Displacement-time equation of a particle moving along x-axis is `x=20+t^3-12t` (SI units)
(a) Find, position and velocity of particle at time t=0.
(b) State whether the motion is uniformly accelerated or not.
(c) Find position of particle when velocity of particle is zero.

Explanation

Let's solve the problem step by step.

### Given:
The displacement-time equation of a particle moving along the x-axis is:
\[ x = 20 + t^3 - 12t \]

### (a) Find the position and velocity of the particle at time \( t = 0 \).

**Step 1: Find the position at \( t = 0 \).**
To find the position, substitute \( t = 0 \) into the displacement equation:
\[ x(0) = 20 + (0)^3 - 12(0) = 20 + 0 - 0 = 20 \text{ meters} \]

**Step 2: Find the velocity at \( t = 0 \).**
Velocity is the derivative of displacement with respect to time:
\[ v = \frac{dx}{dt} = \frac{d}{dt}(20 + t^3 - 12t) \]
Differentiating:
\[ v = 0 + 3t^2 - 12 \]
Now, substitute \( t = 0 \):
\[ v(0) = 3(0)^2 - 12 = 0 - 12 = -12 \text{ meters per second} \]

### (b) State whether the motion is uniformly accelerated or not.

**Step 3: Find the acceleration.**
Acceleration is the derivative of velocity with respect to time:
\[ a = \frac{dv}{dt} = \frac{d}{dt}(3t^2 - 12) \]
Differentiating:
\[ a = 6t \]
Since acceleration depends on time (it increases as \( t \) increases), the motion is **not uniformly accelerated**.

### (c) Find the position of the particle when the velocity of the particle is zero.

**Step 4: Set the velocity equation to zero and solve for \( t \).**
From the velocity equation:
\[ v = 3t^2 - 12 = 0 \]
Solving for \( t \):
\[ 3t^2 = 12 \implies t^2 = 4 \implies t = \pm 2 \]
Since time cannot be negative, we take \( t = 2 \) seconds.

**Step 5: Find the position at \( t = 2 \) seconds.**
Substituting \( t = 2 \) into the displacement equation:
\[ x(2) = 20 + (2)^3 - 12(2) = 20 + 8 - 24 = 20 - 16 = 4 \text{ meters} \]

### Summary of Answers:
(a) Position at \( t = 0 \) is \( 20 \) meters, velocity at \( t = 0 \) is \( -12 \) m/s.
(b) The motion is not uniformly accelerated.
(c) The position when the velocity is zero is \( 4 \) meters.