Final answer:
The horizontal distance a ball travels before returning to its original height in projectile motion is calculated using the time of flight, which can be found by dividing the initial vertical speed by gravity and doubling it. The ball's range is then this flight time multiplied by its constant horizontal speed.
Step-by-step explanation:
The horizontal distance a ball travels before returning to its starting height in projectile motion can be calculated using the time it takes for the ball to reach its peak and fall back down, and the constant horizontal velocity. Since the vertical speed of the ball is given as Vy = 19.6 m/s, we can use the formula for the time of ascent under gravity, which is t = Vy/g, where g is the acceleration due to gravity (approximated as 9.8 m/s2). The total time in air would be twice this ascent time since what goes up must come down and the descent time equals the ascent time because of symmetry in projectile motion.
Given the horizontal speed Vx = 33.9 m/s is constant, the horizontal distance traveled (range) is the horizontal speed multiplied by the total time of flight (R = Vx * Time). To find the total time of flight, we multiply the ascent time by 2 (since there is both an ascent and descent), and then we can multiply this by the horizontal velocity to get the range.
Example: If a ball is kicked horizontally at 16 m/s and vertically at 12 m/s, the time in air can be calculated similarly by finding the time to reach the peak height (using Vy / g), doubling it to get the total flight time, and then using this to find the horizontal distance covered during this time.