To determine the magnitude of the acceleration when the displacement becomes zero, let's proceed step by step with a general analysis.

1. Problem Setup:
   • A particle starts from the origin ($x = 0$) with variable speed $v(t)$, where $v$ is a function of time ($t$).
   • Displacement becomes zero at some later time ($x = 0$ again).
   • We need to find the acceleration ($a$) at this instant.

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Key Concepts:

1. Displacement $x(t)$:

   • Displacement is given by integrating velocity: $x(t) = \int v(t) \, dt.$ The displacement becomes zero when $x(t) = 0$.

2. Acceleration $a(t)$:

   • Acceleration is the rate of change of velocity: $a(t) = \frac{dv(t)}{dt}.$

3. Zero Displacement Condition:

   • For $x(t) = 0$, the velocity profile $v(t)$ must satisfy: $\int_{0}^{t} v(t) \, dt = 0.$

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Step-by-Step Solution:

1. Given $v(t)$:

   • Let $v(t)$ be any general variable-speed function.

2. Condition for Zero Displacement:

   • Displacement is zero if the integral of velocity over the time interval is zero: $\int_{0}^{T} v(t) \, dt = 0,$ where $T$ is the time when displacement returns to zero.

3. Find Acceleration:

   • At the instant $t = T$ when $x = 0$, the acceleration is: $a(T) = \frac{dv(T)}{dt}.$ To find this explicitly, you need the functional form of $v(t)$.

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Example with $v(t) = A \sin(\omega t)$:**

• If $v(t) = A \sin(\omega t)$, then: $x(t) = \int A \sin(\omega t) dt = -\frac{A}{\omega} \cos(\omega t) + C.$ Assuming initial position $x(0) = 0$, constant $C = \frac{A}{\omega}$. Thus: $x(t) = \frac{A}{\omega} (1 - \cos(\omega t)).$ For $x(t) = 0$: $\cos(\omega t) = 1 \implies \omega t = 2n\pi, \, n \in \mathbb{Z}.$ The first return to $x = 0$ is at $t = T = \frac{2\pi}{\omega}$.

• At $t = T$: $a(T) = \frac{dv(T)}{dt} = A \omega \cos(\omega T).$ Substituting $T = \frac{2\pi}{\omega}$, $\cos(\omega T) = 1$, so: $a(T) = A \omega.$

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General Conclusion:

• Without a specific $v(t)$, the acceleration magnitude $a(T)$ is determined as: $a(T) = \frac{dv(T)}{dt}.$

Let me know if you'd like to explore this further with a different $v(t)$ or additional examples!

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Expanding Understanding:

1. What is the physical interpretation of acceleration at zero displacement?
2. How does variable velocity affect the particle's trajectory in general?
3. Can this be generalized to multidimensional motion (e.g., vector displacement)?
4. How does adding an external force alter the results?
5. How can we experimentally determine the velocity function $v(t)$?

Tip: Always ensure boundary conditions (like $x(0) = 0$) are satisfied when solving motion equations.