Equation of displacement for any particle is `s = 3t^(3) +7t^(2) +14t8m`. Its acceleration at time `t = 1` sec is

Options

A. `10 m//s`
B. `16 m//s`
C. `25 m//s`
D. `32 m//s`

Answer

To find the acceleration of the particle at time \( t = 1 \) second, we will follow these steps:

### Step 1: Write down the displacement equation
The displacement \( s \) of the particle is given by:
\[ s = 3t^3 + 7t^2 + 14t + 8 \text{ m} \]

### Step 2: Differentiate the displacement equation to find velocity
The velocity \( v \) is the first derivative of displacement with respect to time:
\[ v = \frac{ds}{dt} = \frac{d}{dt}(3t^3 + 7t^2 + 14t + 8) \]
Using the power rule of differentiation:
\[ v = 9t^2 + 14t + 14 \]

### Step 3: Differentiate the velocity equation to find acceleration
The acceleration \( a \) is the derivative of velocity with respect to time:
\[ a = \frac{dv}{dt} = \frac{d}{dt}(9t^2 + 14t + 14) \]
Again, applying the power rule:
\[ a = 18t + 14 \]

### Step 4: Substitute \( t = 1 \) into the acceleration equation
Now, we need to find the acceleration at \( t = 1 \) second:
\[ a = 18(1) + 14 = 18 + 14 = 32 \text{ m/s}^2 \]

### Final Answer
The acceleration of the particle at \( t = 1 \) second is:
\[ \boxed{32 \text{ m/s}^2} \]

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