Question

A particle moves along the \(x\)-axis and has its displacement \(x\) varying with time \(t\) according to the equation \[ x = c_0 (t^2 - 2) + c(t - 2)^2 \] where \(c_0\) and \(c\) are constants of appropriate dimensions. Which of the following statements is correct?

• A. The acceleration of the particle is \(2c_0\)
• B. The acceleration of the particle is \(2c\)
• C. The initial velocity of the particle is \(4c\)
• D. The acceleration of the particle is \(2(c + c_0)\)

Hint:

For motion along a straight line: \[ v = \frac{dx}{dt}, \quad a = \frac{d^2x}{dt^2} \] Always simplify the displacement equation before differentiating.

Answer 1

The Correct Option is D

In this problem, we need to analyze the particle's motion along the \(x\)-axis given by the displacement equation:

\(x = c_0 (t^2 - 2) + c(t - 2)^2\).

Let's find the acceleration of the particle by following these steps:

1. Determine the velocity of the particle:
   • Velocity is the first derivative of the displacement with respect to time. To find the velocity \(v(t)\), differentiate \(x\) with respect to \(t\): \(v(t) = \frac{dx}{dt} = \frac{d}{dt}[c_0 (t^2 - 2) + c(t - 2)^2]\).
   • Apply the derivative: \(v(t) = \frac{d}{dt}[c_0 t^2 - 2c_0 + c(t^2 - 4t + 4)]\).
   • Simplify by differentiating each term: \(v(t) = 2c_0 t + (2ct - 4c)\).
   • Combine like terms: \(v(t) = (2c_0 + 2c)t - 4c\).
2. Determine the acceleration of the particle:
   • Acceleration is the derivative of velocity with respect to time. Differentiate \(v(t)\) to find the acceleration \(a(t)\): \(a(t) = \frac{dv}{dt} = \frac{d}{dt}[(2c_0 + 2c)t - 4c]\).
   • Using the derivative: \(a(t) = 2c_0 + 2c\).

Therefore, the correct statement is: 
The acceleration of the particle is \(2(c + c_0)\).

Let us analyze the other options to ensure the answer is robust:

• The acceleration of the particle is \(2c_0\): This is incorrect as it does not account for both constants \(c_0\) and \(c\).
• The acceleration of the particle is \(2c\): This is incorrect for the same reason stated above.
• The initial velocity of the particle is \(4c\): This is incorrect. The initial velocity can be determined by substituting \(t = 0\) in \(v(t) = (2c_0 + 2c)t - 4c\).

Thus, the only correct statement is The acceleration of the particle is \(2(c + c_0)\).