Question

What is the ratio of the final and initial velocity of a projectile?

Answer 1

The ratio of the final velocity ($v_f$) to the initial velocity ($v_i$) of a projectile depends on the launch angle and the effects of air resistance.

In the absence of air resistance, the ratio of the final velocity to the initial velocity depends on the vertical and horizontal components of the motion. Assuming the launch angle ($\theta$) is not 90 degrees (vertical launch), we can break down the motion into horizontal and vertical components.

The horizontal component of the motion remains constant throughout the projectile's trajectory. There is no acceleration in the horizontal direction (assuming no external forces like air resistance), so the horizontal velocity ($v_{ix}$) remains the same.

The vertical component of the motion is influenced by gravity, resulting in a change in the vertical velocity ($v_{iy}$) over time. The velocity decreases during the upward motion, becomes zero at the peak (maximum height), and then increases during the downward motion.

At the peak of the projectile's trajectory, the vertical velocity becomes zero. At this point, the only velocity remaining is the horizontal velocity ($v_{ix}$). As the projectile descends, the vertical velocity increases in the downward direction due to the acceleration of gravity.

The final velocity of the projectile can be calculated by combining the horizontal and vertical components of the velocity using vector addition. The magnitude of the final velocity ($v_f$) is given by:

$v_f=\sqrt{{v_{ix}}^2+v_{<span\; class="mord\; mtig}}$