An object is projected with a velocity of `20(m)/(s)` making an angle of `45^@` with horizontal. The equation for the trajectory is `h=Ax-Bx^2` where h is height, x is horizontal distance, A and B are constants. The ratio A:B is (g`=ms^-2)`

Options

A. `1:5`
B. `5:1`
C. `1:40`
D. `40:1`

Explanation

To solve the problem, we need to derive the constants A and B from the trajectory equation of the projectile motion. The trajectory is given as \( h = Ax - Bx^2 \), where \( h \) is the height, \( x \) is the horizontal distance, and \( A \) and \( B \) are constants.

### Step-by-Step Solution:

1. **Identify the Initial Conditions**:
The object is projected with a velocity \( u = 20 \, \text{m/s} \) at an angle \( \theta = 45^\circ \).

2. **Resolve the Initial Velocity**:
The horizontal and vertical components of the initial velocity can be calculated as:
\[ u_x = u \cos \theta = 20 \cos 45^\circ = 20 \times \frac{1}{\sqrt{2}} = 10\sqrt{2} \, \text{m/s} \]
\[ u_y = u \sin \theta = 20 \sin 45^\circ = 20 \times \frac{1}{\sqrt{2}} = 10\sqrt{2} \, \text{m/s} \]

3. **Determine Time of Flight**:
The time of flight \( t \) can be calculated using the formula:
\[ t = \frac{2u_y}{g} = \frac{2 \times 10\sqrt{2}}{10} = 2\sqrt{2} \, \text{s} \]

4. **Calculate Horizontal Distance**:
The horizontal distance \( x \) covered in time \( t \) is:
\[ x = u_x \cdot t = (10\sqrt{2}) \cdot (2\sqrt{2}) = 40 \, \text{m} \]

5. **Find the Height at Horizontal Distance**:
The height \( h \) at any time \( t \) can be expressed as:
\[ h = u_y t - \frac{1}{2} g t^2 \]
Substituting \( t = \frac{x}{u_x} \):
\[ h = 10\sqrt{2} \cdot \frac{x}{10\sqrt{2}} - \frac{1}{2} g \left(\frac{x}{10\sqrt{2}}\right)^2 \]
Simplifying this gives:
\[ h = x - \frac{g}{2(10\sqrt{2})^2} x^2 \]
\[ h = x - \frac{g}{400} x^2 \]

6. **Identify Constants A and B**:
From the trajectory equation \( h = Ax - Bx^2 \), we can identify:
- \( A = 1 \)
- \( B = \frac{g}{400} \)

7. **Calculate the Ratio \( A:B \)**:
Now, we need to find the ratio \( A:B \):
\[ \text{Ratio} = \frac{A}{B} = \frac{1}{\frac{g}{400}} = \frac{400}{g} \]

8. **Substituting the Value of \( g \)**:
Given \( g = 10 \, \text{m/s}^2 \):
\[ \text{Ratio} = \frac{400}{10} = 40 \]

### Final Answer:
The ratio \( A:B \) is \( 40:1 \).