To determine the horizontal displacement of the football, we can break the motion into horizontal (x-axis) and vertical (y-axis) components and analyze them separately.

Given:

• Initial velocity: $\mathbf{v}_0 = 7.00 \, \hat{i} + 23.0 \, \hat{j} \, \text{m/s}$
  • Horizontal velocity: $v_{0x} = 7.00 \, \text{m/s}$
  • Vertical velocity: $v_{0y} = 23.0 \, \text{m/s}$

Key Steps to Solve:

1. Time of flight (T):
   The football follows a parabolic trajectory. The time of flight depends on the vertical motion, so we can use kinematic equations for the vertical direction to find the total time.

   In the vertical direction, acceleration $a_y$ due to gravity is $-9.81 \, \text{m/s}^2$. Using the following kinematic equation for vertical displacement:

   $y(t) = v_{0y} t + \frac{1}{2} a_y t^2$

   When the ball returns to the ground, $y(t) = 0$. Solving for $t$:

   $0 = (23.0 \, \text{m/s}) t + \frac{1}{2} (-9.81 \, \text{m/s}^2) t^2$

   This simplifies to:

   $t \left( 23.0 - 4.905 t \right) = 0$

   The non-zero solution is:

   $t = \frac{23.0}{4.905} \approx 4.69 \, \text{seconds}$

   Therefore, the total time of flight (T) is 4.69 seconds.

2. Horizontal displacement (range):
   The horizontal velocity remains constant because there's no acceleration in the horizontal direction (ignoring air resistance). The horizontal displacement $x$ can be calculated using:

   $x = v_{0x} \cdot T$

   Substituting the known values:

   $x = (7.00 \, \text{m/s}) \cdot (4.69 \, \text{seconds}) = 32.83 \, \text{m}$

Final Answer:

The horizontal displacement of the football is 32.83 meters.

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Would you like more details on the steps or related concepts? Here are some further questions to explore:

1. How does gravity affect projectile motion?
2. What happens if the initial velocity is increased?
3. How would air resistance change the horizontal displacement?
4. How do we calculate the maximum height of the football?
5. What would happen if the football were kicked at a different angle?

Tip: In projectile motion, the horizontal velocity remains constant (if we neglect air resistance), while the vertical velocity is affected by gravity.