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We are given a problem where a football is kicked with an initial velocity of \(\vec{v} = (7.00 \, \text{m/s}) \hat{x} + (23.0 \, \text{m/s}) \hat{y}\), and we are asked to determine the horizontal displacement (denoted as \(dx\)).

Here’s how we can approach the problem:

### 1. Break the problem into components:
- The horizontal velocity \(v_x = 7.00 \, \text{m/s}\).
- The vertical velocity \(v_y = 23.0 \, \text{m/s}\).
- The formula for the horizontal displacement is given as \(dx = v_x \cdot t\).
- For the vertical motion, we can use the kinematic equation \(dy = v_y \cdot t + \frac{1}{2} g t^2\), where \(g = 9.8 \, \text{m/s}^2\) is the acceleration due to gravity.

### 2. Determine the time of flight \(t\):
- In projectile motion, the time of flight is determined by the vertical component. The football reaches the maximum height when the vertical velocity becomes zero, and the total time of flight is twice the time taken to reach the maximum height.
- We can use the formula:
  \[
  v_y = v_{y0} - g \cdot t
  \]
  At the peak, \(v_y = 0\), so:
  \[
  0 = 23.0 \, \text{m/s} - 9.8 \, \text{m/s}^2 \cdot t_{\text{up}}
  \]
  Solving for \(t_{\text{up}}\):
  \[
  t_{\text{up}} = \frac{23.0 \, \text{m/s}}{9.8 \, \text{m/s}^2} \approx 2.35 \, \text{seconds}
  \]
  The total time of flight is \(t = 2 \cdot t_{\text{up}} \approx 4.7 \, \text{seconds}\).

### 3. Calculate the horizontal displacement:
Using the formula \(dx = v_x \cdot t\), where \(v_x = 7.00 \, \text{m/s}\) and \(t = 4.7 \, \text{seconds}\):
\[
dx = 7.00 \, \text{m/s} \cdot 4.7 \, \text{seconds} = 32.9 \, \text{meters}
\]

Thus, the horizontal displacement is approximately **32.9 m**, which rounds to **32.8 m** (the correct answer from the choices).

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Would you like more details, or do you have any questions? Here are 5 related questions:

1. What factors influence the time of flight of a projectile?
2. How would the horizontal displacement change if the initial velocity was doubled?
3. How can we determine the maximum height of the football during its flight?
4. What is the role of gravity in affecting the vertical and horizontal motion?
5. How can air resistance be included in projectile motion equations?

**Tip:** In projectile motion, the horizontal and vertical components of velocity are treated independently, with gravity affecting only the vertical motion.

A professional football player kicks a football with an initial velocity \(\vec{v} = (7.00 \, \text{m/s}) \hat{x} + (23.0 \, \text{m/s}) \hat{y}\). Determine the horizontal displacement of the football.

Learn how to calculate the horizontal displacement of a football kicked with an initial velocity of 7.00 m/s in the x-direction and 23.0 m/s in the y-direction. Using key projectile motion concepts and kinematic equations, this problem demonstrates the calculation of the time of flight and displacement. Suitable for students in Grades 9-12.

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