A particle is projected with a velocity of 30 m/s at an angle `theta` with the horizontal . Where ` theta = tan^(-1) (3/4)` . After 1 second, direction of motion of the particle makes an angle ` alpha` with the horizontal then ` alpha` is given by

Options

A. `tan alpha = 1/3`
B. `tan alpha = 1/2`
C. `tan alpha = 2`
D. ` tan alpha= 1`

Answer

To solve the problem step-by-step, we will analyze the motion of the particle projected at an angle and determine the angle \( \alpha \) after 1 second.

### Step 1: Determine the angle \( \theta \)
Given:
\[ \theta = \tan^{-1}\left(\frac{3}{4}\right) \]
From this, we can form a right triangle where the opposite side is 3 and the adjacent side is 4. The hypotenuse can be calculated using the Pythagorean theorem:
\[ \text{Hypotenuse} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 2: Calculate the components of the initial velocity
The initial velocity \( V_0 = 30 \, \text{m/s} \). The components of the velocity can be calculated as:
\[ V_{0x} = V_0 \cos \theta = 30 \cdot \frac{4}{5} = 24 \, \text{m/s} \]
\[ V_{0y} = V_0 \sin \theta = 30 \cdot \frac{3}{5} = 18 \, \text{m/s} \]

### Step 3: Analyze the horizontal motion
In the horizontal direction, there is no acceleration. Therefore, the horizontal component of the velocity remains constant:
\[ V_x = V_{0x} = 24 \, \text{m/s} \]

### Step 4: Analyze the vertical motion
In the vertical direction, we can use the equation of motion:
\[ V_y = V_{0y} - g \cdot t \]
Where \( g \) (acceleration due to gravity) is approximately \( 10 \, \text{m/s}^2 \) and \( t = 1 \, \text{s} \):
\[ V_y = 18 - 10 \cdot 1 = 8 \, \text{m/s} \]

### Step 5: Calculate the resultant velocity after 1 second
After 1 second, the components of the velocity are:
- Horizontal component: \( V_x = 24 \, \text{m/s} \)
- Vertical component: \( V_y = 8 \, \text{m/s} \)

### Step 6: Determine the angle \( \alpha \)
The angle \( \alpha \) can be found using the tangent function:
\[ \tan \alpha = \frac{V_y}{V_x} = \frac{8}{24} = \frac{1}{3} \]

### Step 7: Calculate \( \alpha \)
To find \( \alpha \):
\[ \alpha = \tan^{-1}\left(\frac{1}{3}\right) \]

### Final Answer
Thus, the angle \( \alpha \) after 1 second is:
\[ \alpha = \tan^{-1}\left(\frac{1}{3}\right) \]