The longest recorded pass in an NFL game traveled 83 yards...

The longest recorded pass in an NFL game traveled 83 yards in the air from the quarterback to the receiver. Assuming that the pass was thrown at the optimal $45^{\circ}$ angle, what was the speed at which the ball left the quarterback's hand?

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Key Concepts

Projectile Motion

Projectile motion is the two-dimensional motion of an object that is launched into the air and moves under the influence of gravity alone. It involves both horizontal and vertical components, which are analyzed separately to understand the overall trajectory.

Range Formula

The range formula provides the horizontal distance traveled by a projectile when launched with an initial velocity at a given angle. Under ideal conditions and neglecting air resistance, this formula connects the initial speed, the launch angle, and the acceleration due to gravity to determine how far the projectile will go.

Optimal Launch Angle

The optimal launch angle of 45° maximizes the horizontal distance a projectile can cover when air resistance is neglected. This is because at 45°, the product of the horizontal and vertical components of the initial velocity is optimized, leading to the maximum range.

Acceleration due to Gravity

Gravitational acceleration is a constant that describes the rate at which objects accelerate downwards due to the Earth’s gravity. In projectile motion problems, it plays a key role in determining how long the projectile remains in the air and, consequently, how far it travels horizontally.

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