Tom Brady passes a football – at a velocity of 50 ft/s at an angle of 40 degrees to the horizontal – toward an intended receiver 30 yd straight downfield. The pass is released 5 ft above the ground. Assume that the receiver is stationary, is isolated and that he can catch the ball from the ground (a shoestring catch) up to a height of 7 ft when the ball arrives at his position. Will the pass be completed? If not will it be short or long? If the conditions (launch height, distance to receiver, height for catch) are exactly the same but the quarterback can only launch the football at 45 degrees to the horizontal what is the range of speeds he must give the ball at launch so as to make a completion?

Question

Tom Brady passes a football – at a velocity of 50 ft/s at an angle of 40 degrees to the horizontal – toward an intended receiver 30 yd straight downfield. The pass is released 5 ft above the ground. Assume that the receiver is stationary, is isolated and that he can catch the ball from the ground (a shoestring catch) up to a height of 7 ft when the ball arrives at his position. Will the pass be completed? If not will it be short or long? If the conditions (launch height, distance to receiver, height for catch) are exactly the same but the quarterback can only launch the football at 45 degrees to the horizontal what is the range of speeds he must give the ball at launch so as to make a completion?

Answer 1

using the data, first find the time of flight using horizontal velocity.

Then, knowing the time in air, find the final height.

Answer 2

To determine if the pass will be completed, we need to analyze the motion of the football and see if it reaches the receiver's catchable height (up to 7 ft) when it reaches his position. Let's break down the problem into different steps:

Step 1: Convert units
The given velocity of 50 ft/s needs to be converted into yards per second (since the distance is given in yards). 1 yard is equal to 3 feet, so 50 ft/s is equal to 16.67 yd/s.

Step 2: Initial vertical and horizontal velocities
Given that the pass is released at a 40-degree angle to the horizontal, we can find the initial vertical and horizontal velocities of the football using trigonometry.
Vertical velocity (V_y): V_y = V * sin(theta) = 16.67 yd/s * sin(40 degrees)
Horizontal velocity (V_x): V_x = V * cos(theta) = 16.67 yd/s * cos(40 degrees)

Step 3: Time of flight
To determine the time of flight (t), which is the time it takes for the football to reach the receiver, we can use the equation of motion:
h = (1/2) * g * t^2 + V_y * t + h_0
where:
h = target height (7 ft)
g = acceleration due to gravity (32.2 ft/s^2)
h_0 = initial height (5 ft)

We can rearrange the equation to find t:
(1/2) * g * t^2 + V_y * t + h_0 - h = 0

Step 4: Quadratic equation
We can solve the quadratic equation to find the time of flight. The quadratic equation is:
(1/2) * g * t^2 + V_y * t + h_0 - h = 0
By plugging in the values for g, V_y, h_0, and h, we can solve for t.

Step 5: Calculate horizontal distance
Once we have the time of flight (t), we can calculate the horizontal distance (x) the football travels using:
x = V_x * t

Step 6: Compare the horizontal distance to the distance to the receiver
If the horizontal distance (x) is equal to or greater than the distance to the receiver (30 yd), the pass will be completed. If it is shorter, the pass will fall short and not be completed.

Now let's go through the steps again with a launch angle of 45 degrees and find the range of speeds needed for completion.

Repeat steps 2 to 4 with a launch angle of 45 degrees, and find the time of flight (t). Then calculate the horizontal distance (x) using V_x * t. Adjust the launch speed to find the minimum required speed that results in x being equal to or greater than the distance to the receiver (30 yd).

By following these steps, we can determine if the pass will be completed and find the necessary speed for completion in the case of a 45-degree launch angle.