Velocity-time equation of a particle moving in a straight line is, `v=(10+2t+3t^2)` (SI units) Find
(a) displacement of particle from the mean position at time `t=1s,` if it is given that displacement is 20m at time `t=0`.
(b) acceleration-time equation.

Explanation

To solve the problem step by step, we will address both parts of the question: (a) finding the displacement of the particle at \( t = 1 \, s \) and (b) determining the acceleration-time equation.

### Part (a): Finding Displacement at \( t = 1 \, s \)

1. **Given the Velocity Equation**: The velocity of the particle is given by the equation:
\[ v = 10 + 2t + 3t^2 \]

2. **Relate Velocity to Displacement**: We know that velocity is the derivative of displacement with respect to time:
\[ \frac{dx}{dt} = v \]
Therefore, we can express displacement as:
\[ dx = v \, dt \]

3. **Integrate to Find Displacement**: To find the displacement \( x \), we integrate the velocity function with respect to time:
\[ x = \int v \, dt = \int (10 + 2t + 3t^2) \, dt \]

4. **Perform the Integration**:
\[ x = 10t + \frac{2t^2}{2} + \frac{3t^3}{3} + C \]
Simplifying this gives:
\[ x = 10t + t^2 + t^3 + C \]

5. **Determine the Constant of Integration \( C \)**: We know that at \( t = 0 \), the displacement \( x = 20 \, m \):
\[ x(0) = 10(0) + (0)^2 + (0)^3 + C = 20 \]
Thus, \( C = 20 \).

6. **Final Displacement Equation**: The displacement equation is:
\[ x = t^3 + t^2 + 10t + 20 \]

7. **Calculate Displacement at \( t = 1 \, s \)**:
\[ x(1) = (1)^3 + (1)^2 + 10(1) + 20 = 1 + 1 + 10 + 20 = 32 \, m \]

### Part (b): Finding the Acceleration-Time Equation

1. **Acceleration Definition**: Acceleration is the derivative of velocity with respect to time:
\[ a = \frac{dv}{dt} \]

2. **Differentiate the Velocity Equation**: We differentiate the given velocity equation:
\[ v = 10 + 2t + 3t^2 \]
Taking the derivative:
\[ a = \frac{d}{dt}(10 + 2t + 3t^2) = 0 + 2 + 6t \]

3. **Final Acceleration Equation**: Therefore, the acceleration-time equation is:
\[ a = 2 + 6t \]

### Summary of Results:
- (a) The displacement of the particle from the mean position at \( t = 1 \, s \) is \( 32 \, m \).
- (b) The acceleration-time equation is \( a = 2 + 6t \).