Instantaneous Acceleration

What instantaneous acceleration means

Just as instantaneous velocity is the rate of change of position at a specific moment, instantaneous acceleration is the rate of change of velocity at a specific moment.

Average acceleration over an interval tells you the overall change in velocity. Instantaneous acceleration tells you how rapidly the velocity is changing at one particular instant.

$a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt}$

Two equivalent expressions

Since $v = \frac{dx}{dt}$, we can write:

$a = \frac{dv}{dt} = \frac{d}{dt}\left(\frac{dx}{dt}\right) = \frac{d^2x}{dt^2}$

Acceleration is the second derivative of position with respect to time. This is a compact and powerful way to express it — position differentiated once gives velocity, differentiated again gives acceleration.

Geometric meaning

On a velocity-time graph ($v$ vs $t$), the instantaneous acceleration at any moment is the slope of the tangent to the curve at that point.

• Steep upward slope → large positive acceleration (velocity increasing rapidly)
• Gentle upward slope → small positive acceleration
• Horizontal tangent → zero acceleration (velocity momentarily not changing)
• Downward slope → negative acceleration (velocity decreasing)

Finding instantaneous acceleration from functions

From a velocity function: Differentiate $v(t)$:

$a(t) = \frac{dv}{dt}$

Example: $v = 6t + 2$ m/s

$a = \frac{dv}{dt} = 6 \text{ m/s}^2$

Constant — this is uniform acceleration.

From a position function: Differentiate $x(t)$ twice:

$a(t) = \frac{d^2x}{dt^2}$

Example: $x = 3t^2 + 2t + 1$ m

$\frac{dx}{dt} = 6t + 2, \quad \frac{d^2x}{dt^2} = 6 \text{ m/s}^2$

Same result — consistent.

Example with variable acceleration: $x = t^3 - 3t^2 + 2t$

$v = \frac{dx}{dt} = 3t^2 - 6t + 2$

$a = \frac{dv}{dt} = 6t - 6 \text{ m/s}^2$

Here acceleration changes with time. At $t = 0$: $a = -6$ m/s². At $t = 2$ s: $a = +6$ m/s². The equations of motion cannot be used for this motion.

The hierarchy of derivatives

$x(t) \xrightarrow{\frac{d}{dt}} v(t) \xrightarrow{\frac{d}{dt}} a(t)$

Going the other way — integrating:

$a(t) \xrightarrow{\int dt} v(t) \xrightarrow{\int dt} x(t)$

This is the foundation of kinematics. Every kinematic formula is a consequence of these relationships.

A useful identity

By the chain rule:

$a = \frac{dv}{dt} = \frac{dv}{dx} \cdot \frac{dx}{dt} = v\frac{dv}{dx}$

This form $a = v\frac{dv}{dx}$ is used when acceleration is given as a function of position rather than time — it eliminates $t$ and directly relates $v$ and $x$. This is precisely how the third equation of motion $v^2 = u^2 + 2as$ is derived using calculus.