Projectile Motion

Projectile Motion CCHS Physics

Projectile Properties?

Projectile Motion Describe the motion of an object in TWO dimensions Keep it simple by considering motion close to the surface of the earth for the time being (g = -9.8 m/s2 = constant in y direction) Neglect air resistance to make it simpler Assume the rotation of the Earth has no effect

Projectile Motion

vertical component v vy  Velocity Components at various points of the Trajectory Net velocity vx Horizonal component Above: Vectors are added in geometric Fashion. Launch speed = Return Speed. Speed is minimum at apex of parabolic trajectory.

Projectile Motion The ball is in free fall vertically and moves at constant speed horizontally!!!

Video

What’s wrong with this picture ? Answer: It never happens ! Only when there is no gravity.

A History of Projectile Motion Aristotle: The canon ball travels in a straight line until it lost its ‘impetus’. Galileo: a result of Free Fall Motion along y-axis and Uniform Motion along x-axis.

What’s the similarity between a freely-falling ball and a projectile ? y uniform motion Projectile motion vertical motion x A dropped ball falls in the same time as a ball shot horizontally. Along the vertical, their motions are identical (uniformly accelerated motion (free-fall). Along the horizontal, notice the ball fired horizontally covers the Same distance in the same unit time intervals (uniform motion along x)

Projectile Motion Equations

Projectile Motion = Sum of 2 Independent Motions • 1. Along x, the projectile travels with constant velocity. • vx=vxo x = vxot • Along y, the projectile travels in free-fall fashion. • vy = vyo – gt y = vyot – (1/2) gt2 , g= 9.8 m/s2 • Projectile motion = a combination of uniform motion along x and • uniformly accelerated motion (free fall) along y.

Everyday Examples of Projectile Motion • Baseball being thrown • Water fountains • Fireworks Displays • Soccer ball being kicked • Ballistics Testing

V = 50 m/s 125 m Type I A ball is kicked off a 125 m cliff with a horizontal velocity of 50 m/s. What is the range of the ball? 255 m -125 m 50 m/s 0 m/s 0 m/s2 -9.8 m/s2 5.1 s 5.1 s d =255 m

v = 50 m/s 37° v = 50 m/s vy = 50sin37 = 30 m/s 37° vx = 50cos37 = 40 m/s Type II A ball is kicked of a with a velocity of 50 m/s at an angle of 37° from the horizontal. What is the range of the ball? 244 m 0 m 40 m/s 30 m/s 0 m/s2 -9.8 m/s2 6.1 s 6.1 s d =244 m

Note, this type requires the use of the quadratic equation v = 50 m/s 37° 125 m Type III A ball is kicked off a 125 m cliff with a velocity of 50 m/s at an angle of 37° from the horizontal. What is the range of the ball? 360 m -125 m 40 m/s 30 m/s 0 m/s2 -9.8 m/s2 9.0 s 9.0 s d = 360 m

Horizontal Range

How to maximize horizontal range: keep the object off the ground for as long as possible. This allows the horizontal motion to be a maximum since x = vxt Make range longer by having a greater initial velocity velocity Maximum Range

vi  R Range Equation What is total t? To solve, set vertical displacement = 0. Trig Identity: 2sincos = sin(2) REMEMBER: ONLY VALID WHEN VERTICAL DISPLACEMENT IS ZERO (Type II problems).

Projectile Motion • We can see that complementary angles have the same range because sin = sin2

At what angle do I launch for Maximum Range ? Need to stay in air for the longest time, and with the fastest horizontal velocity component Answer: 45°

What happens when we add air resistance? Adds a new force on the ball The force is in the opposite direction to the ball’s velocity vector and is proportional to the velocity at relatively low speeds Need calculus to sort out the resulting motion Lowers the angle for maximum range Projectile Motion

Projectile Motion