Section 3 7 Projectile Motion A projectile is

Section 3 -7: Projectile Motion

A projectile is an object moving in two dimensions under the influence of Earth's gravity; its path is a parabola.

Projectile Motion • Projectile Motion of an object that is projected into the air at an angle. • Near the Earth’s surface, the acceleration a on the projectile is downward and equal to a = g = 9. 8 m/s 2 – Goal: Describe motion after it starts. • Galileo: Analyzed horizontal & vertical components of motion separately. • Today: Displacement D & velocity v are vectors Components of motion can be treated separately

Projectile Motion • Simplest example: Ball rolls across table, to the edge & falls off edge to floor. Leaves table at time t = 0. Analyze y part of motion & x part of motion separately. • y part of motion: Down is positive & origin is at table top: y 0 = 0. Initially, no y component of velocity: vy 0 = 0 vy = gt, y = (½)g t 2 • x part of motion: Origin is at table top: xf = 0. No x component of acceleration(!): ax = 0. Initially x component of velocity is: vx vx = vx 0 , x = v 0 t

Ball Rolls Across Table & Falls Off t = 0 here Can be understood by analyzing horizontal vertical motions separately. Take down as positive. Initial velocity has an x component ONLY! That is vy 0 = 0. At any point, v has both x & y components. Kinematic equations tell us that, at time t, vx = vx 0, vy = gt x = vx 0 t y = (½)gt 2

• Summary: Ball rolling across table & falling. • Vector velocity v has 2 components: vx = vx 0 , vy = gt • Vector displacement D has 2 components: x = vx 0 t , y = (½)g t 2

The speed in the x-direction is constant; in the y-direction the object moves with constant acceleration g. Photo shows two balls that start to fall at the same time. The on the right has an initial speed in the x-direction. It can be seen that vertical positions of the two balls are identical at identical times, while the horizontal position of the yellow ball increases linearly.

• PHYSICS: y part of motion: vy = gt , y = (½)g t 2 SAME as free fall motion!! An object projected horizontally will reach the ground at the same time as an object dropped vertically from the same point! (x & y motions are independent)

General Case: Object is launched at initial angle θ 0 with the horizontal. Analysis is similar to before, except the initial velocity has a vertical component vy 0 0. Let up be positive now! vx 0 = v 0 cosθ 0 vy 0 = v 0 sinθ 0 but, acceleration = g downward for the entire motion! Parabolic shape of path is real (neglecting air resistance!)

• General Case: Take y positive upward & origin at the point where it is shot: x 0 = y 0= 0 vx = v 0 cosθ 0, vy = v 0 sinθ 0 • Horizontal motion: NO ACCELERATION IN THE x DIRECTION! vx = vx 0 , x = vx 0 t • Vertical motion: vy = vy 0 - gt , y = vy 0 t - (½)g t 2 (vy) 2 = (vy 0)2 - 2 gy – If y is positive downward, the - signs become + signs. ax = 0, ay = -g = -9. 8 m/s 2

Summary: Projectile Motion Projectile motion is motion with constant acceleration in two dimensions, where the acceleration is g and is down.

Solving Problems Involving Projectile Motion 1. Read the problem carefully, &choose the object(s) you are going to analyze. 2. Sketch a diagram. 3. Choose an origin & a coordinate system. 4. Decide on the time interval; this is the same in both directions, & includes only the time the object is moving with constant acceleration g. 5. Solve for the x and y motions separately. 6. List known & unknown quantities. Remember that vx never changes, & that vy = 0 at the highest point. 7. Plan how you will proceed. Use the appropriate equations; you may have to combine some of them.