When there is just a horizontal velocity for a projectile and another case in which there is a horizontal velocity and vertical velocity for a projectile that launches at an angle, are the horizontal velocities going to remain the same?

The horizontal velocities of the projectile for both cases are going to remain the same.

Explanation:

Taking the horizontal direction as x-axis and the vertical component as y-axis.

Let V be the object's velocity and θ be the angle formed with the horizontal.

Therefore,  

The horizontal component of velocity is,

Due to the force of gravity the vertical component of velocity changes. But there is no influence of force in the horizontal component other than the air resistance.

Hence, the horizontal component of the velocities of the projectile remains the same.

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