Let us say the altitude h at which the rocket flies is very small compared to radius of Earth. Let us also assume that rocket flies above the equator and along the equator. Let us assume that given time duration is in seconds.
Let the rocket fly from west to East, along the rotational direction of Earth. Then the relative linear speed of rocket wrt to an object stationary on Earth will be:
Relative linear speed = v = 2 π Re / time duration = 2π *6400 km / 3601 sec
= 11.16 km/sec
angular velocity of a stationary object on Earth's surface = 2π/(24*3600) rad/sec
ω₀ = 7.272 * 10^-7 rad/sec
Its linear speed will be V₀ = 465.42 meters/sec
Speed of rocket (wrt to Sun, an inertial frame of reference)
= 11.16 + 0.465 km/sec = 11.625 km/sec
Suppose the rocket flies in the direction East to West against rotation of Earth. Then rocket actually does not cover full circumference in its orbit. An object stationary on Earth also moves some distance in time of 3601. But the relative speed remains same as before.
we find the absolute speed of rocket wrt Sun,
distance covered by rocket in 3601 sec =
v * 3601 + (3,601/86,400) * 2 π 6400 = 2 π 6400
then v = 10.70 km /sec