Kinematic equations will be used to solve non-horizontally launched projectiles will be demonstrated. A non-horizontally launched projectile is a projectile that begins its motion with an initial velocity that is both horizontal and vertical. To treat such problems, concepts will have to be combined with the kinematic equations for projectile motion. There are two sets of kinematic equations - a set of equations for the horizontal components of motion and a similar set for the vertical components of motion. For the horizontal components of motion, the equations are

x = v_ix•t + 0.5*a_x*t²

v_fx = v_ix + a_x•t

v_fx² = v_ix² + 2*a_x•x

where

x = horiz. displacement

a_x = horiz. acceleration

t = time

v_fx = final horiz. velocity

v_ix = initial horiz. velocity

Of these three equations, the top equation is the most commonly used. The other two equations are seldom (if ever) used. An application of projectile concepts to each of these equations would also lead one to conclude that any term with a_x in it would cancel out of the equation since a_x = 0 m/s/s.

For the vertical components of motion, the three equations are

y = v_iy•t + 0.5*ay*t²

v_fy = v_iy + ay•t

v_fy² = v_iy² + 2*ay•y

where

y = vert. displacement

ay = vert. acceleration

t = time

v_fy = final vert. velocity

v_iy = initial vert. velocity

In each of the above equations, the vertical acceleration of a projectile is known to be -9.8 m/s/s (the acceleration of gravity).

The v_ix and v_iy values in each of the above sets of kinematic equations can be determined by the use of trigonometric functions. The initial x-velocity (v_ix) can be found using the equation v_ix = v_i•cosine(Theta) where Theta is the angle that the velocity vector makes with the horizontal. The initial y-velocity (v_iy) can be found using the equation v_iy = v_i•sine(Theta) where Theta is the angle that the velocity vector makes with the horizontal.

Here is a reminder about trig functions:

Examples of this type of problem are

1. A football is kicked with an initial velocity of 25 m/s at an angle of 45-degrees with the horizontal. Determine the time of flight, the horizontal distance, and the peak height of the football.

2. A long jumper leaves the ground with an initial velocity of 12 m/s at an angle of 28-degrees above the horizontal. Determine the time of flight, the horizontal distance, and the peak height of the long-jumper.

Examples of this type of problem with solutions:**

3. An object is launched at a velocity of 20 m/s in a direction making an angle of 25° upward with the horizontal.

a) What is the maximum height reached by the object?

b) What is the total flight time (between launch and touching the ground) of the object?

c) What is the horizontal range (maximum x above ground) of the object?

d) What is the magnitude of the velocity of the object just before it hits the ground?

4. A projectile is launched from point O at an angle of 22° with an initial velocity of 15 m/s up an incline plane that makes an angle of 10° with the horizontal. The projectile hits the incline plane at point M. (See graphic below)

a) Find the time it takes for the projectile to hit the incline plane.

b) Find the distance OM.

Want to read more about non-horizontal launch projectiles? Click on the link below for videos, notes and practice problems: