If the velocity of a particle is given by `v=(180-16x)^((1)/(2))(m)/(s)`, then its acceleration will be

Options

A. zero
B. `8(m)/(s^2)`
C. `-8(m)/(s^2)`
D. `4(m)/(s^2)`

Explanation

To find the acceleration of a particle when its velocity is given by the equation \( v = (180 - 16x)^{1/2} \) m/s, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the relationship between acceleration, velocity, and displacement**:
The acceleration \( a \) can be expressed in terms of velocity \( v \) and displacement \( x \) as:
\[ a = \frac{dv}{dt} = \frac{dv}{dx} \cdot \frac{dx}{dt} \]
Here, \( \frac{dx}{dt} \) is the velocity \( v \).

2. **Differentiate the velocity with respect to displacement**:
We need to find \( \frac{dv}{dx} \). Start by differentiating \( v \):
\[ v = (180 - 16x)^{1/2} \]
Using the chain rule, we have:
\[ \frac{dv}{dx} = \frac{1}{2}(180 - 16x)^{-1/2} \cdot (-16) \]
Simplifying this gives:
\[ \frac{dv}{dx} = -\frac{8}{(180 - 16x)^{1/2}} \]

3. **Substitute \( \frac{dx}{dt} \) with \( v \)**:
Since \( \frac{dx}{dt} = v \), we can substitute this into the equation for acceleration:
\[ a = \frac{dv}{dx} \cdot v \]
Substituting \( \frac{dv}{dx} \):
\[ a = -\frac{8}{(180 - 16x)^{1/2}} \cdot (180 - 16x)^{1/2} \]

4. **Simplify the expression**:
The \( (180 - 16x)^{1/2} \) terms cancel out:
\[ a = -8 \, \text{m/s}^2 \]

### Final Answer:
The acceleration of the particle is:
\[ a = -8 \, \text{m/s}^2 \]