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 much time (after it is thrown) is required for the football to return to its original level? How does this compare with the time calculated in part (a)?

Verified step by step guidance

1. First, identify the vertical motion of the football. The initial upward velocity component is given as 12.0 m/s. Use the kinematic equation for vertical motion: $v=u+\mathrm{at}$, where $v$ is the final velocity (0 m/s at the peak), $u$ is the initial velocity (12.0 m/s), $a$ is the acceleration due to gravity (-9.8 m/s²), and $t$ is the time to reach the peak.
2. Solve the equation $v=u+\mathrm{at}$ for $t$ to find the time to reach the peak. Rearrange to get $t=-u/a$. Substitute $u=12.0$ m/s and $a=-9.8$ m/s².
3. The time to reach the peak is half the total time for the football to return to its original level. Therefore, multiply the time to reach the peak by 2 to find the total time of flight.
4. Compare the total time calculated here with the time calculated in part (a). Ensure you understand the context of part (a) to make a meaningful comparison.
5. Reflect on the symmetry of projectile motion: the time to ascend to the peak is equal to the time to descend back to the original level, assuming no air resistance.

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.

Vertical Motion Under Gravity

The vertical component of projectile motion is influenced by gravity, which causes a constant acceleration of approximately 9.81 m/s² downward. The time taken for an object to return to its original level can be calculated using kinematic equations, considering the initial vertical velocity and gravitational acceleration.

Time of Flight

Time of flight refers to the total time an object remains in the air during projectile motion. For symmetrical projectile motion, the time to reach the peak is equal to the time to descend back to the original level. This can be calculated by doubling the time taken to reach the peak, which is found using the initial vertical velocity and gravity.