Final answer:
The final velocity of a projectile is determined by combining its constant horizontal velocity and altered vertical velocity. The vertical velocity changes direction upon impact if the projectile lands at the same elevation it was launched from. The magnitude and angle with the horizontal can be calculated using trigonometric functions.
Step-by-step explanation:
The final velocity of a projectile object launched across its range can be determined by combining the final horizontal and vertical velocities. Since horizontal velocity remains constant throughout projectile motion, and assuming negligible air resistance, the final horizontal velocity (vx) will be equal to the initial horizontal velocity. However, the final vertical velocity (vy) will be affected by gravity and can be found using the initial vertical velocity and the time spent in the air (which is derived from the vertical displacement and acceleration due to gravity).
The final vertical velocity will have the same magnitude but opposite direction to the initial vertical velocity if the object lands at the same elevation from which it was launched. Both velocities can then be combined to find the total final velocity (magnitude and direction) at the point of impact. The direction of this velocity will be at an angle with the horizontal, which can be found using trigonometric relations.
If we assume level ground and ignore air resistance, the range (R) of a projectile is given by R = (vo2sin(2 hetao))/g, where vo is the initial speed, hetao is the initial angle, and g is the acceleration due to gravity. This equation indicates that for a given initial velocity, the range is maximized when the initial launch angle is 45 degrees, a fact that is significantly altered if air resistance is taken into account. On the moon, due to weaker gravity, the range would be much larger for the same initial velocity.