A particle is moving in a circel of radius `R = 1m` with constant speed `v =4 m//s`. The ratio of the displacement to acceleration of the foot of the perpendicular drawn from the particle onto the diameter of the circel is

Options

A. (a)`1//16(second)^(2)`
B. (b)`1//2(second)^(2)`
C. (c)`2(second)^(2)`
D. (d)`16(second)^(2)`

Explanation

To solve the problem step by step, we will analyze the motion of the particle and calculate the required ratio of displacement to acceleration.

### Step 1: Understand the motion of the particle
The particle is moving in a circle of radius \( R = 1 \, \text{m} \) with a constant speed \( v = 4 \, \text{m/s} \). The motion in a circle can be described using angular parameters.

**Hint:** Remember that the angular speed \( \omega \) can be related to the linear speed \( v \) and the radius \( R \) by the equation \( v = \omega R \).

### Step 2: Calculate the angular speed
Using the relationship \( v = \omega R \):
\[ \omega = \frac{v}{R} = \frac{4 \, \text{m/s}}{1 \, \text{m}} = 4 \, \text{rad/s} \]

**Hint:** The angular speed \( \omega \) is a measure of how fast the particle is moving around the circle.

### Step 3: Define the displacement
The displacement \( x \) of the foot of the perpendicular from the particle to the diameter of the circle can be expressed as:
\[ x = R \cos \theta \]
where \( \theta \) is the angle between the radius to the particle and the vertical diameter.

**Hint:** The displacement is the horizontal distance from the center of the circle to the foot of the perpendicular.

### Step 4: Calculate the acceleration
The acceleration \( a \) of the particle in circular motion can be expressed as:
\[ a = -\omega^2 x \]
Substituting \( x = R \cos \theta \):
\[ a = -\omega^2 (R \cos \theta) = -\omega^2 R \cos \theta \]

**Hint:** The negative sign indicates that the acceleration is directed towards the center of the circle.

### Step 5: Calculate the ratio of displacement to acceleration
Now we can find the ratio of displacement \( x \) to acceleration \( a \):
\[ \text{Ratio} = \frac{x}{a} = \frac{R \cos \theta}{-\omega^2 R \cos \theta} \]
This simplifies to:
\[ \text{Ratio} = \frac{1}{-\omega^2} \]

**Hint:** Notice that \( R \cos \theta \) cancels out, provided it is not zero.

### Step 6: Substitute the value of \( \omega \)
Substituting \( \omega = 4 \, \text{rad/s} \):
\[ \text{Ratio} = \frac{1}{-(4)^2} = \frac{1}{-16} \]

**Hint:** The negative sign indicates the direction of acceleration, but for the ratio, we can consider the magnitude.

### Final Result
The ratio of displacement to acceleration is:
\[ \text{Ratio} = -\frac{1}{16} \]

### Conclusion
The correct option based on the calculation is option A, which corresponds to the ratio of displacement to acceleration.