The displacement-time relationship for a particle is given by X = a0 + a1t + a2t^2, where a0, a1, and a2 are constants. To determine the acceleration of the particle, we need to differentiate the displacement equation twice with respect to time.

Differentiating once:
The first derivative of displacement with respect to time gives us the velocity of the particle. Taking the derivative of the displacement equation, we get:

v = dX/dt = 0 + a1 + 2a2t

Differentiating twice:
To find the acceleration, we need to differentiate the velocity equation with respect to time. Taking the derivative of the velocity equation, we get:

a = dv/dt = d²X/dt² = 0 + 0 + 2a2

The acceleration of the particle is given by 2a2, which means it is a constant value and independent of time. This indicates that the particle is undergoing uniform acceleration.

Explanation:

1. Displacement equation: X = a0 + a1t + a2t^2
- The displacement-time relationship is given by this equation, where a0, a1, and a2 are constants.
- It describes the position of the particle as a function of time.

2. Differentiating once: Determining velocity
- To find the velocity, we differentiate the displacement equation once with respect to time.
- The derivative of a constant term (a0) is zero, so it does not contribute to the velocity.
- Differentiating the linear term (a1t) gives us a1.
- For the quadratic term (a2t^2), we apply the power rule of differentiation, which gives us 2a2t.
- Combining all the derivatives, we get the velocity equation: v = a1 + 2a2t.

3. Differentiating twice: Finding acceleration
- To determine the acceleration, we differentiate the velocity equation with respect to time.
- The derivative of the constant term (a1) is zero, so it does not contribute to the acceleration.
- The derivative of the linear term (2a2t) is 2a2, as it follows the power rule of differentiation.
- Thus, the acceleration equation is a = 2a2, representing a constant value.

Conclusion:
The acceleration of the particle is given by 2a2, which means it is constant and does not depend on time. This implies that the particle experiences uniform acceleration throughout its motion.

2a0