The dispacement of a body is given by `4s=M +2Nt^(4)`, where `M` and `N` are constants.
The velocity of the body at any instant is .

Options

A. `(M+2Nt^(4))/(4)`
B. `2N`
C. ` (M+2N)/(4)`
D. `2Nt^(3)`

Explanation

To find the velocity of the body given the displacement equation \( 4s = M + 2Nt^4 \), we will follow these steps:

### Step 1: Rearrange the Displacement Equation
The displacement \( s \) can be expressed as:
\[ s = \frac{M + 2Nt^4}{4} \]

### Step 2: Differentiate Displacement with Respect to Time
To find the velocity \( v \), we need to differentiate \( s \) with respect to time \( t \):
\[ v = \frac{ds}{dt} \]

### Step 3: Apply the Chain Rule
Using the chain rule for differentiation:
\[ v = \frac{d}{dt} \left( \frac{M + 2Nt^4}{4} \right) \]
Since \( M \) is a constant, its derivative is zero. Thus, we focus on the term \( 2Nt^4 \):
\[ v = \frac{1}{4} \cdot \frac{d}{dt}(2Nt^4) \]

### Step 4: Differentiate \( 2Nt^4 \)
Now, we differentiate \( 2Nt^4 \):
\[ \frac{d}{dt}(2Nt^4) = 2N \cdot 4t^3 = 8Nt^3 \]

### Step 5: Substitute Back into the Velocity Equation
Substituting this back into the equation for velocity:
\[ v = \frac{1}{4} \cdot 8Nt^3 = 2Nt^3 \]

### Final Answer
Thus, the velocity of the body at any instant is:
\[ v = 2Nt^3 \]

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