The acceleration of a bullet as it leaves the gun barrel depends on several factors, including the firearm, ammunition, and barrel length. On average, a bullet can accelerate from zero to several thousand feet per second within a few milliseconds (thousandths of a second).

Here's a general idea of how fast a bullet accelerates:

1. Initial Acceleration: Immediately after the firing pin strikes the primer and ignites the propellant, the pressure inside the gun barrel rapidly increases. This pressure acts on the base of the bullet, propelling it forward with tremendous force. The acceleration in this initial phase can be extremely high, causing the bullet to gain speed quickly.

2. Pressure Curve: The pressure inside the barrel rises to a peak and then decreases as the bullet travels down the barrel. The acceleration decreases accordingly as the pressure drops.

3. Barrel Length: The length of the gun barrel plays a crucial role in acceleration. Longer barrels allow more time for the propellant gases to act on the bullet, providing a greater impulse and therefore higher velocity. This means that bullets fired from longer barrels generally achieve higher velocities compared to those fired from shorter barrels.

4. Projectile Mass: The mass of the bullet also affects acceleration. Heavier bullets require more force to accelerate than lighter bullets. Therefore, lighter bullets typically accelerate faster than heavier bullets when fired from the same firearm.

5. Propellant Charge: The amount and type of propellant used in the ammunition influence the acceleration. Higher propellant charges generate greater pressure, resulting in increased acceleration.

For reference, some high-powered rifle cartridges can propel bullets to velocities exceeding 3,000 feet per second (914 meters per second) within a barrel length of a few inches. It's important to note that these figures are approximate and can vary based on specific firearm and ammunition combinations.