Is Acceleration due to gravity independent of horizontal velocity?
The answer to the first question is yes providing that the mass of each object is very small compared with the mass of the earth. Remember that gravity is a mutual attraction, that is to say that the earth and an object fall towards each other. To be really silly, the earth and Jupiter would move towards each other with a greater acceleration than the earth and a feather. The answer to the second is of course not. By horizontally I hope you mean parallel to a tangent to the earth’s surface. This being the case, without the influence of gravity the bullet fired from a gun would move away from the earth’s surface because the earth is curved. So, we start with one bullet that is not moving away from the surface and one that is. The fired bullet will hit the surface after the other one.
When a bullet is dropped down it is pulled by the earth and its speed increases as it falls down; or the speed increases as time of travel increases. If the bullet is fired horizontally, the earth does not stop its pulling action. Therefore as in the above case, it is pulled down and its speed in the vertically downward direction increases as time of travel increases. At the same time it is moving horizontally with a speed because of the firing. Therefore, the bullet falls down as well as moves horizontally. The combined path is a parabolic path and the time of touching the ground is the time it will take when dropped. The time for the bullet to reach the ground will be small, if dropped down instead of firing. However, if fired from that height horizontally, it covers longer horizontal distance during that small time because of its high speed.
The answer to your first question is YES, all objects, irrespective of mass, will reach the ground at the same time when dropped from the same height at the same time. But we must assume that there is NO air resistance, i.e. nothing else in the vertical direction that would counteract the uniform acceleration due to gravity. So the answer to your second question is also YES given the same assumption that there is NO air resistance, and that the bullet fired from the gun travels a distance that is comparatively short relative to the curvature of the Earth (I think this is a VERY safe assumption. I don’t know what the hell the previous responder is talking about). Both bullets will reach the ground at exactly the SAME time. This is because the equation of motion for both bullets are independent of their mass. See below: Bullet 1 (fired from a gun horizontally): x = v(x)t + 1/2 a(x)t^2, v(x) is velocity in the x direction and a(x) is acceleration in the x direction y = v(y)t + 1/2 gt^2, v
Perpendicular vectors are independent. Over an infinite plain, two objects dropped with identical vertical velocities in a uniform gravitational field in a vacuum will hit the ground at the exact same time regardless of mass or horizontal velocity. In other words, yes. We can, for the purposes of this example, ignore the curvature of the Earth and air resistance. Gravity will be accelerating both object’s equally towards the ground, they will hit at the same time if both we dropped at the same height.
