What is Projectile Motion?

Instructional Objectives: Students will be able to: Define Projectile Motion Distinguish between the different types of projectile motion Apply the concept to a toy car and measure its velocity

Vocabulary Projectile – any object projected through the air, whose motion is due to the force of gravity acting on the object and the object’s inertia. Inertia – the tendency of an object at rest to remain at rest or an object in motion to remain in motion with uniform speed in a straight line.

Vocabulary Trajectory - the name given to the path followed by a projectile Range – the horizontal displacement of a projectile Launch angle – the angle, with respect to the horizontal, at which an object is projected into the air. trajectory θ range

Projectile Motion When an object is thrown or launched into the air, it undergoes projectile motion. The pathway of any projectile is a parabola (excluding air resistance).

Types of Projectile Motion Horizontal Motion of a ball rolling freely along a level surface Horizontal velocity is ALWAYS constant Vertical Motion of a freely falling object Force due to gravity Vertical component of velocity changes with time Parabolic Path traced by an object accelerating only in the vertical direction while moving at constant horizontal velocity

Equations Used to Analyze Projectile Motion For horizontal motion: Use the constant speed equation, v = Δx/t For vertical motion: Use the Big 4 kinematics equations for uniformly accelerated motion Assume no air resistance Assume “g” is constant throughout the object’s motion

Vertically Launched Projectiles An object dropped or thrown vertically downward or projected vertically upward is considered a projectile. The motion of such an object is analyzed using the equations for free-fall.

Horizontally Launched Projectiles HORIZONTAL AND VERTICAL MOTIONS ARE ALWAYS INDEPENDENT OF EACH OTHER! Horizontal motion is due to inertia. Vertical motion is due to gravity.

Analyzing Horizontal and Vertical Motions Horizontal Motion (x) Vertical Motion (y) vix ≠ 0 vfx ≠ 0 viy = 0 vfy ≠ 0 vix = vfx g = -9.81 m/s2 a = 0 To find ∆y or t, ∆x = range use equations for free-fall vconstant = ∆x/t vf = vi + gt vconstant = vix vf2 = vi2 + 2g∆y ∆x = vixt ∆y = vit + ½gt2 ∆y = ½ (vi + vf) t

The distance a horizontally projected object falls vertically in every second is equal to the distance a freely-falling object falls vertically in every second. 0 m 5.0 m 10. m 15 m 20. m Initial horizontal velocity = 5.0 m/s Initial vertical velocity = 0 m/s Total time of “flight” = 4.0 s Horizontal distance traveled /second = 5.0 m Horizontal distance traveled in 4.0 s = 20. m 4.9 m 19.6 m 44.1 m 78.4 m

Projectiles Launched at an Angle viy=0 vfy=0 Δy viy vix Range, Δx

vi is the initial velocity. In the previous diagram: θ is the launch angle. vi is the initial velocity. vix is the x-component of the initial velocity. viy is the y-component of the initial velocity. vfy is the y-component of the final velocity and equals zero at the top of the trajectory. vfx is the x-component of the final velocity and is equal to vix throughout the object’s motion. (No acceleration horizontally!) ∆y is the vertical displacement of the object.

Analyzing Horizontal and Vertical Motions vix = vi cos θ viy = vi sin θ vfx = vix vfy = 0 a = 0 a = g Δx = 2 vixt Δy = viyt + ½ gt2 Δx = 2 vitcos θ Δy = vitsin θ + ½ gt2 vfy2 = viy2 + 2gΔy 0 = (visin θ)2 + 2gΔy Δy = ½ vi2sin2θ 2g tup = tdown ttotal = tup + tdown =2t

What’s Important: The trajectory has a symmetrical shape. The height of the trajectory is the highest point reached by the object. At the top of the trajectory, the object’s vertical velocity (vfy) = 0. Time up = time down Total time in the air = 2t The range = 2 vixt = 2 vitcos θ θ up = θ down

Projectile Motion After breaking the initial velocity into x and y components separately, the pathway of the object can be tracked by using a series of equations:

Examples of Projectile Motion Launching a Cannon ball

Projectile Motion Equations Variables: Y = distance in meters vertically Vy = y component of velocity Viy = y component of initial velocity Vix = x component of initial velocity g = acceleration of gravity t = time during flight

Equations X- Component Y- Component Vectors X or Horizontal: Velocity: Vx remains constant; therefore vi(x) = vf(x) (Newton’s 1st law of motion) X- Component Y- Component Vectors Note: g= 9.8 m/s2

Factors Affecting Projectile Motion What two factors would affect projectile motion? Angle Initial velocity Initial Velocity Angle

Projectile Motion In projectile problems, vertical and horizontal motions are completely independent of one another. For example: An object will accelerate downward at the exact same rate regardless of whether it is dropped, thrown forward slowly, or thrown forward very quickly. Likewise: An object will continue to move forward the same distance every second whether it is rolling along a flat surface or moving vertically.

Projectile Motion Time is the only variable which will be the same for the two motions. The time it takes for an object to hit the ground is the same amount of time it has to move forward.

Projectile Motion Equations One method for solving for horizontally projected objects: 1) Write down all variables, and identify them as horizontal or vertical. 2) Solve one dimension (x or y, depending on what is given in the problem) for time. 3) Use that time in the other dimension to find unknown quantity

Class Exercise An object is fired from the ground at 100 meters per second at an angle of 30 degrees with the horizontal Calculate the horizontal and vertical components of the initial velocity After 2.0 seconds, how far has the object traveled in the horizontal direction? How high is the object at this point?

Solution Part a Part b Part c

Projectile Launched at an Upward Angle from an Elevated Surface

θ Treat problem as 2 separate motions. θ

Remember: For ANY problem: You cannot interchange x and y components for initial and final velocities! You cannot interchange ∆x and ∆y values! You must solve for horizontal and vertical quantities separately! The only link between horizontal and vertical motion is TIME !