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Angular velocity is a crucial concept when studying rotational motion. It describes how quickly an object spins around an axis. In our exercise, the football spins with an angular velocity of \(7.7 \text{ rev/s}\), which means it makes 7.7 complete turns around its central axis every second. This rotational speed remains constant throughout the ball's flight unless any external forces act on it to change its rate of spin.
Angular velocity is generally measured in radians per second (rad/s), but in this problem, it's given in revolutions per second (rev/s). Remember that one full revolution equals \(2\pi\) radians. Thus, if needed, you can convert between these units by multiplying the number of revolutions by \(2\pi\) to get radians.
Some key points to understand about angular velocity:

• It is a vector quantity, meaning it has both magnitude and direction. The direction is determined by the right-hand rule.
• For a perfectly spiraling football, the axis is along the length of the ball itself.
• In our context, angular velocity helps us calculate how many spins the ball completes during its flight.

Time of flight is a fundamental aspect of projectile motion, which explains how long an object remains airborne. When the projectile launches and lands at the same height, we can use a special formula to determine this time. The equation is \( t = \frac{2v_0 \sin \theta}{g} \). Here, \(v_0\) represents the initial speed, \(\theta\) is the launch angle, and \(g\) is the acceleration due to gravity.
In our football problem, we have:

• Initial speed \(v_0 = 19 \text{ m/s}\)
• Launch angle \(\theta = 55^\circ\), which is converted to radians for calculations.
• Gravity \(g = 9.8 \text{ m/s}^2\).

Plugging these values into the formula gives a time of flight, \(t \approx 3.18 \text{ seconds}\). This helps us understand how long the ball stays in the air from the moment it leaves the quarterback's hand until it is caught. With time of flight, we can explore further calculations like horizontal distance or, as in this case, the number of revolutions.

The number of revolutions gives us a sense of how many complete turns an object makes as it moves through the air. In the context of our spiraling football, it allows us to visualize its spinning motion from launch to catch.
To find the number of revolutions a projectile makes while in the air, multiply the angular velocity by the time of flight. Using the equation \( \text{Number of Revolutions} = \text{Angular Velocity} \times \text{Time of Flight} \), we have:

• Angular velocity is \(7.7 \text{ rev/s}\)
• Time of flight is \(3.18 \text{ seconds}\)

Multiplying these gives us approximately \(24.486\) revolutions.
Keep these points in mind:

• The longer the time of flight, the more revolutions the object tends to make.
• The higher the angular velocity, the more revolutions occur in a given period.
• This calculation helps understand how angular motion relates to projectile motion in practical scenarios like sports or physics experiments.