In projectile motion, the range refers to the horizontal distance covered by the projectile, while the maximum height represents the highest point it reaches in its trajectory. Both these quantities depend on the initial velocity and the launch angle of the projectile.

To calculate the range of a projectile, we can use the equation:

Range = (initial velocity)^2 * sin(2 * launch angle) / g.

Here, g represents the acceleration due to gravity. From this formula, it is evident that the range is affected by the square of the initial velocity and the sine of twice the launch angle. As the launch angle approaches 45 degrees, the range becomes maximum. Additionally, a higher initial velocity will also result in a greater range.

On the other hand, the maximum height of a projectile can be determined using the equation:

Maximum height = (initial velocity)^2 * (sin(launch angle))^2 / (2g).

Similarly, we see that the maximum height depends on the square of the initial velocity and the square of the sine of the launch angle. The maximum height is achieved when the launch angle is 90 degrees, i.e., the projectile is launched vertically upwards.

Understanding these calculations and their dependence on initial velocity and launch angle allows us to make accurate predictions and analyze the motion of projectiles.