Passage VI Two particles collide when they are at same position at same time.
Displacement time equation of two particles moving along x-axis are `x_(1)=(8+3sinomegat)m`
and `x_(2)=(4cosomegat)m`
Here, `omega=pirad/s`
The two particles will collide after time t=...................s.

Options

A. 1
B. 2
C. 4
D. None of the above.

Answer

To solve the problem, we need to find the time at which the two particles collide, meaning their displacements \( x_1 \) and \( x_2 \) are equal. The displacement equations are given as:

\[ x_1 = 8 + 3 \sin(\omega t) \]
\[ x_2 = 4 \cos(\omega t) \]

where \( \omega = \pi \) rad/s.

### Step 1: Set the equations equal to each other

To find the time when the two particles collide, we set their displacement equations equal to each other:

\[ 8 + 3 \sin(\pi t) = 4 \cos(\pi t) \]

### Step 2: Rearrange the equation

Rearranging the equation gives us:

\[ 3 \sin(\pi t) - 4 \cos(\pi t) + 8 = 0 \]

### Step 3: Isolate the trigonometric terms

We can isolate the trigonometric terms:

\[ 3 \sin(\pi t) = 4 \cos(\pi t) - 8 \]

### Step 4: Divide by \(\cos(\pi t)\)

To solve for \( t \), we can divide both sides by \( \cos(\pi t) \) (assuming \(\cos(\pi t) \neq 0\)):

\[ 3 \tan(\pi t) = 4 - \frac{8}{\cos(\pi t)} \]

### Step 5: Analyze the ranges of \( x_1 \) and \( x_2 \)

Now, we need to analyze the ranges of \( x_1 \) and \( x_2 \):

- For \( x_1 \):
- Minimum: \( 8 - 3 = 5 \)
- Maximum: \( 8 + 3 = 11 \)

- For \( x_2 \):
- Minimum: \( 4 \cdot (-1) = -4 \)
- Maximum: \( 4 \cdot 1 = 4 \)

### Step 6: Determine if there is an overlap in ranges

The range of \( x_1 \) is from 5 to 11, and the range of \( x_2 \) is from -4 to 4. Since there is no overlap between these ranges, the two particles will never collide.

### Conclusion

Thus, the two particles will not collide at any time \( t \). Therefore, the answer is:

\[ \text{The two particles will collide after time } t = \text{none of these (they never collide)}. \]