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 motion of the ball using kinematic equations. Here’s a step-by-step solution:

### Step 1: Understand the given information
- Distance to the goal post (horizontal range, R) = 40 yards = 120 feet (since 1 yard = 3 feet)
- Height of the goal post (H) = 10 feet
- Initial speed of the ball (u) = 64 ft/s
- Angle of projection (α) = 45 degrees
- Acceleration due to gravity (g) = 32.2 ft/s² (converted from 9.8 m/s²)

### Step 2: Calculate the time of flight (t)
The horizontal range formula is given by:
\[ R = u \cdot \cos(\alpha) \cdot t \]

We need to find t:
\[ t = \frac{R}{u \cdot \cos(\alpha)} \]

Substituting the values:
- R = 120 ft
- u = 64 ft/s
- \(\cos(45^\circ) = \frac{1}{\sqrt{2}} \approx 0.7071\)

Now, calculate t:
\[ t = \frac{120}{64 \cdot 0.7071} \]
\[ t \approx \frac{120}{45.25} \approx 2.65 \text{ seconds} \]

### Step 3: Calculate the vertical position (y) of the ball at time t
The vertical motion can be described by the equation:
\[ y = u \cdot \sin(\alpha) \cdot t - \frac{1}{2} g t^2 \]

Substituting the values:
- \(\sin(45^\circ) = \frac{1}{\sqrt{2}} \approx 0.7071\)

Now, calculate y:
\[ y = 64 \cdot 0.7071 \cdot 2.65 - \frac{1}{2} \cdot 32.2 \cdot (2.65)^2 \]
\[ y \approx 64 \cdot 0.7071 \cdot 2.65 - 16.1 \cdot 7.0225 \]
\[ y \approx 64 \cdot 1.874 - 113.5 \]
\[ y \approx 120.0 - 113.5 \]
\[ y \approx 6.5 \text{ feet} \]

### Step 4: Compare the height of the ball with the height of the goal post
The height of the ball at the time it reaches the goal post is approximately 6.5 feet, which is less than the height of the goal post (10 feet).

### Conclusion
Since the height of the ball (6.5 feet) is less than the height of the goal post (10 feet), the ball will not reach the goal post.