A kicker is setting up for a field goal. The goal is 120 feet away and 10 feet high. The kicker can give an initial speed of 75 ft/s to the ball. What range of angles of elevation are possible to kick the ball so it passes over the goal?
To determine the range of angles of elevation that will allow the ball to pass over the goal, we need to calculate the trajectory of the ball. This can be done using projectile motion equations.
The motion of the ball in the horizontal direction is not affected by gravity, so we can use the formula:
Range = (Initial Velocity * Time of Flight) * cos(θ)
where Range is the distance traveled in the horizontal direction, Initial Velocity is the initial velocity of the ball, Time of Flight is the total time the ball is in the air, and θ is the angle of elevation.
In this case, the Range is given as 120 feet, and the Initial Velocity is 75 ft/s.
The motion of the ball in the vertical direction is affected by gravity, so we can use the formula:
Height = (Initial Velocity * Time of Flight) * sin(θ) - (0.5 * g * (Time of Flight)^2)
where Height is the maximum height reached by the ball, g is the acceleration due to gravity (approximately 32 ft/s^2), and all other variables are the same as before.
In this case, the Height is given as 10 feet.
To find the range of angles, we need to solve these two equations simultaneously. Here's the step-by-step process:
1. Let's start with the equation for the Range:
120 = (75 * Time of Flight) * cos(θ)
2. Next, let's solve for the Time of Flight:
Time of Flight = 120 / (75 * cos(θ))
3. Now, substitute this expression for Time of Flight in the equation for the Height:
10 = (75 * (120 / (75 * cos(θ)))) * sin(θ) - (0.5 * 32 * ((120 / (75 * cos(θ))))^2)
4. Simplify the above equation:
10 = (96 / cos(θ)) * sin(θ) - (2.56 * ((120 / (75 * cos(θ))))^2)
5. Rearrange the equation to isolate cos(θ):
cos(θ) = 96 / (10 * sin(θ) + 2.56 * (120 / (75 * cos(θ)))^2)
6. This equation cannot be solved directly for cos(θ), so we need to use numerical methods or a graphing calculator to find the range of values for cos(θ) that satisfy the equation.
By graphing the equation or using numerical methods, we can find the range of angles of elevation that will allow the ball to pass over the goal.