In a soccer practice sesson of the football is kept at the centre of the field 40 yards from the 10 ft hight goalposts. A goal is attempted by kicking the football at a speed of 64 ft/s at angle of `45^0` to the horizontal. Will the ball reach the goal post?

Explanation

To determine if the football will reach the goal post, we need to analyze the projectile motion of the ball. Here’s a step-by-step solution:

### Step 1: Convert the distance to feet
The distance from the football to the goal post is given as 40 yards. We need to convert this distance into feet:
- **Conversion**: 1 yard = 3 feet
- **Calculation**: 40 yards = 40 × 3 = 120 feet

### Step 2: Break down the initial velocity into components
The football is kicked at a speed of 64 ft/s at an angle of 45 degrees. We need to find the horizontal (x) and vertical (y) components of the initial velocity:
- **Horizontal component (u_x)**:
\[ u_x = u \cdot \cos(45^\circ) = 64 \cdot \frac{1}{\sqrt{2}} = \frac{64}{\sqrt{2}} \approx 45.25 \text{ ft/s} \]
- **Vertical component (u_y)**:
\[ u_y = u \cdot \sin(45^\circ) = 64 \cdot \frac{1}{\sqrt{2}} = \frac{64}{\sqrt{2}} \approx 45.25 \text{ ft/s} \]

### Step 3: Calculate the time of flight to reach the goal post
Using the horizontal motion equation, we can find the time (t) it takes for the ball to travel 120 feet:
- **Horizontal motion equation**:
\[ x = u_x \cdot t \]
- Rearranging for time:
\[ t = \frac{x}{u_x} = \frac{120}{45.25} \approx 2.65 \text{ seconds} \]

### Step 4: Calculate the height of the ball at time t
Now we need to find out how high the ball will be after 2.65 seconds using the vertical motion equation:
- **Vertical motion equation**:
\[ y = u_y \cdot t - \frac{1}{2} g t^2 \]
where \( g \) (acceleration due to gravity) is approximately 32.2 ft/s².
- Substituting the values:
\[ y = 45.25 \cdot 2.65 - \frac{1}{2} \cdot 32.2 \cdot (2.65)^2 \]
- Calculate:
\[ y \approx 120.86 - 0.5 \cdot 32.2 \cdot 7.0225 \]
\[ y \approx 120.86 - 113.59 \approx 7.27 \text{ feet} \]

### Step 5: Compare the height with the goal post height
The height of the goal post is 10 feet. The calculated height of the ball when it reaches the goal post is approximately 7.27 feet.

### Conclusion
Since the height of the ball (7.27 feet) is less than the height of the goal post (10 feet), the ball will **not** reach the goal post.

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