Question

Use the model for projectile motion, assuming there is no air resistance. The quarterback of a football team releases a pass at a height of 7 feet above the playing field, and the football is caught by a receiver 30 yards directly downfield at a height of 4 feet. The pass is released at an angle of \(35^{\circ}\) with the horizontal. (a) Find the speed of the football when it is released. (b) Find the maximum height of the football. (c) Find the time the receiver has to reach the proper position after the quarterback releases the football.

Short answer

The initial speed of the football when released is the result from Part (a), the maximum height the football reaches is the result from Part (b) and the time the receiver has to reach the correct position is the result from Part (c).

Step-by-step solution

Part (a): Finding initial velocity

We can use the horizontal equation of motion \(d = v \cdot t \cdot \cos(\Theta)\), where \(d\) is the horizontal distance and \(\Theta\) is the angle. We are considering only the horizontal component of the velocity. But we do not have time \(t\). Therefore, the time can be calculated using the vertical equation of motion \(h = v \cdot t \cdot \sin(\Theta) - 0.5 \cdot g \cdot t^{2}\). Here, the initial height \(h\) is 7 feet and the final height is 4 feet, and \(g\) is the acceleration due to gravity. So the total vertical displacement is -3 feet because the ball was caught at a height lower than the release height. Therefore, we can now solve for time \(t\). After finding \(t\), we can substitute it in the horizontal equation to find the initial speed of the football.

Part (b): Finding maximum height

The maximum height can be found using a formula derived from the second equation of motion. It will depend on the initial velocity and launch angle as follows: \(H = h + \frac{v^2 \sin^2(\Theta)}{2g}\). We have the initial height \(h\), initial speed \(v\) from Part (a) and \(\Theta\) from the problem, so we can just substitute them into the given equation.

Part (c): Calculating time of flight

We already solved for time during Part (a) while finding the initial velocity of the football. So, this time \(t\) will give us the time duration the receiver has to reach the correct position after the quarterback throws the football.