A rookie quarterback throws a football with an initial upward velocity component of 12.0 m/s and a horizontal velocity component of 20.0 m/s. Ignore air resistance. How far has the football traveled horizontally during this time?

Verified step by step guidance

1. Identify the horizontal motion of the football as uniform motion since there is no air resistance affecting it. The horizontal velocity remains constant at 20.0 m/s.
2. Determine the time of flight for the football. Since the vertical motion is independent of the horizontal motion, use the vertical component to find the time. The vertical motion is subject to gravity, so use the kinematic equation: $v$_f = v_i + at, where $v$_f is the final vertical velocity (0 m/s at the peak), $v$_i is the initial vertical velocity (12.0 m/s), and $a$ is the acceleration due to gravity (-9.8 m/s²).
3. Solve for the time to reach the peak using the equation: $t=\; (v$_f - v_i)/a. Since the time to reach the peak is half of the total time of flight, multiply the result by 2 to find the total time the football is in the air.
4. Calculate the horizontal distance traveled using the formula for distance in uniform motion: $d=v$_ht, where $v$_h is the horizontal velocity (20.0 m/s) and $t$ is the total time of flight calculated in the previous step.
5. Substitute the values into the distance formula to find the horizontal distance traveled by the football during its flight.

Key Concepts

Projectile Motion

Projectile motion involves the motion of an object thrown into the air, subject to only the acceleration of gravity. It can be analyzed by separating the motion into horizontal and vertical components. The horizontal motion is uniform, while the vertical motion is uniformly accelerated due to gravity.

Horizontal Velocity

Horizontal velocity refers to the constant speed at which an object moves along the horizontal axis. In projectile motion, this component remains unchanged throughout the flight, as there is no horizontal acceleration when air resistance is ignored. It is crucial for calculating the horizontal distance traveled.

Time of Flight

Time of flight is the total time an object remains in the air during projectile motion. It is determined by the vertical motion, specifically the time taken for the object to reach its peak and return to the ground. This duration is essential for calculating how far the object travels horizontally.