Draw: (i) Velocity-time graph and (ii) Acceleration-time graph in simple harmonic motion.

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![Figure — Hard draw gate ('Draw: (i) Velocity-time graph and (ii) Acceleration-time graph in SHM'). Catalog fig-13-13 sh](/figures/oscillations/displacement-velocity-acceleration-particle-shm-have-fig-13-13.svg)

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> [!TLDR]
> For $x=A\sin\omega t$: $v(t)=A\omega\cos\omega t$ (a cosine wave) and $a(t)=-A\omega^2\sin\omega t$ (an inverted sine wave, i.e. exactly $-\omega^2 x$).
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> Starting from displacement $x(t)=A\sin\omega t$:
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> **(i) Velocity-time graph:** $v=\dfrac{dx}{dt}=A\omega\cos\omega t$. This is a cosine curve of amplitude $A\omega$:
> - At $t=0$: $v=A\omega$ (maximum, since displacement is zero and the particle is moving fastest through the mean position).
> - At $t=T/4$: $v=0$ (particle momentarily at rest at maximum displacement).
> - At $t=T/2$: $v=-A\omega$ (maximum speed in the opposite direction).
> - At $t=3T/4$: $v=0$ again.
> Shape: a smooth cosine wave, leading the displacement curve by a quarter cycle ($\pi/2$ phase).
>
> **(ii) Acceleration-time graph:** $a=\dfrac{dv}{dt}=-A\omega^2\sin\omega t = -\omega^2 x$. This is a negative-sine curve of amplitude $A\omega^2$:
> - At $t=0$: $a=0$.
> - At $t=T/4$: $a=-A\omega^2$ (maximum restoring acceleration, directed back toward the mean position, since displacement is maximum positive there).
> - At $t=T/2$: $a=0$. …

Q.Draw:

Concept understanding — Simple Harmonic Motion

Simple Harmonic Motion: The Natural Rhythm of Things

The Intuition: A Restoring Force That Fights Displacement

The Precise Statement

What Does This Motion Look Like?

The Mathematical Description (Derived from $F = -kx$ )

Q.Draw:

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