For a body travelling with uniform acceleration, its final velocity is `v=sqrt(180-7x)` , where x is the distance travelled by the body. Then the acceleration is

Options

A. `-8m//s^2`
B. `-3.5m//s^2`
C. `-7m//s^2`
D. `180m//s^2`

Answer

To find the acceleration of a body traveling with uniform acceleration, given that its final velocity \( v \) is expressed as \( v = \sqrt{180 - 7x} \), we can follow these steps:

### Step 1: Square the velocity equation
Start with the equation for velocity:
\[ v = \sqrt{180 - 7x} \]
Square both sides to eliminate the square root:
\[ v^2 = 180 - 7x \]

### Step 2: Differentiate both sides with respect to \( x \)
Now, differentiate both sides with respect to \( x \):
\[ \frac{d(v^2)}{dx} = \frac{d(180 - 7x)}{dx} \]
Using the chain rule on the left side:
\[ 2v \frac{dv}{dx} = -7 \]

### Step 3: Solve for \( \frac{dv}{dx} \)
Rearranging the equation gives:
\[ \frac{dv}{dx} = \frac{-7}{2v} \]

### Step 4: Use the relationship for acceleration
We know that acceleration \( a \) can be expressed as:
\[ a = v \frac{dv}{dx} \]
Substituting \( \frac{dv}{dx} \) from the previous step:
\[ a = v \left(\frac{-7}{2v}\right) \]
This simplifies to:
\[ a = \frac{-7}{2} \]

### Step 5: Final result
Thus, the acceleration is:
\[ a = -3.5 \, \text{m/s}^2 \]

### Summary
The acceleration of the body is \( -3.5 \, \text{m/s}^2 \).

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