A particle is projected from horizontal XZ plane with velocity `(u_(x)hati+u_(y)hati+u_(z)hatk)` from origin (+ yaxis is upward). Find time when velocity of particle will be at `37^(@)` with horizontal.

Options

A. `(u_(y)sqrt(u_(x)^(2)+u_(z)^(2)))/(g)`
B. `(3u_(y)-4sqrt(u_(x)^(2)+u_(z)^(2)))/(3g)`
C. `(3u_(y)+4sqrt(u_(x)^(2)+u_(z)^(2)))/(3g)`
D. `(u_(y)+sqrt(u_(x)^(2)+u_(z)^(2)))/(3g)`

Explanation

To solve the problem, we need to analyze the motion of the particle projected from the horizontal XZ plane with a given initial velocity. The velocity vector is given as:

\[ \vec{u} = u_x \hat{i} + u_y \hat{j} + u_z \hat{k} \]

### Step 1: Understand the Components of Velocity
The velocity vector can be broken down into its components along the x, y, and z axes. The particle is projected from the origin, and the y-axis is considered upward. The components of the initial velocity are:
- \( u_x \) along the x-axis
- \( u_y \) along the y-axis (upward)
- \( u_z \) along the z-axis

### Step 2: Determine the Velocity at Time \( t \)
As the particle moves under the influence of gravity, the velocity components will change over time. The vertical component \( u_y \) will be affected by gravity. The velocity at any time \( t \) can be expressed as:

\[ \vec{v}(t) = u_x \hat{i} + (u_y - gt) \hat{j} + u_z \hat{k} \]

where \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)).

### Step 3: Find the Angle with the Horizontal
The angle \( \theta \) that the velocity vector makes with the horizontal can be found using the tangent of the angle:

\[ \tan(\theta) = \frac{v_y}{\sqrt{v_x^2 + v_z^2}} \]

Substituting the components of the velocity:

\[ \tan(\theta) = \frac{u_y - gt}{\sqrt{u_x^2 + u_z^2}} \]

### Step 4: Set Up the Equation for \( 37^\circ \)
We need to find the time when the angle \( \theta \) is \( 37^\circ \). Therefore, we set up the equation:

\[ \tan(37^\circ) = \frac{u_y - gt}{\sqrt{u_x^2 + u_z^2}} \]

Using the known value \( \tan(37^\circ) \approx 0.7536 \):

\[ 0.7536 = \frac{u_y - gt}{\sqrt{u_x^2 + u_z^2}} \]

### Step 5: Rearranging the Equation
Now, we can rearrange the equation to solve for \( t \):

\[ u_y - gt = 0.7536 \sqrt{u_x^2 + u_z^2} \]

This gives us:

\[ gt = u_y - 0.7536 \sqrt{u_x^2 + u_z^2} \]

Thus,

\[ t = \frac{u_y - 0.7536 \sqrt{u_x^2 + u_z^2}}{g} \]

### Step 6: Conclusion
This equation gives us the time \( t \) when the velocity of the particle will be at \( 37^\circ \) with the horizontal.