To understand this concept, let's consider the motion of a projectile. When a projectile is launched, its velocity can be broken into two components: a horizontal (x-axis) component and a vertical (y-axis) component. These two components are independent of each other. This means that what happens in the horizontal motion does not affect the vertical motion, and vice versa.
Now, let's take a look at each of the options given in the question:
a. The range of the projectile is actually affected by both the horizontal and vertical components because the range is the horizontal distance traveled by the projectile, and this will be influenced by the initial velocity and the angle of projection. So, the range is not independent of the horizontal component.
b. The initial velocity of the projectile is the speed at which it is launched. It is this velocity that is broken down into horizontal and vertical components. Therefore, the initial velocity is directly related to the horizontal component, as the horizontal component is derived from the initial velocity and the angle of launch.
c. The vertical component of a projectile's velocity is affected by gravity, which causes the projectile to accelerate downward. The horizontal motion, however, is not affected by gravity if we neglect air resistance. So, the horizontal component of a projectile’s velocity remains constant (in ideal conditions) and is independent of the vertical component, which changes over time due to gravity.
d. Time affects both the horizontal and vertical components in terms of where the projectile is at a given point in time, but it does not have a direct effect on the independence of the horizontal component from the vertical component. The horizontal velocity component stays consistent over time unless acted upon by another force, such as air resistance (which, in an idealized situation, is often neglected).
Therefore, the correct answer is "c", the horizontal component of a projectile's velocity is independent of the vertical component of its velocity.
