, $v_y$ = v×sinθ = v×sin×50 = 53.62 ft/s

$v_y$ = u$_y$ - g·t

At maximum height $v_y$ = 0, therefore;

u$_y$ = g·t and

t =  u$_y$/g

Where:

u$_y$ = Initial vertical velocity = 53.62 ft/s

g = Acceleration due to gravity = 32.1740 ft/s²

∴ t = 53.62/32.1740 = 1.66666 s

Total time of flight = 2 × 1.66666 = 3.33332 s

Max height is given by the following relation;

v$_y$² = u$_y$² - 2·g·s

v$_y$ = 0 ft/s at maximum height, therefore;

u$_y$² = 2·g·s

53.62² = 2×32.1740×s

s = 53.62² ÷(2×32.1740) = 44.69 ft

Horizontal velocity = 70×cos(50) = 44.995 ft/s

Hence the time it takes the ball to reach 42 yards is given by the following relation;

42 yards = 126 ft

$Time, t = \frac{Distance}{Velocity} = \frac{126}{44.995} = 2.8 \, s$

Height of ball at 2.8 s is given by;

Direction of ball at 2.8 s = downwards

Hence time  after peak = 2.8 - 1.666 = 1.1336429 s

Position at 2.8 s is given by

s = u$_y$ ·t + 1/2·g·t²

u$_y$ = 0 ft/s as ball is on second half of flight

∴ s = 1/2·g·t² = 1/2×32.1740×1.1336429²= 20.674 ft below maximum height

∴ Height above ground =  Maximum height - 20.674 ft = 44.69 - 20.674

Height above ground = 24.01 ft

Hence the ball clears the crossbar by 24.01 - 10 = 14.01 ft

The best option is therefore;

The kick is good! The ball clears the crossbar by nearly 15 feet.