The quarterback of a football team releases a pass at a height of 7 feet above the playing field, and the football is caught at a height of 4 feet, 30 yards directly downfield. The pass is released at an angle of 35 degrees with the horizontal. The parametric equations for the path of the football are given by x=0.82vot and y=7+0.57vot-16t^2 where vo is the speed of the football (in feet per second) when it is released. Find the speed of the football when it is released.
I'll just use v for readability. We need to solve
Since the ball was caught 30 yards (90 feet) downfield,
.82vt = 90
t = 109.76/v
Now, knowing what t is in terms of v, we can solve
7+0.57v(109.76/v)-16(109.76/v)^2 = 4
v = 54.22
.4067
We can solve this problem by using the given information and the parametric equations for the path of the football.
From the given information, we know that the pass is released at a height of 7 feet and caught at a height of 4 feet. So, the change in height is 4 - 7 = -3 feet.
We can use the equation for the vertical motion of the football to find the time it takes for the football to travel from the release point to the catching point:
y = 7 + 0.57vot - 16t^2
Since the release and catching points have the same horizontal position, we can set x = 0 on the path of the football. Thus, x = 0.82vot = 0, which implies that vot = 0.
Substituting these values into the equation for y gives:
4 = 7 + 0.57(0) - 16t^2
Rewriting this equation, we have:
-3 = -16t^2
Dividing both sides of the equation by -16, we get:
t^2 = 3/16
Taking the square root of both sides, we have:
t = +/- sqrt(3/16)
Since time cannot be negative in this context, we can discard the negative square root.
So, t = sqrt(3/16) seconds.
Now, we can use the equation for horizontal motion to find the speed of the football when it is released:
x = 0.82vot
Setting x = 30 yards = 90 feet and solving for vo gives:
90 = 0.82vo(sqrt(3/16))
Dividing both sides of the equation by 0.82(sqrt(3/16)), we get:
vo = 90 / [0.82(sqrt(3/16))]
Simplifying this expression, we find:
vo ≈ 29.5 feet per second
Therefore, the speed of the football when it is released is approximately 29.5 feet per second.
To find the speed of the football when it is released, we can use the information given in the problem:
1. We know that the quarterback releases the pass at a height of 7 feet above the playing field, and the football is caught at a height of 4 feet. This means that the vertical distance the football travels is 7 - 4 = 3 feet.
2. The pass is released at an angle of 35 degrees with the horizontal. This angle is measured from the ground. We can use this angle to determine the initial vertical and horizontal velocities of the football.
3. The horizontal distance between the release point and the catch point is given as 30 yards, which is equivalent to 90 feet.
Now, let's break down the problem and solve it step by step:
Step 1: Determine the initial vertical and horizontal velocities of the football.
The vertical component of the initial velocity is given by:
Voy = vo * sin(θ)
The horizontal component of the initial velocity is given by:
Vox = vo * cos(θ)
Where:
Voy is the initial vertical velocity,
Vox is the initial horizontal velocity,
vo is the speed of the football when it is released, and
θ is the angle of release (35 degrees).
Step 2: Determine the time taken to reach the catch point.
We know that the horizontal distance traveled by the football is 90 feet.
Using the equation for horizontal distance, we have:
x = Vox * t
where x is the horizontal distance, Vox is the initial horizontal velocity, and t is time.
Substituting the values, we have:
90 = vo * cos(35) * t
Solving for t, we get:
t = 90 / (vo * cos(35))
Step 3: Determine the vertical position of the football at time t.
Using the equation for vertical position, we have:
y = 7 + Voy * t - 16 * t^2
where y is the vertical position, Voy is the initial vertical velocity, and t is time.
Substituting the values, we have:
4 = 7 + vo * sin(35) * t - 16 * t^2
Step 4: Determine the vertical time of flight.
We know that the vertical distance traveled by the football is 3 feet. This occurs during the time of flight.
Using the equation for vertical distance, we have:
y = Voy * t - 16 * t^2
Substituting the values, we have:
3 = vo * sin(35) * t - 16 * t^2
Step 5: Solve the system of equations.
From Steps 3 and 4, we have a system of two equations with two unknowns (vo and t). Solve this system of equations to find the values of vo and t.
Once you solve the system of equations, you will get the values of vo and t. Then you can substitute these values into any of the original equations (e.g., horizontal distance equation) to find the speed of the football when it is released (vo).