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Projectile Motion
Definition: Projectile -a moving object given initial velocity that then moves only under
the force of gravity
-the path followed by a projectile is called a trajectory (parabolic)
-shape of trajectory and horizontal motion depend on frame of reference of the observer
-vertical motion does not depend on frame of reference of observer
-vertical and horizontal motion of a projectile are independent of each other!
Neglecting air resistance... let x be horizontal displacement, let vx be initial horizontal
velocity, & let t be time
x  vx  t
{remember...
v xfinal  v xinitial }
For falling objects, g = -9.81 m/s2…let y be vertical displacement, let vy be initial
vertical velocity, let t be time
1 2
gt
2
 v y  gt
y  vyt 
v y final
Example:
1. A stone is thrown horizontally at 15 m/s from the top of a cliff 44 m high.
a. How long does the stone take to reach the bottom of the cliff?
b. How far from the base of the cliff does the stone strike the ground?
c. Sketch the trajectory of the stone.
Objects Launched at an Angle
-horizontal component, vx, is constant
-since gravity acts on the projectile...vy becomes less as the ball reaches the top of the
trajectory
-at the top of trajectory, vy = 0 m/s and vx is still constant
-after the top of trajectory, vy increases and becomes -ve (downward)
-vertical speed, vx, is the same throughout
Definition: Range (R) -horizontal distance from the point of bounce until projectile
returns to surface height
Problem-Solving Tips for Objects Launched at an Angle
1. Determine vx and vy components of initial velocity.
2. Solve parts of the problem involving each component separately.
3. Use symmetry of the trajectory (parabola).
a) Launching and landing heights are equal
b) Rising and falling times are equal
c) Horizontal distances moved in each half of trajectory are equal
4. Remember, ignore air resistance!
Any velocity (or any other vector quantity!) can be resolved or "broken down" into a
horizontal (vx) component and a vertical (vy) component ---> vector resolution
Vector resolution is done by using trigonometry! YEAH!
v x  v  cos 
v
vy
v y  v  sin
vx
-when the vx and vy are connected "head-to-tail" ---> a right triangle is formed
-the hypotenuse of the triangle is initial velocity v
Example:
A football player kicks a football from ground level with a velocity of magnitude 35 m/s
at an angle of 30 ° above the horizontal. Find
a. its "hang time" or the time in the air.
b. the distance the ball travels before it hits the ground.
c. its maximum height.
Repeat this problem using angles of 45 ° and 60 ° above the horizontal.
Formulas for Projectile Motion:
Horizontally…
Vertically…
x  vx  t
v x  v  cos 
1 2
gt
2
 v y  gt
y  vyt 
v y final