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Projectile Motion

Projectile Motion

_{Projectile motion describes the motion of an object which is thrown, or projected, over the ground.  We will discuss projectiles which are accelerating at a uniform rate which implies no air resistance (air friction) or propulsion during flight.}

_{A typical projectile is affected ONLY by gravity once it is traveling. Since gravity acts in the "downward" direction, we can conclude that there is zero acceleration in the horizontal direction of travel; the horizontal component of velocity remains constant for the entire flight of the projectile.}

Using the kinematics equations, we can solve for displacement, velocity, acceleration and time values by treating the whole problem as two independent problems; the horizontal part of the problem can be treated separately from the vertical part of the problem producing the common value of time in each case. 

In most cases, the two directions of the problem will be solved for time simultaneously so that the actual value of time is not needed (not always).There are different conditions that should be considered in projectile motion:

Level Ground:
The initial and final velocities are the same in magnitude (and same angle relative to the ground).

**Note:  The horizontal component of velocity remain constant (v_x = 6 cos 60°) for the entire trip.  The vertical component of velocity changes from v_y = 6 sin 60° to v'_y = -6 sin 60°.

From the Top of a Cliff:
The vertical displacement (from top to bottom) is in the same direction as the acceleration of gravity (also called gravitational field strength) so should have the same sign for the initial conditions.The initial and final velocities ARE NOT the same (magnitude nor angle).

**Note:  The horizontal component of velocity remain constant (v_x = 6 cos 60°) for the entire trip.  The vertical component of velocity changes from v_y = 6 sin 60° to v'_y = -10 sin 75°.

From the Ground UP onto a Cliff:
The vertical displacement (from bottom to top) is NOT in the same direction as the acceleration of gravity (also called gravitational field strength) so should have the opposite sign for the initial conditions.The initial and final velocities ARE NOT the same (magnitude nor angle).

**Note:  The horizontal component of velocity remain constant (v_x = 6 cos 60°) for the entire trip.  The vertical component of velocity changes from v_y = 6 sin 60° to v'_y = - 4 sin 40°.

**Note:  When reading a problem, be sure of what the problem is asking for

Velocity at Any Time During Flight:
Since the horizontal component of the velocity of a projectile remains constant (in a frictionless environment), the resultant velocity of the projectile at a given time can be found by adding the horizontal component at that time to the calculated vertical component at that time.

Projectile Motion Review (.pps)

Navigation

Navigation in a plane or a boat is accomplished using vectors.  In order to navigate from one place to another, the operator of a plane or a boat must consider ALL of the factors that affect the motion of the craft.  Each factor, itself, contributes to the motion as a vector which must be added to the other factors to result in the desired effect.

Plane Navigation Factors:

• Wind speed vector with its direction  (v_w)
• Airspeed vector with its direction (v_a)
• Ground speed vector with its direction (v_g)

The ground speed vector will always represent the overall effect of the wind speed and the airspeed on the plane; the ground speed is defined as the sum of the wind speed and the air speed:

v_g =  v_a + v_w

_{Remember:  All of these vectors have directions associated with them and MUST be considered in the vector sum. Since the overall velocity of an aircraft can be represented by the above vector sum, any one of the vectors in the relationship can be found by rearranging the equation:}

Example:

v_a =  v_g -  v_w
OR
v_w =  v_g -  v_a

_{The trick is to identify which variable is the one that is asked for.  Once the problem is identified, and the vectors are labeled and drawn, the missing vector can be found by performing the vector sum discussed in the Vector section.}

_{Boat Navigation Factors:}

• Velocity vector for the water (in a river) including its direction (v_w)
• Velocity vector for the boat in still water including  its direction (v_b)
• Overall Velocity vector including its direction (v)

The overall velocity vector will always represent the overall effect of the velocity vector for the water and the velocity vector for the boat in still water; the overall velocity is defined as the sum of the two other vectors:

v =  v_b + v_w

_{Remember:  All of these vectors have directions associated with them and MUST be considered in the vector sum. Since the overall velocity of a boat can be represented by the above vector sum, any one of the vectors in the relationship can be found by rearranging the equation:}

Example:

v_b =  v -  v_w
OR
v_w =  v -  v_b

_{Again, the trick is to identify which variable is the one that is asked for.  Once the problem is identified, and the vectors are labeled and drawn, the missing vector can be found by performing the vector sum discussed in the Vector section.}

Navigation and Relative Velocity Review (.pps)