Final answer:
To show that at the range (X=R), the height (Y) is zero and at half the range (X=R/2), the height (Y) is at maximum height (H), we analyze the projectile's horizontal and vertical motions which are constant and affected by gravity, respectively.
Step-by-step explanation:
When analyzing projectile motion, we often break it down into its horizontal and vertical components. Since air resistance is negligible, the horizontal motion is constant and the vertical motion is influenced by gravity. The trajectory of the projectile can be described by equations that relate distance (x), height (y), and time (t).
To show part (a) that at X=R, Y=0, we understand that R represents the range of the projectile motion, which is the horizontal distance it covers before returning to the same vertical position it was launched from, which is Y=0. Essentially, it is the point where the projectile lands.
For part (b), at X=R/2, Y=H, we are looking at the midpoint of the range. At this point, the value of Y would be at its maximum, which corresponds to the peak of the projectile's arc, known as maximum height (H). The vertical velocity at this point is zero due to the effect of gravity pulling the projectile back down after reaching the peak of its flight.
This can be found by using the equation 0 = Voy - gt, where Voy is the initial vertical velocity, g is the acceleration due to gravity, and t is the time to reach this maximum height.