"A baseball thrown from the outfield, a golf ball or a thrown football all will follow a trajectory (path) that has horizontal as well as vertical components. "

"If we neglect the air friction (drag) on the ball, the velocity in the horizontal direction remains constant."

"The velocity in the vertical direction varies because the ball is free falling with the acceleration of gravity g (9.81 meters per second). The acceleration of gravity is always directed toward the earth. "

"Displacement :"

"X=Xo+Vox*t "

"Y=Yo+Voy-(1/2)*g*t^2 "

"Velocity as a function of time (t):"

"Vx=Vox "

"Vy=Voy-g*t "

"Here we see that the velocity is a combination of constant motion in the x-direction and a constantly accelerating motion in the y-direction. "

"All baseball players, golfers and football players know that the effects of the wind are not negligible, but for now we ill ignore these effects and look at the basic causes of the motion. "

"This keeps the mathematics simpler. In your junior and senior year you will deal with the effects of drag on motion of bodies. "

"Initial velocity of the projectile is: "

"Vox=Vo*cos(theta) "

"Voy=Vo*sin(theta) "

"We can assume that the projectile starts at the origin of the coordinate system (x=0,y=0). Then, the position of the projectile as a function of time is given by the two equations: "

"X=Vox*t "

"Y=Voy*t-(1/2)*g*t^2 "

"The range is the horizontal displacement when the projectile returns to ground level (y=0). The range can be calculated from the formula: "

"R=(Vo^2*sin(2*theta))/g "

"Use the EES max/min function to prove that the maximum range is obtained when the projectile motion is initiated at a 45

degree angle."

"Given:"

Xo=0

Yo=0

Vo=50 "m/s"

g=9.81 "acceleration of gravity"

"Equations:"

X=Xo+Vox*t;

Y=Yo+Voy*t-(1/2)*g*t^2

Vx=Vox

Vy=Voy-g*t

Vox=Vo*cos(theta)

Voy=Vo*sin(theta)

R=(Vo^2*sin(2*theta))/g