Motion in two dimensions must involve acceleration one way or another. If there is no acceleration, an object continues to move at the same speed in the same direction; hence it moves along the same line in one dimension.
Vectors in more than one dimension are best handled by decomposing them into perpendicular dimensions. This applies, of course, to motion in two dimensions.
It is easiest to use the direction (if it is known) of acceleration as one of the two perpendicular dimensions. This way, we do not need to decompose acceleration, and calculations are made easier.
Free fall problems always involve the acceleration of gravity, which is 9.8 m/s [downward].
Projectile motion
The trajectory (also known as the world line) of a projectile is its position in space at every given point in time. You may think of it as the path the projectile takes after it has been launched.
The trajectory of any projectile in free fall is always parabolic (i.e., it is part of a parabola, whether this part includes the vertex or not).
If the trajectory of a projectile includes the vertex of the parabola, then the vertex is the highest point in the trajectory; otherwise, the highest point is either where it is launched or where it lands (depending on the context of the problem).
The altitude \(h\) of the projectile is the highest it gets above the level of the ground.
The flight time of the projectile is how long it takes from the moment the projectile is launched to just before it hits the ground.
The range \(R\) of the projectile is the horizontal displacement it undergoes during its flight time.
Projectile is launched from the same level as its landing point.
Projectile is launched from a higher level than its landing point.
Projectile is launched from a lower level than its landing point.
Analysis
All kinematic vector quantities (displacement, velocity and acceleration) are decomposed into perpendicular, horizontal and vertical components.
Vectors in each dimension are handled separately from those in the other dimension (i.e., as if the other did not exist).
Scalars (time in the case of kinematics) are the same in every dimension. Time is the link between the two dimensions in projectile motion.
Implied givens
There is only acceleration in the vertical dimension, which is the acceleration of gravity (9.8 m/s2 [down]). There is no horizontal acceleration.
When the launch velocity is horizontal, the initial vertical velocity is zero.
The highest point of the trajectory is the vertex if the trajectory contains the vertex. At that point, the vertical velocity is zero.
Caveats
In two dimensions, the horizontal velocity is never zero.
The velocity ‘just before the projectile hits the ground’ is not zero. Projectile motion is necessarily in free fall. When a projectile starts to touch the ground, it is no longer in free fall.
The words ‘initial’ and ‘final’ just mean two points in time. The initial happens first in time before the final. Any two points can be chosen as initial and final. The initial is not necessarily the point of launch, and the final is not necessarily the end of the trajectory.
