Problem 12 - University Physics with Modern Physics

Question

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. (a) How much time is required for the football to reach the highest point of the trajectory? (b) How high is this point? (c) 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)? (d) How far has the football traveled horizontally during this time? (e) Draw \(x-t, y-t, v_x-t\), and \(v_y-t\) graphs for the motion.

Step-by-Step Solution

Answer

(a) 1.22 s; (b) 7.35 m; (c) 2.44 s; (d) 48.8 m; Graphs as described.

Step 1: Identify Known Values

The initial upward velocity \( v_{0y} \) is 12.0 m/s, and the horizontal velocity \( v_{0x} \) is 20.0 m/s. The acceleration due to gravity \( g \) is approximately 9.8 m/s².

Step 2: Calculate Time to Reach Highest Point (Part a)

To find the time to reach the highest point, use the formula \( v_y = v_{0y} - gt \). At the highest point, \( v_y = 0 \). Solve for \( t \):\[ 0 = 12.0 - 9.8t \]\[ t = \frac{12.0}{9.8} \approx 1.22 \text{ s} \]

Step 3: Calculate Maximum Height (Part b)

The maximum height \( h \) can be found using the formula \( h = v_{0y} t - \frac{1}{2} g t^2 \). Substitute \( t \) from Step 2:\[ h = 12.0 \cdot 1.22 - \frac{1}{2} \cdot 9.8 \cdot (1.22)^2 \]\[ h \approx 7.35 \text{ m} \]

Step 4: Calculate Total Time of Flight (Part c)

The time to return to the original level is twice the time to reach the highest point (because the upward and downward paths are symmetric in time):\[ t_{total} = 2 \times 1.22 \approx 2.44 \text{ s} \]

Step 5: Determine Horizontal Distance Traveled (Part d)

The horizontal distance \( d \) can be calculated using the horizontal velocity and total flight time:\[ d = v_{0x} \cdot t_{total} \]\[ d = 20.0 \times 2.44 \approx 48.8 \text{ m} \]

Step 6: Plotting Motion Graphs (Part e)

1. **Position-Time Graphs (x-t, y-t):** - The \( x-t \) graph is a straight line with a positive slope (constant velocity). - The \( y-t \) graph is a parabola opening downwards, peaking at 1.22 s.2. **Velocity-Time Graphs (v_x-t, v_y-t):** - The \( v_x-t \) graph is a horizontal line (20.0 m/s). - The \( v_y-t \) graph is a straight line that decreases linearly from 12.0 m/s to -12.0 m/s.

Key Concepts

Vertical and Horizontal Components of Motion

In projectile motion, understanding the vertical and horizontal components is crucial. These components describe how an object moves along the x-axis and y-axis. For the quarterback's throw, the football has two initial velocity components: vertical and horizontal.
The initial vertical velocity, denoted as \( v_{0y} \), is 12.0 m/s. This component is influenced by gravity. It decreases as the ball moves upwards and reaches zero at the peak. The horizontal velocity component, denoted as \( v_{0x} \), is 20.0 m/s. This remains constant because, in ideal projectile motion, there's no air resistance to slow the ball down.

• The vertical motion is affected by the acceleration due to gravity, which is approximately \( 9.8 \text{ m/s}^2 \).
• Horizontal motion maintains constant velocity due to the zero horizontal acceleration.

These components act independently, with gravity only acting on the vertical velocity component.

Time of Flight Calculations

The time of flight is the total time a projectile spends in the air. For the quarterback’s throw, time calculations can help determine various key metrics. Calculating the time to reach the highest point requires setting the vertical velocity to zero since it stops for an instant at the peak.
The formula is \( v_y = v_{0y} - gt \). Setting \( v_y = 0 \) helps find the time to the peak, which is approximately 1.22 seconds.
For the total flight time to return to its starting level, the projectile's ascent time is mirrored by its descent time. Hence, the total time of flight is double this value, yielding about 2.44 seconds.

• At maximum height, the upward and downward times are equal due to symmetry.
• This means time of flight can be calculated simply by doubling the ascent time.

This symmetry simplifies calculations and enables quick analysis of projectile motion.

Graphing Motion in Physics

Graphing motion provides a visual representation of projectile movement. It can clarify time points such as when a ball reaches its peak or returns to its original position.
The position-time graphs for the quarterback’s throw involve two key graphs: \( x-t \) and \( y-t \).
The \( x-t \) graph is straightforward—it shows a linear relationship since horizontal velocity is constant. This line is angled upwards as time progresses.
The \( y-t \) graph is more complex, showing a parabolic trajectory. Reaching the highest point at 1.22 seconds, it peaks before descending symmetrically.

• Velocity-time graphs also offer insights.
• The \( v_x-t \) remains flat at 20.0 m/s, a constant velocity over time.
• The \( v_y-t \) graph has a downward slope, starting at 12.0 m/s and descending.

These graphs visually outline the motion's dynamics, aiding in understanding the interaction of forces and velocities.

Effect of Gravity on Projectile

Gravity plays a pivotal role in projectile motion, influencing it primarily through its vertical component. It acts as a force that continuously accelerates the projectile downward, pulling it back to Earth.
In our scenario, this constant acceleration alters the football’s initial vertical velocity of 12.0 m/s. As the football rises, gravity decreases this velocity until it reaches zero at the apex. Then, as the football descends, gravity increases the velocity again, but in the opposite direction.

• Gravity causes the projectile to follow a curved trajectory rather than a straight line.
• It is responsible for the symmetrical rise and fall in projectile motion.

Understanding gravity's impact helps predict and calculate crucial traits of the projectile's path, such as its peak height and time in the air.

Symmetry in Projectile Trajectories

Projectile motion showcases symmetry that simplifies predictions and calculations regarding the object’s path.
For the football thrown by the quarterback, the time taken to ascend to its maximum height equals the time taken to descend. This symmetry means that if you know how long the projectile takes to rise, the total flight time is simply double that.
Additionally, the maximum height occurs at half the total flight time, and the velocities at corresponding points during ascent and descent are equal in magnitude but opposite in direction.

• Symmetry helps in calculating the total flight time simply by doubling the ascent time.
• It implies that vertical velocities mirror each other pre and post-peak.

This symmetry also helps in troubleshooting and refining predictions regarding how projectiles move in ideal conditions.