Given these assumptions, the following steps are then used to analyze projectile motion:
Step 1.Resolve or break the motion into horizontal and vertical components along the x- and y-axes. These axes are perpendicular, so
and
are used. The magnitude of the components of displacement
along these axes are
and
The magnitudes of the components of the velocity
are
and
where
is the magnitude of the velocity and
is its direction, as shown in
[link] . Initial values are denoted with a subscript 0, as usual.
Step 2.Treat the motion as two independent one-dimensional motions, one horizontal and the other vertical. The kinematic equations for horizontal and vertical motion take the following forms:
Step 3.Solve for the unknowns in the two separate motions—one horizontal and one vertical. Note that the only common variable between the motions is time
. The problem solving procedures here are the same as for one-dimensional
kinematics and are illustrated in the solved examples below.
Step 4.Recombine the two motions to find the total displacementand velocity
. Because the
x - and
y -motions are perpendicular, we determine these vectors by using the techniques outlined in the
Vector Addition and Subtraction: Analytical Methods and employing
and
in the following form, where
is the direction of the displacement
and
is the direction of the velocity
:
Total displacement and velocity
A fireworks projectile explodes high and away
During a fireworks display, a shell is shot into the air with an initial speed of 70.0 m/s at an angle of
above the horizontal, as illustrated in
[link] . The fuse is timed to ignite the shell just as it reaches its highest point above the ground. (a) Calculate the height at which the shell explodes. (b) How much time passed between the launch of the shell and the explosion? (c) What is the horizontal displacement of the shell when it explodes?
Strategy
Because air resistance is negligible for the unexploded shell, the analysis method outlined above can be used. The motion can be broken into horizontal and vertical motions in which
and
. We can then define
and
to be zero and solve for the desired quantities.