When a gun is fired the bullet accelerates while traveling along the barrel of the gun. it takes 0.001s for a bullet to travel the length of the barrel. The acceleration of the bullet is 200000 m/s. how fast is the bullet when it leaves the gun?

Since a time interval is given we know this is the average acceleration. The formula for the average acceleration is
a=(Delta v)/(Delta t)=(v_f-v_i)/(Delta t)
Solve for the final velocity
a=(v_f-v_i)/(Delta*t)
a*Delta t=v_f-v_i
a*Delta t+v_i=v_f
Now plug in all the values to solve for the final velocity.
(200,000)(m/s^2)*(0.001) (m/s)+0 (m/s)=v_f
v_f = 200 (m/s)
http://hyperphysics.phy-astr.gsu.edu/hbase/acca.html