The equations of motion of a projectile are given by `x=36t m and 2y =96t-9.8t^(2)m`. The angle of projection is

Options

A. `sin^(-1)((4)/(5))`
B. `sin^(-1)((3)/(5))`
C. `sin^(-1)((4)/(5))`
D. `sin^(-1)((3)/(4))`

Explanation

To find the angle of projection of the projectile given the equations of motion, we can follow these steps:

### Step 1: Identify the equations of motion
The equations of motion provided are:
- \( x = 36t \) (horizontal motion)
- \( 2y = 96t - 9.8t^2 \) (vertical motion)

### Step 2: Simplify the vertical motion equation
From the vertical motion equation, we can express \( y \):
\[ y = 48t - 4.9t^2 \]

### Step 3: Determine the horizontal and vertical components of velocity
The horizontal component of velocity \( v_x \) can be found by differentiating \( x \) with respect to time \( t \):
\[ v_x = \frac{dx}{dt} = \frac{d}{dt}(36t) = 36 \text{ m/s} \]

The vertical component of velocity \( v_y \) can be found by differentiating \( y \) with respect to time \( t \):
\[ v_y = \frac{dy}{dt} = \frac{d}{dt}(48t - 4.9t^2) = 48 - 9.8t \text{ m/s} \]

### Step 4: Calculate the vertical velocity at \( t = 0 \)
At time \( t = 0 \):
\[ v_y = 48 - 9.8(0) = 48 \text{ m/s} \]

### Step 5: Use the components of velocity to find the angle of projection
The angle of projection \( \theta \) can be calculated using the formula:
\[ \tan \theta = \frac{v_y}{v_x} \]
Substituting the values:
\[ \tan \theta = \frac{48}{36} = \frac{4}{3} \]

### Step 6: Convert \( \tan \theta \) to \( \sin \theta \)
To find \( \theta \), we can use the relationship between \( \tan \) and \( \sin \). We know:
\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \]
In a right triangle where \( \tan \theta = \frac{4}{3} \):
- Opposite side = 4
- Adjacent side = 3

Using the Pythagorean theorem to find the hypotenuse \( h \):
\[ h = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Thus:
\[ \sin \theta = \frac{4}{5} \]

### Step 7: Final expression for the angle of projection
Therefore, the angle of projection \( \theta \) can be expressed as:
\[ \theta = \sin^{-1}\left(\frac{4}{5}\right) \]

### Conclusion
The angle of projection is \( \theta = \sin^{-1}\left(\frac{4}{5}\right) \).

---