Projectile Motion in Three Dimensions Projectile Motion in Three Dimensions An object is launched subject to gravity and air resistance. a̰

= r̰

=

ḭ +

j̰ + z t k̰

v̰

= r̰

=

ḭ +

j̰ + z t k̰ + v̰ 0

r̰

=

ḭ +

j̰ + z t k̰ + v̰ 0

+ r̰ 0

Maximum Height The highest point the object reaches. At this point, the object has no vertical velocity. Solving the k̰ component of the velocity vector equal to 0 will find the time. Time of Flight Time taken from launch until the object lands. If the - -plane represents the ground, solving the k̰ component of the position vector equal to 0 will find the time of flight. Range The distance from the where the object is launched to where it lands. Find the magnitude of the ḭ and j̰ components of the position vector at the time when the object lands. Landing Speed The magnitude of the velocity at the time when the object lands. Landing Angle Since the direction of motion is parallel to the velocity, the tangent of the landing angle is the ratio of the k̰ component of the velocity vector at the time when the object lands and the magnitude of the ḭ and j̰ component of the velocity vector at the time when the object lands. tan

=  

+ v̰ 0 + v̰

0

+

+ v̰

0