The ratio of the final velocity (vf) to the initial velocity (vi) of a projectile depends on the specific conditions of the projectile's motion. Assuming we are considering projectile motion in a uniform gravitational field (neglecting air resistance), we can determine the relationship between vf and vi.
In projectile motion, the horizontal component of velocity remains constant throughout the motion, while the vertical component experiences acceleration due to gravity. The vertical and horizontal motions are independent of each other.
The horizontal component of velocity (vix) remains unchanged, so we have vix = vfx, where vix is the initial horizontal velocity and vfx is the final horizontal velocity.
In the vertical direction, the final vertical velocity (vfy) can be determined using the equation:
vfy = viy + gt
where viy is the initial vertical velocity, g is the acceleration due to gravity, and t is the time of flight.
If we assume the projectile lands at the same vertical position from which it was launched, then vfy = -viy (equal magnitudes but opposite in direction). This is because the vertical velocity changes sign when the projectile reaches its maximum height and returns to the same level.
Therefore, we can rewrite the equation as:
-viy = viy + gt
Simplifying the equation gives:
2viy = -gt
Now, let's consider the ratio of the final and initial velocities in the vertical direction:
vf / vi = vfy / viy
Since vfy = -viy, we can substitute these values:
vf / vi = -viy / viy = -1
Hence, the ratio of the final velocity to the initial velocity of a projectile, considering only the vertical component of motion in a uniform gravitational field, is -1. This means that the final vertical velocity magnitude is equal to the initial vertical velocity magnitude but in the opposite direction.
It's important to note that this ratio only considers the vertical component of velocity. The ratio of the final and initial velocities in the horizontal direction remains 1 since the horizontal velocity remains constant throughout the motion.