Question

Sanjay attempts a 50-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 64 feet per second, at an angle of 34° with the horizontal.Is Sanjay's attempt successful? If not, how many feet too low is the ball?

Answer 1

Let us draw a sketch to understand the situation

We will use some rules here

`\begin{gathered} v_x=vcos\theta=64cos(34) \\ d_x=v_xt=64cos(34)t \end{gathered}`

Since the horizontal distance is 50 yards

Since 1 yard = 3 feet, then

`d_x=50*3=150feet`

We will use it to find the time t

`\begin{gathered} d_x=150 \\ 64cos(34)t=150 \\ t=(150)/(64cos(34))\text{ s} \end{gathered}`

Now, we will find the vertical distance (h) by using this rule

`\begin{gathered} v_y=vsin\theta=64sin(34) \\ d_y=h=v_yt-(1)/(2)at^2=64sin(34)t-(1)/(2)(32)t^2 \end{gathered}`

Note that: a is the acceleration of gravity which is 32 ft/s^2

We will substitute t by its value

`h=64sin(34)((150)/(64cos(34)))-16((150)/(64cos(34)))`

We can simplify it by using sin34/cos34 = tan34, and 1/cos34 = sec34

But I will put it on the calculator to find the final answer

`h=55.94\text{ ft}`

Since the height of the crossbar is 10 feet, then

Sanjay's attempt successful