Question

Sanjay attempts a 49-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.Sanjay kicks the ball from the ground with an initial velocity of 73 feet per second, at an angle of 34° with the horizontal.What is true of Sanjay's attempt?A. The kick is not successful. The ball is approximately 8 feet too low.B. The kick is not successful. The ball is approximately 5 feet too low.C. The kick is good! The football clears the crossbar by approximately 5 feet.D. The kick is not successful. The ball is approximately 2 feet too low.

Answer 1

To determine what is true about Sanjay's attempt, the first thing we should do is calculate the vertical component of the initial velocity of the ball.

To do so, we must remember the following relation:

`\begin{gathered} V=V_0 \\ V_(vertical)=V_0\cdot sin(\theta) \\ V_(horizontal)=V_0\cdot cos(\theta) \end{gathered}`

The horizontal and vertical equations are linear and quadratic, respectively, as follows:

`\begin{gathered} y=y_0+V_(0-Vertical)\cdot t+(1)/(2)at² \\ x=x_0+V_(0-Horizontal)\cdot t \end{gathered}`

Isolating t in the second equation, for x0 = 0, because it is going to be the initial position, then we substitute it in the first equation:

`\begin{gathered} t=(x)/(V_(0-Horizontal))\Rightarrow y=y_0+x(V_(0-Vertical))/(V_(0-Horizontal))+(1)/(2)a((x)/(V_(0-Horizontal)))² \\ \\ y=x(V_(0-Vertical))/(V_(0-Horizontal))+(1)/(2)a((x)/(V_(0-Horizontal)))² \end{gathered}`

We used y0 = 0, because it starts from the ground.

Now, we know that x = 49 yd = 147 ft, let's assume gravity equal to -32.2 yd/s², and the other values are given. By substituting the values, we will find the value of the height of the ball when it finishes traveling the 49-yard field, as follows:

`y=147\cdot(V_0sin(34°))/(V_0cos(34°))-(1)/(2)32.2((147)/(73\cdot cos(34°)))²`

Performing the calculation, we have:

`\begin{gathered} y=147*(V_(0)s\imaginaryI n(34))/(V_(0)cos(34))-(1)/(2)*32.2((147)/(73cos(34)))² \\ \\ y=99.15-16.1\cdot(2.42)² \\ \\ y=99.15-94.29 \\ \\ y=4.9\text{ ft} \end{gathered}`

From the solution developed above, we are able to conclude that the ball passed 5 ft below the required height to be successful.

For this reason, the corect answer for the question is:

B. The kick is not successful. The ball is approximately 5 feet too low.