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 given its velocity as \( v = (180 - 16x)^{1/2} \) m/s, we can follow these steps:

### Step 1: Understand the relationship between acceleration, velocity, and position
Acceleration \( a \) can be expressed in terms of velocity \( v \) and position \( x \) using the chain rule:
\[ a = \frac{dv}{dt} = \frac{dv}{dx} \cdot \frac{dx}{dt} \]
Here, \( \frac{dx}{dt} = v \), so we can rewrite the equation as:
\[ a = v \cdot \frac{dv}{dx} \]

### Step 2: Differentiate the velocity function with respect to \( x \)
Given \( v = (180 - 16x)^{1/2} \), we need to find \( \frac{dv}{dx} \).

Using the chain rule:
\[ \frac{dv}{dx} = \frac{1}{2}(180 - 16x)^{-1/2} \cdot (-16) \]
This simplifies to:
\[ \frac{dv}{dx} = -8(180 - 16x)^{-1/2} \]

### Step 3: Substitute \( v \) and \( \frac{dv}{dx} \) into the acceleration formula
Now we can substitute \( v \) and \( \frac{dv}{dx} \) into the acceleration equation:
\[ a = v \cdot \frac{dv}{dx} = (180 - 16x)^{1/2} \cdot (-8(180 - 16x)^{-1/2}) \]

### Step 4: Simplify the expression
When we multiply these two expressions:
\[ a = (180 - 16x)^{1/2} \cdot (-8(180 - 16x)^{-1/2}) = -8 \]

### Conclusion
Thus, the acceleration \( a \) is:
\[ a = -8 \, \text{m/s}^2 \]

### Final Answer
The acceleration of the particle is \( -8 \, \text{m/s}^2 \).

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